AppendixA Auxiliary Results

A.1 Equivalence Relations; Groups

A relation: x ∼ y among the points of a space X is an equivalence relation if it is reflexive, symmetric, and transitive, that is, if (i) x ∼ x for all x ∈X; (ii) x ∼ y implies y ∼ x; (iii) x ∼ y, y ∼ z implies x ∼ z.

Example A.1.1 Consider a class of statistical decision procedures as a space, of which the individual procedures are the points. Then the relation defined by δ ∼ δ if the procedures δ and δ have the same risk function is an equivalence relation. As another example consider all real-valued functions defined over the real line as points of a space. Then f ∼ g if f(x)=g(x) a.e. is an equivalence relation.

Given an equivalence relation, let Dx denote the set of points of the space that are equivalent to x. Then Dx = Dy if x ∼ y, and Dx ∩Dy = 0 otherwise. Since by (i) each point of the space lies in at least one of the sets Dx, it follows that these sets, the equivalence classes defined by the relation ∼, constitute a partition of the space. AsetG of elements is called a group if it satisfies the following conditions. (i) There is defined an operation, group multiplication, which with any two elements a, b ∈ G associates an element c of G. The element c is called the product of a and b and is denoted by ab. A.2. Convergence of Functions; Metric Spaces 693

(ii) Group multiplication obeys the associative law

(ab)c = a(bc).

(iii) There exists an element e ∈ G, called the identity, such that

ae = ea = a for all a ∈ G.

(iv) For each element a ∈ G, there exists an element a−1 ∈ G,itsinverse, such that

aa−1 = a−1a = e.

Both the identity element and the inverse a−1 of any element a can be shown to be unique.

Example A.1.2 The set of all n × n orthogonal matrices constitutes a group if matrix multiplication and inverse are taken as group multiplication and inverse respectively, and if the identity matrix is taken as the identity element of the group. With the same specification of the group operations, the class of all non- singular n × n matrices also forms a group. On the other hand, the class of all n × n matrices fails to satisfy condition (iv).

If the elements of G are transformations of some space onto itself, with the group product ba defined as the result of applying first transformation a and following it by b, then G is called a transformation group. Assumption (ii) is then satisfied automatically. For any transformation group defined over a space X the relation between points of X given by

x ∼ y if there exists a ∈ G such that y = ax is an equivalence relation. That it satisfies conditions (i), (ii), and (iii) required of an equivalence follows respectively from the defining properties (iii), (iv), and (i) of a group. Let C be any class of 1 : 1 transformations of a space, and let G be the class ±1 ±1 ±1 of all finite products a1 a2 ...am ,witha1,...,am ∈ C, m = 1, 2, . . . , where each of the exponents can be +1 or −1 and where the elements a1, a2, . . . need not be distinct. Then it is easily checked that G is a group, and is in fact the smallest group containing C.

A.2 Convergence of Functions; Metric Spaces

When studying convergence properties of functions it is frequently convenient to consider a class of functions as a realization of an abstract space F of points f in which convergence of a sequence fn to a limit f, denoted by fn → f, has been defined.

Example A.2.1 Let µ be a measure over a measurable space (X , A). 694 AppendixA. Auxiliary Results

(i) Let F be the class of integrable functions. Then fn converges to f in the 1 mean if

|fn − f| dµ → 0. (A.1)

(ii) Let F be a uniformly bounded class of measurable functions. The sequence is said to converge to f weakly if

fnpdµ→ fpdµ (A.2)

for all functions p that are integrable µ.

(iii) Let F be the class of measurable functions. Then fn converges to f pointwise if

fn(x) → f(x) a.e. µ. (A.3)

A subset of F0 is dense in F if, given any f ∈F, there exists a sequence in F0 having f as its limit point. A space F is separable if there exists a countable dense subset of F. A space F such that every sequence has a convergent subsequence whose limit point is in F is compact.2 AspaceF is a metric space if for every pair of points f, g in F there is defined a metric (or distance) d(f,g) ≥ 0 such that (i) d(f,g) = 0 if and only if f = g; (ii) d(f,g)=d(g, f); (iii) d(f,g)+d(g, h) ≥ d(f,h) for all f, g, h. The space is a pseudometric space if (i) is replaced by (i) d(f,f) = 0 for all f ∈F. A pseudometric space can be converted into a metric space by introducing the equivalence relation f ∼ g if d(f,g) = 0. The equivalence classes F , G, . . . then constitute a metric space with respect to the metric D(F, G)=d(f,g) where f ∈ F , g ∈ G. In any pseudometric space a natural convergence definition is obtained by putting fn → f if d(fn,f) → 0.

Example A.2.2 The space of integrable functions of Example A.2.1(i) becomes a pseudometric space if we put d(f,g)= |f − g| dµ and the induced convergence definition is that given by (1).

1Here and in the examples that follow, the limit f is not unique. More specifically, if fn → f,thenfn → g if and only if f = g (a.e. µ). Putting f ∼ g when f = g (a.e. µ), uniqueness can be obtained by working with the resulting equivalence classes of functions rather than with the functions themselves. 2The term compactness is more commonly used for an alternative concept. which coincides with the one given here in metric spares. The distinguishing term sequential compactness is then sometimes given to the notion defined here. A.2. Convergence of Functions; Metric Spaces 695

Example A.2.3 Let P be a family of probability distributions over (X , A). Then P is a metric space with respect to the metric d(P, Q)= sup|P (A) − Q(A)|. (A.4) A∈A

Lemma A.2.1 If F is a separable pseudometric space, then every subset of F is also separable.

Proof. By assumption there exists a dense countable subset {fn} of F.Let 1 S = f : d(f,f ) < , m,n n m and let A be any subset of F. Select one element from each of the intersections A ∩ Sm,n that is nonempty, and denote this countable collection of elements by A0.Ifa is any element of A and m any positive integer, there exists an element fnm such that d(a, fnm ) < 1/m. Therefore a belongs to Sm,nm , the intersection

A∩Sm,nm is nonempty, and there exists therefore an element of A0 whose distance to a is < 2/m. This shows that A0 is dense in A, and hence that A is separable.

Lemma A.2.2 A sequence fn of integrable functions converges to f in the mean if and only if

fn dµ → fdµ uniformly for A ∈A. (A.5) A A Proof. That (1) implies (5) is obvious, since for all A ∈A % % % % % % % fn dµ − fdµ% ≤ |fn − f| dµ. A A Conversely, suppose that (5) holds, and denote by An and An the set of points x for which fn(x) >f(x) and fn(x)

|fn − f| dµ = (fn − f) dµ − (fn − f) dµ → 0 . An An

Lemma A.2.3 A sequence fn of uniformly bounded functions converges to a bounded function f weakly if and only if

fn dµ → fdµ for all A with µ(A) < ∞. (A.6) A A Proof. That weak convergence implies (6) is seen by taking for p in (2) the indicator function of a set A, which is integrable ifµ(A) < ∞. Conversely (6) implies that (2) holds if p is any simple function s = aiIAi with all the µ(Ai) < ∞. Given any integrable function p, there exists, by the definition of the integral, such a simple function s for which |p − s| dµ < /3M, where M is a bound on the |f|’s. We then have % % % % % % % % % % % % % % % % % % % % % % % % % (fn − f)pdµ% ≤ % fn(p − s) dµ% + % f(s − p) dµ% + % (fn − f)sdµ% . 696 AppendixA. Auxiliary Results

The first two terms on the right-hand side are </3, and the third term tends to zero as n tends to infinity. Thus the left-hand side is <for n sufficiently large, as was to be proved.

3 Lemma A.2.4 Let f and fn, n =1, 2, . . . , be nonnegative integrable functions with

fdµ= fn dµ =1.

Then pointwise convergence of fn to f implies that fn → f in the mean.

− Proof. If gn = fn − f, then g ≥−f, and the negative part gn =max(−gn, 0) | −|≤ → satisfies gn f.Sincegn(x) 0 (a.e. µ), it follows from Theorem 2.2.2(ii) of − → + Chapter 2 that gn dµ 0, and gn dµ then also tends to zero, since gn dµ = + − 0. Therefore |gn| dµ = (gn + gn ) dµ → 0, as was to be proved. Let P and Pn, n = 1, 2, . . . be probability distributions over (X , A)with densities pn and p with respect to µ. Consider the convergence definitions

(a) pn → p (a.e. µ); (b) |pn − p| dµ → 0; (c) gpn dµ → gp dµ for all bounded measurable g; and (b ) Pn(A) → P (A) uniformly for all A ∈A; (c ) Pn(A) → P (A) for all A ∈A. Then Lemmas A.2.2 and A.2.4 together with a slight modification of Lemma A.2.3 show that (a) implies (b) and (b) implies (c), and that (b) is equivalent to (b) and (c) to (c). It can further be shown that neither (a) and (b) nor (b) and (c) are equivalent.4

A.3 Banach and Hilbert Spaces

AsetViscalledavector space (or linear space) over the reals if there exists a function + on V × V to V and a function · on R × V to V which satisfy for x, y, z ∈ V , (i) x + y = y + x. (ii) (x + y)+z = z +(y + z). (iii) There is a vector 0 ∈ V : x +0= x for all x ∈ V . (iv) λ(x + y)=λx + λy for any λ ∈ R. (v) (λ1 + λ2)x = λ1x + λ2x for λi ∈ R. (vi) λ1(λ2x)=(λ1λ2)x for λi ∈ R. (vii) 0 · x =0, 1 · x = x.

3Scheff´e (1947). 4Robbins (1948). A.3. Banach and Hilbert Spaces 697

The operation + is called addition by scalars and · is multiplication by scalars. A nonnegative real-valued function defined on a vector space is called a norm if (i) x = 0 if and only if x =0. (ii) x + y≤x + y. (iii)λx = |λ|x. A vector space with norm is a then a metric space if we define the metric d to be d(x, y)=x − y. A sequence {xn} of elements in a normed vector space V is called a Cauchy sequence if, given >0, there is an N such that for all m, n ≥ N,wehave xn − xm <. A Banach space is a normed vector space that is complete in the sense that every Cauchy sequence {xn} satisfies xn − x→0forsomex ∈ V .

Example A.3.1 (Lp spaces.) Let µ be a measure over a measurable space p (X , A). Fix p>0 and L [X ,µ] denote the measurable functions f such that |f|pdµ < ∞. If we identify equivalence classes of functions that are equal al- most everywhere µ, then, for p ≥ 1, this vector space becomes a normed vector space by defining 1/p p f = fp = |f| dµ .

In this case, the triangle inequality

f + gp ≤fp + gp is known as Minkowski’s inequality. Moreover, this space is a Banach space.5

A Hilbert space H is a Banach space for which there is defined a function x, y on H × H to R, called the inner product of x and y, satisfying, for xi, y ∈ H, λi ∈ R, (i) λ1x1 + λ2x2,y = λ1x1,y + λ2x2,y . (ii) x, y = y, x . (iii) x, x = x2 . Two vectors x and y of H are called orthogonal if x, y =0.Acollection H0 ⊂ H of vectors is called an orthogonal system if any two elements in H0 are orthogonal. An orthogonal system is orthonormal if each vector in it has norm 1. An orthonormal system H0 is called complete if x, h = 0 for all h ∈ H0 implies x =0. In a separable Hilbert space, every orthonormal system is countable and there exists a complete orthonormal system. Letting {h1,h2,...} denote a complete orthonormal system, Parseval’s identity says that, for any x ∈ H, ∞ 2 2 x = [x, hj ] . (A.7) j=1

Example A.3.2 (L2 spaces.) In example A.3.1 with p = 2, the equivalence classes of square integrable functions is a Hilbert space with inner product given

5For proofs of the results in this section, see Chapter 5 of Dudley (1989). 698 AppendixA. Auxiliary Results by

f1,f2 = f1f2dµ .

If X is [0, 1] and µ is Lebesgue measure,√ then a complete orthonormal system is given by the functions fj (u)= 2sin(πju), j =1, 2,.... Therefore, for any square integrable function f, Parseval’s identity yields 1 ∞ 1 2 f 2(u)du =2 f(u)sin(πju)du . 0 0 j=1

A.4 Dominated Families of Distributions

Let M be a family of measures defined over a measurable space (X , A). Then M is said to be dominated by a σ-finite measure µ defined over (X , A) if each member of M is absolutely continuous with respect to µ. The family M is said to be dominated if there exists a σ-finite measure dominating it. Actually, if M is dominated there always exists a finite dominating measure. For suppose that M X ∪ is dominated by µ and that = Ai,withµ(Ai) finite for all i.Ifthesets i Ai are taken to be mutually exclusive, the measure ν(A)= µ(A ∩ Ai)/2 µ(Ai) also dominates M and is finite.

Theorem A.4.16 A family P of probability measures over a Euclidean space (X , A) is dominated if and only if it is separable with respect to the metric (4) or equivalently with respect to the convergence definition

Pn → P if Pn(A) → P (A) uniformly for A ∈A.

Proof. P { } Suppose first that is separable and that the sequence Pn is dense n in P, and let µ = Pn/2 . Then µ(A)=0impliesPn(A) = 0 for all n, and hence P (A) = 0 for all P ∈P. Conversely suppose that P is dominated by a measure µ, which without loss of generality can be assumed to be finite. Then we must show that the set of integrable functions dP/dµ is separable with respect to the convergence definition (5) or, because of Lemma A.2.2, with respect to convergence in the mean. It follows from Lemma A.2.1 that it suffices to prove this separability for the class F of all functions f that are integrable µ.Sinceby the definition of the integral every integrable function can be approximated in the mean by simple functions, it is enough to prove this for the case that F is the class of all simple integrable functions. Any simple function can be approximated in the mean by simple functions taking on only rational values, so that it is sufficient to prove separability of the class of functions riIAi where the r’s are rational and the A’s are Borel sets, with finite µ-measure since the f’s are integrable. It is therefore finally enough to take for F the class of functions IA, which are indicator functions of Borel sets with finite measure. However, any such set can be approximated by finite unions of disjoint rectangles with rational end

6Berger (1951b). A.4. Dominated Families of Distributions 699 points. The class of all such unions is denumerable, and the associated indicator functions will therefore serve as the required countable dense subset of F.

An examination of the proof shows that the Euclidean nature of the space (X , A) was used only to establish the existence of a countable number of sets Ai ∈Asuch that for any A ∈Awith finite measure there exists a subsequence Ai with µ(Ai) → µ(A). This property holds quite generally for any σ-field A which has a countable number of generators, that is, for which there exists a countable 7 number of sets Bi such that A is the smallest σ-field containing the Bi. It follows that Theorem A.4.1 holds for any σ-field with this property. Statistical applications of such σ-fields occur in sequential analysis, where the sample space X is the union X = ∪iXi of Borel subsets Xi of i-dimensional Euclidean space. In these problems, Xi is the set of points (x1,...,xi) for which exactly i observations are taken. If Ai is the σ-field of Borel subsets of Xi, one can take for A,theσ- field generated by the Ai, and since each Ai possesses a countable number of generators, so does A. If A does not possess a countable number of generators, a somewhat weaker conclusion can be asserted. Two families of measures M and N are equivalent if µ(A) = 0 for all µ ∈Mimplies ν(A) = 0 for all ν ∈N and vice versa.

Theorem A.4.28 A family P of probability measures is dominated by a σ-finite measure if and only if P has a countable equivalent subset.

Proof. Suppose first thatP has a countable equivalent subset {P1,P2,...}. Then n P is dominated by µ = Pn/2 . Conversely, let P be dominated by a σ-finite measure µ, which without loss of generality can be assumed to be finite. Let Q ∈P be the class of all probability measures Q of the form ciPi, where Pi ,the c’s are positive, and ci = 1. The class Q is also dominated by µ, and we denote by q a fixed version of the density dQ/dµ. We shall prove the fact, equivalent to the theorem, that there exists Q0 in Q such that Q0(A)=0impliesQ(A)=0 for all Q ∈Q. Consider the class C of sets C in A for which there exists Q ∈Qsuch that q(x) > 0a.e.µ on C and Q(C) > 0. Let µ(Ci) tend to sup µ(C), let qi(x) > 0 C∗ a.e. on Ci, and denote the union of the Ci by C0. Then q0 (x) ciqi(x) agrees a.e. with the density of Q0 = ciQi and is positive a.e. on C0, so that C0 ∈ C. Suppose now that Q0(A) = 0, let Q be any other member of Q, and let C = {x : q(x) > 0}. Then Q0(A ∩ C0) = 0, and therefore µ(A ∩ C0) = 0 and Q(A ∩ C0)=0.AlsoQ(A ∩ C˜0 ∩ C˜) = 0. Finally, Q(A ∩ C˜0 ∩ C) > 0 would lead to µ(C0 ∪ [A ∩ C˜0 ∩ C]) >µ(C0) and hence to a contradiction of the relation ∩ ˜ ∩ ∪ ∩ ˜ ∩ µ(C0)=supC µ(C), since A C0 C and therefore C0 [A C0 C] belongs to C.

7A proof of this is given for example by Halmos (1974, Theorem B of Section 40). 8Halmos and Savage (1949). 700 AppendixA. Auxiliary Results A.5 The Weak Compactness Theorem

The following theorem forms the basis for proving the existence of most powerful tests, most stringent tests, and so on.

Theorem A.5.19 (Weak compactness theorem). Let µ be a σ-finite mea- sure over a Euclidean space, or more generally over any measurable space (XA) for which A has a countable number of generators. Then the set of measurable functions φ with 0 ≤ φ ≤ 1 is compact with respect to the weak convergence (2).

Proof. Given any sequence {φn}, we must prove the existence of a subsequence

{φnj } and a function φ such that

lim φni pdµ= φp dµ

∗ ∗ for all integrable p.Ifµ is a finite measure equivalent to µ, thenp is integrable µ∗ if and only if p =(dµ∗/dµ)p∗ is integrable µ, and φp dµ = φp∗ dµ∗ for all φ. We may therefore assume without loss of generality that µ is finite. Let {pn} be a sequence of p’s which is dense in the p’s with respect to convergence in the mean. The existence of such a sequence is guaranteed by Theorem A.4.1 and the remark following it. If

Φn(p)= φnpdµ,

the sequence Φn(p) is bounded for each p. A subsequence Φnk can be extracted such that Φnk (pm) converges for each pm by the following diagonal process. Consider first the sequence of numbers {Φn(p1)} which possesses a convergent 1 1 2 2 subsequence Φn1 (p ), Φn2 (p ),.... Next the sequence Φn1 (p ), Φn2 (p ),... has 2 2 a convergent subsequence Φn1 (p ), Φn2 (p ),.... Continuing in this way, let n1 = n1, n2 = n2 , n3 ,.... Then n1

that Φni (p) converges for all p. Denote its limit by Φ(p), and define a set function Φ∗ over A by putting ∗ Φ (A)=Φ(IA). Then Φ∗ is nonnegative and bounded, since for all A,Φ∗(A) ≤ µ(A). To see that it is also countably additive let A = ∪Ak, where the Ak are disjoint. Then ∗ ∗ ∪ Φ (A)=limΦni ( Ak) and % % % % % % % % % m % % − ∗ % ≤ − ∗ % φni dµ Φ (Ak)% % φni dµ Φ (Ak)% ∪ % ∪m % Ak k=1Ak k=1

9Banach (1932). The theorem is valid even without the assumption of a countable number of generators; see N¨olle and Plachky (1967) and Aloaglu’s theorem, given for example in Royden (1988). A.5. The Weak Compactness Theorem 701 % % % ∞ % % ∗ % + % φni dµ − Φ (Ak)% . % ∪∞ % k=m+1Ak k=m+1 m Here the second term is to be taken as zero in the case of a finite sum A = ∪k=1Ak, ∞ and otherwise does not exceed 2µ(∪k=m+1Ak), which can be made arbitrarily small by taking m sufficiently large. For any fixed m the first term tends to zero as i tends to infinity. Thus Φ∗ is a finite measure over (X , A). It is furthermore absolutely continuous with respect to µ,sinceµ(A) = 0 implies Φni (IA) = 0 for ∗ all i, and therefore Φ(IA)=Φ (A) = 0 We can now apply the Radon–Nikodym theorem to get Φ∗(A)= φdµ for all A, A with 0 ≤ φ ≤ 1. We then have

φni dµ → φdµ for all A, A A and weak convergence of the φni to φ follows from Lemma A.2.3. References

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Agresti, A., 127, 129, 133, 134, 135, Baker, R., 446 168, 318 Balakrishnan, N., 156, 159, 193, 197, Aiyar, R. J., 272 281, 306, 307, 446 Akritas, M., 318 Banach, S., 700 Albers, W., 272, 451, 582, 690 Barankin, E. W., 47 Albert, A., 286 Bar-Lev, S., 118, 201 Alf, E., 590 Barlow, R. E., 287 Andersen, S. L., 210 Barnard, G. A., 175, 413 Anderson, T. W., 90, 218, 306, 318 Barndorff-Nielsen, O., 47, 55, 106, Andersson, S., 218 214, 398, 403, 517 Anscombe, F., 474 Barnett, V., 6 Antille, A., 248 Barron, A., 630 Arbuthnot, J., 107, 149 Bartholomew, D. J., 287 Arcones, M., 655, 678 Bartlett, M. S., 95, 517 Armsen, P., 127 Basu, D., 106, 210, 395, 397, 398, 410, Arnold, S., 239, 274, 292-293, 318, 411, 412 374 Basu, S., 462, 481 Arrow, K., 58 Bayarri, J., 108 Arthur, K. H., 306 Bayarri, M., 175 Arvesen, J. N., 300 Becker, B., 109 Athreya, K., 655 Becker, N., 397 Atkinson, A., 169, 293, 318 Bednarski, T., 328 Behnen, K., 582 Babu, G., 655, 690 Bell, C. B., 118, 241 Bahadur, R., 54, 210, 466, 481, 575, Bell, C. D., 210 582 Benichou, J., 526, 549 Bain, L. J., 200, 201 Bening, V., 582 758 Author Index

Benjamini, Y., 354, 374, 445 Brockwell, P. J., 451 Bennett, B., 127, 171 Bromeling, L. D., 304 Bentkus, V., 604 Bross, I. D. J., 127 Beran, R., 481, 526, 539, 582, 589, Brown, K. G., 304 629, 657, 658, 668, 671, 672, Brown, L. D., 18, 18, 47, 55, 69, 71, 673, 679, 689, 690 108, 115, 141, 157, 237, 308, Berger, A., 320, 698 336, 347, 408, 409, 414, 415, Berger, J., 15, 16, 18, 27, 95, 108, 173, 435, 561, 647, 668 175, 331, 400, 414, 415, 526 Brown, M. B., 448, 480 Berger, R., 108, 287, 561 Brownie, C., 409, 446 Berk, R., 220, 226, 241 Brunk, H. D., 287 Bernardo, J., 16 Brunner, E., 318 Bernoulli, D., 107 Buehler, R., 175, 196, 408, 408, 412, Best, D., 616, 630 413, 414, 414 Bhapkar, V. P., 135 Burkholder, D. L., 54 Bhat, U., 145 Bhattacharya, P. K., 248 Caba˜na, A., 629 Bhattacharya, R., 460, 481, 668 Caba˜na, E., 629 Bickel, P., 11, 27, 226, 241, 474, 481, Cai, T., 18, 435, 647, 668 488, 517, 539, 571, 582, 654, Carroll, R. J., 318 677, 678, 679, 690 Casella, G., vii, 5, 13, 17, 21, 55, 108, Billingsley, P., 42, 55, 117, 147, 185, 124, 157, 173, 174, 201, 292, 223, 256, 424, 427, 451, 476, 335, 336, 395, 396, 408, 415, 480, 611 506, 507, 548, 561, 679 Birch, M. W., 135 Castillo, J., 144 Birnbaum, A., 99, 126, 276, 400, 414 Chakraborti, S., 146, 245, 251, 286, Birnbaum, Z. W., 108, 256, 442 290, 442 Bishop, Y. M. M., 135, 525 Chalmers, T. C., 57 Blackwell, D., 16, 21, 27, 40, 95, 118 Chambers, E. A., 134 Blair, R. C., 539 Chapman, D. G., 108 Bloomfield, P., 378, 384 Chatterjee, S., 169 Blyth, C. R., 6, 75, 108, 167, 168 Chebyshev, P., 481 Bohrer, R., 384 Chen,H.J.,629 Boldrick, J., 109, 391 Chen, L., 445 Bondar, J. V., 334, 415 Chernoff, H., 231, 526, 598-599, 630 Bondessen, L., 13 Chhikara, R. S., 100, 197, 197 Boos, D., 108, 210, 248, 446, 481 Chmielewski, M. A., 314 Boschloo, R. D., 127 Choi, K., 318 Bose, R. C., 391 Choi, S., 582 Boukai, B., 175 Chou, Y. M., 306 Bowker, A. H., 524 Choy, K., 582 Box, G. E. P., 210, 293, 304, 421, 474, Christensen, R., 318 480 Cima, J. A., 384 Box, J. F., 27 Clinch, J. C., 448, 480 Brain, C. W., 629 Cochran, W. G., 448 Braun, H., 391 Cohen, A., 57, 69, 135, 201, 210, 239, Breiman, L., 118 287, 316, 318, 341, 629 Bremner, J. M., 287 Cohen, J., 281 Bretagnolle, J., 655, 678 Cohen, L., 95 Author Index 759

Conover, W. J., 446, 481 Dvoretzky, A., 95, 442 Coull, B., 168 Dykstra, R., 287 Cox, D., 214 Cox, D. R., 6, 108, 134, 134, 220, 397, Eaton, M., 211, 218, 227, 318, 331 414, 474 Edelman, D., 445, 462 Cram´er, H., 27, 481, 506, 526, 629 Edgeworth, F. Y., 107, 481 Cressie, N., 445, 629 Edgington, E. S., 690 Cs¨org¨o, S., 655 Edwards, A. W. F., 129, 175 Cvitanic, J., 338 Efron, B., 190, 195, 439, 481, 626, Cyr, J. L., 481 648, 668, 672, 686, 690 Eisenhart, C., 146 D’Agostino, R., 408, 589, 616, 628, Elfving, G., 54 629 Engelhardt, M. E., 200, 201 Dantzig, G. B., 78, 108 Eubank, R., 607, 616, 630 Darmois, G., 57 DasGupta, A., 18, 276, 435, 462, 481, Falk, M., 451 647, 668 Fan, J., 526, 607, 630 Davenport, J. M., 231 Faraway, J., 391 David, H. A., 243 Farrell, R., 108, 218, 331 Davis, B. M., 127 Fears, T., 526, 549 Davis, R. A., 451 Feddersen, A. P., 408 Davison, A., 690 Feinberg, S. E., 135, 525 Dawid, A. P., 106, 411 Feller, W., 5, 256, 459, 464, 604 Dayton, C., 373 Fenstad, G. U., 448 de Leeuw, J., 11 Ferguson, T. S., 5, 16, 18, 27 de Moivre, A., 480 Fienberg, S., 134, 210 Dempster, A. P., 175 Finch, P. D., 132 Deshpande, J. V., 629 Finner, H., 140, 354, 373, 391 Deuchler, G., 276 Finney, D. J., 127 Devroye, L., 443 Fisher, R. A., 27, 97, 107, 108, 127, de Wet, T., 589, 630 149, 175, 210, 408, 414, 415, Diaconis, P., 180, 270, 318, 626, 639, 526, 597, 598, 630, 635 649 Folks, J. L., 100, 197, 197 DiCiccio, T., 517, 686, 691 Forsythe, A., 190, 207, 448, 480 Dobson, A., 318 Fourier, J. B. J., 108 Doksum, K. A., 27, 474, 629 Franck, W. E., 259 Donev, A., 293 Fraser, D. A. S., 106, 141, 175 Donoghue, J., 366 Freedman, D., 131, 654, 678, 690 Donoho, D., 481 Freeman, M. F., 474 Draper, D., 286 Freiman, J. A., 57 Drost, F., 630 Fris´en, M., 138 Dubins, L. E., 40 Fuller, W., 451 Ducharme, G., 671 Dudley, R., 55, 424, 472 480, 486, Gabriel, K. R., 188, 190, 368, 374, 385 571, 697 Gail, M., 526, 549 Dudoit, S., 109, 391, 690 Galambos, J., 629 D¨umbgen, L. 629 Gan, L., 507 Duncan, D. B., 391 Garside, G. R., 127 Durbin, J., 245, 414, 442, 616, 629 Gart, J. J., 134 760 Author Index

Garthwaite, P., 190 Harville, D. A., 304 Gastonis, C., 135 Has’minskii, R., 506 Gastwirth, J. L., 248, 450 Hastie, T., 318 Gauss, C. F., 27, 108, 480 Hayter, T., 391 Gavarett, J., 107 Haytner, A., 391 George, E. I., 415 Hedges, L., 109 Ghosh, J., 46, 200, 220, 481, 517 Hegazy, Y. A. S., 629 Ghosh, M., 149, 323 Hegemann, V., 290, 291 Gibbons, J., 108, 146, 245, 251, 286, Heritier, S., 526 290, 442 Hettmansperger, T., 286, 287, 290, Giesbrecht, F., 293 318, 446, 448 Gin´e, E., 655, 658 Higgins, J. J., 539 Giri, N., 322, 335 Hillier, G. H., 480 Girshick, M. A., 16, 27, 141 Hinkley, D., 401, 474, 690 Glaser, R. E., 200, 481 Hipp, C., 47 Gleser, L. J., 197, 442 Hochberg, Y., 353, 354, 368, 373, 374, Gokhale, D. V., 131 384, 391 Good, P., 180, 210, 690 Hocking, R. R., 292, 293, 316 Goodman, L. A., 129 Hodges, J. L., Jr., 157, 157, 424, 582 Gordon, I., 397 Hoeffding, W., 118, 210, 276, 481, G¨otze, F., 677, 679 629, 637, 690 Goutis, C., 201, 408 Hoel, P. G., 149 Green, B. F., 180 Hogg, R. V., 159 Green, J. R., 629 Holland, P. W., 135, 525 Greenwood, P., 598, 630 Holm, S., 350, 374, 375, 391 Grenander, U., 108 Holmes, S., 180, 639, 649 Groeneboom, P., 539, 540, 582 Hooper, P. M., 220, 239 Guenther, W. C., 146 Horst, C., 127 Guillier, C. L., 272 Hotelling, H., 108, 276, 318, 391, 448, Gumpertz, M., 293 474, 480 Gupta, A., 339 Hsu, C. F., 188 Hsu, C. T., 208 Haberman, S. J., 129, 134 Hsu, J., 391, 561 Hackney, O. P., 292 Hsu, P., 127, 318 Hadi, A., 169 Huang, J. S., 323 H´ajek, J., 33, 245, 256, 286, 486, 487, Huber, P. J., 328, 347, 480 525, 526, 582, 588, 590 Hughes-Oliver, J., 210 Hald, A., 480 Hung, K., 145 Hall, P., 75, 446, 460, 481, 517, 629, Hunt, G., 276, 347 655, 664, 665, 668, 679, 690, Hunter, J. S., 293 691 Hunter, W. G., 293 Hall, W., 109, 190, 220, 582 Hutchinson, D. W., 167 Hallin, M., 582 Hwang, J., 108, 157, 197, 336, 415, Halmos, P. R., 95, 331, 499 561 Hamada, M., 293 Hamilton, J. D., 451 Ibragimov, I., 506 Hartigan, J. A., 190, 206, 207, 211, Ibragimov, J. A., 323 691 Inglot, T., 582, 607, 630 Hartley, H. O., 281 Ingster, Y., 597, 622, 624 Author Index 761

Isaacson, S. L., 341 Knight, K., 655 Jagers, P., 279 Knott, M., 616 James, A. T., 448, 480 Koehn, U., 210 James, G. S., 448, 480 Kohne, W., 451 Jammalamadaka, S., 630 Kolassa, J., 668 Janssen, A., 582, 583, 616, 617, 621, Kolmogorov, A., 20, 585, 629 622, 683, 690 Kolodziejczyk, S., 317 Jensen, J., 517 Koopman, B., 47 Jiang, J., 507 Korn, E., 374 Jing, B., 481 Koschat, M., 197 Jockel, K., 443 Koshevnik, Y., 582 Johansen, S., 119, 448, 480 Kotz, S., 98, 99, 114, 127, 145, 156, John, R. D., 180, 188 159, 193, 197, 209, 281, 306, Johnson, D. E., 290, 291 307, 435 Johnson, M. E., 446, 481 Kowalski, J., 18 Johnson, M. M., 446, 481 Koziol,J.A.,248 Johnson, N. L., 98, 99, 114, 127, 156, Krafft, O., 86 159, 193, 197, 209, 281, 306, Kraft, C., 320 307, 435 Kruskal, W., 95, 129, 276 Johnson, N. S., 131 Kuebler, R. R., 57 Johnstone, I. M., 71, 115, 308 K¨unsch, H. R., 687 Joshi, V., 239 Kusunoki, U., 391

Kabe,D.G.,93 Lahiri, S. N., 451, 687, 690 Kakutani, S., 582 Lambert, D., 109, 328, 447, 690 Kalbfleisch, J. D., 395 Lane, D., 131 Kallenberg, W., 140, 272, 339, 582, Laplace, P. S., 27, 107, 108, 480 607, 630 LaRiccia, V., 607, 616 Kanoh, S., 391 Latscha, R., 127 Kappenman, R. F., 440 Laurent, A. G., 93 Karatzas, I., 338 Lawless, J. F., 400 Kariya, T., 314 Layard,M.W.J.,300 Karlin, S., 5, 22, 69, 71, 231, 323 Le Cam, L., 21, 54, 487, 488, 489, Kasten, E. L., 127 507, 525, 526, 533, 549, 550, Kemp, A., 114, 127, 435 582 Kemperman, J., 210 Ledwina, T., 272, 582, 607, 630 Kempthorne, O., 293 L´eger, C., 653 Kempthorne, P., 11 Lentner, M. M., 196 Kendall, M. G., 272, 629, 630 Levit, B., 582 Kent, J., 629 Levy,K.J.,255 Kersting, G., 248 Lexis, W., 107, 108 Kesselman, H. J., 448, 480 Liang, K. Y., 403, 525 Khmaladze, E., 629 Lieberman, G. J., 127 Khurshid, A., 127 Lin, S., 630 Kiefer, J., 276, 293, 306, 316, 322, Lindley, D. V., 95 335, 409, 413, 442 Linnik, Y, V., 335 King, M. L., 480 Liseo, B., 526 Klaassen, C., 488, 571, 582 Littell, R. C., 109 Klett, G. W., 165, 201, 252 Liu, H., 287 762 Author Index

Liu, R. Y., 108, 665, 687 Miller, J., 304, 316 Loh, W.-Y., 220, 323, 481, 626, 667, Miller, R. G., 293, 390 668 Milliken, G. A., 127 Loomis, L. H., 331 Milnes, P., 334 Lorenzen, T. J., 293 Miwa, T., 391 Lou, W., 146 Montgomery, D., 293 Louv, W. C., 109 Morgan, W. A., 208 Low, M., 582 Morgenstem, D., 210 Lyapounov, A. M., 95 Morimoto, H., 21, 46 Mosteller, F., 141, 421 Maatta, J., 415 M¨ott¨onen, J., 318, 582 MacGibbon, K. G., 71, 115, 308 Mudholkar, G. S., 439 Mack, C., 127 M¨uller, C., 480 Mack, G. A., 290 Munk, A., 157 Madansky, A., 200 Murphy, S., 526 Maitra, A. P., 47 Murray, L. W., 300 Mandelbaum, A., 118 Muyot, M., 318 Mann, H., 597 Manoukian, E. B., 481 Nachbin, L., 331 Mantel, N., 549 Naiman, D. Q., 384 Marasinghe, M. C., 290 Narula, S. C., 255 Marcus, R., 374, 385 Neill, J., 318 Marden, J., 18, 108, 109, 135, 141, Nelder, J., 318 287, 318, 403, 404 Neuhaus, G., 582, 629 Mardia, K. V., 159 Neyman, J., 27, 107, 108, 108, 149, Maritz, J. S., 203 210, 211, 480, 597, 599, 600, Marshall, A. W., 308, 323 630 Mart´ın, A., 127 Nicolaou, A., 174 Martin, M., 668 Niederhausen, H., 442 Mason, D., 655 Nikitin, Y., 539, 582, 622, 629 Massart, P., 442 Nikulin, M., 598, 630 Massey, F. J., 586, 587 Noether, G., 582 Mathew, M., 300 Nogales, A., 220 Mattner, L., 118 N¨olle, G., 700 McCullagh, P., 318, 590, 668 McCulloch, C., 293 Od´en, A., 179 McDonald, L. L., 127 Oja, H., 318, 582 McKean, J., 286-287, 287, 446, 448 Ojeda, M., 297 McLaughlin, D. H., 421 Olkin, I., 109, 308, 323 McShane, L., 374 Olshen, R. A., 408 Mee, R., 268 Oosterhoff, J., 534, 540, 582, 630 Meeks, S. L., 408 Ord, J., 27, 316, 439, 597 Meng, X., 108 Owen, A., 442, 526, 673, 690 Michel, R., 121 Owen, D. B., 127, 193, 306, 447 Milbrodt, H., 616, 622 Oyola, J., 220 Millar, W., 526, 629, 658 Miller, F. L., 616 Pace, L., 220 Miller, F. R., 318 Pachares, J., 114 Miller, G., 145 Padmanabhan, A., 446 Author Index 763

Patel, J. K., 209 Rao, R., 460, 481, 668 Pauls, T., 690 Rayner, J., 616, 630 Pawitan, Y., 526, 549 Read, C. B., 209 Pearson, E. S., 27, 107, 108, 149, 210, Read, T., 629 281, 480 Reid, C., 27, 108 Pearson, K., 107, 526, 590, 629 Reid, N., 214 Pedersen, K., 47 Reinhardt, H. E., 86 Pena, E., 630 Reiser, B., 201 Pereira, B., 220 Riani, M., 169, 318 Perez, P., 220 Richmond, J., 384, 390 Peritz, E., 374, 379, 385 Rieder, H., 328, 582 Perlman, M., 157, 233, 287, 403, 404, Ripley, B., 443 506, 561 Ritov, Y., 488, 571, 582 Pesarin, F., 448, 690 Robbins, H., 696 Peters, D., 582 Robert, C., 15, 108, 173, 173, 175 Pfanzagl, J., 65, 67, 162, 231, 581, Robertson, T., 287 582 Robins, J., 109 Piegorsch, W. W., 384 Robinson, G., 188, 202, 231, 408, 408, Pierce, D. A., 414, 415 415 Pitman, E. J. G., 47, 210, 276, 414, Robinson, J., 180, 188, 630, 690 582 Rojo, J., 23, 101 Plachky, D., 118, 700 Ronchetti, E., 287, 526 Plackett, R. L., 134, 525 Rosenstein, R. B., 306 Pliss, V. A., 335 Rosenthal, R., 58 Politis, D. N., 658, 674, 676, 679, 680, Ross, S., 5 687, 690, 691 Roters, M., 354, 391 Pollard, D., 424, 471, 484, 519, 585 Rothenberg, T. J., 157, 448, 480 Pollard, K., 690 Roussas, G., 525, 543, 582 Polonik, W., 629 Roy, K. K., 21, 54 Posten, H. O., 447 Roy, S. N., 391 Pratt, J. W., 99, 150, 200, 245, 413, Royden,H.L.,700 418 Rubin, D. B., 58 Prescott, P., 287 Rubin, H., 69, 248, 450 Price, B., 169 Rukhin, A., 220 Puig, P., 144 Runger, G., 211, 474 Pukelsheim, F., 293 Ruppert, D., 318 R¨uschendorf, L., 118 Quenouille, M., 691 Quesenberry, C. P., 260, 263, 616, 629 Sackrowitz, H., 108, 210, 237, 287, Quine, M. P., 630 341, 629 Sahai, H., 127, 297 Radlow, R., 590 Salaevskii, O. V., 335 Ramachandran, K. V., 166 Salaevskii, Y., 335 Ramamoorthi, R. V., 21, 54 Salvan, A., 220 Ramsey, P. H., 447 Samuel-Cahn, E., 108 Randles, R., 251, 286, 539, 582, 589, Sanathanan, L., 58 630 Sarkar, S. K., 354 Rao, C. R., 11, 526, 630 Savage, L. J., 16, 27, 58, 141, 414, Rao, P., 488 466, 481, 499, 575 764 Author Index

Schafer, G., 95 Small, C., 507 Scheff´e, H., 149, 166, 231, 293, 316, Smirnov, N. V., 256 317, 318, 374, 448, 480, 696 Smith, A., 16 Schervish, M., 27 Smith, D. W., 300 Schick, A., 582 Smith, H., 57 Scholz, F. W., 109 Sophister (G. Story), 480 Schuirmann, D., 561 Speed, F. M., 292 Schwartz, R., 276, 306, 316 Speed, T., 210, 318 Schweder, T., 109 Spiegelhalter, D. J., 629 Seal, H. L., 317 Spjøtvoll, E., 109, 293, 300, 370, 374 Searle, S., 293 Sprott, D. A., 106 Seber, G. A. F., 318 Spurrier, J. D., 391, 639 Seidenfeld, T., 175 Srivastava, M. S., 481 Self, S. G., 525 Starbuck, R. R., 260, 263 Sellke, T., 95, 108 Staudte, R., 108 Sen, P., 33, 210, 245, 256, 286, 525, Stein, C. M., 89, 210, 237, 276, 281, 582, 588, 590 335, 336, 347, 539, 570, 582 Serfling, R. H., 245, 539, 574, 582, Stephens, M., 589, 616, 628, 629 648 Stern, S., 517 Serroukh, A., 582 Stigler, S., 107, 480 Severini, T., 174 Still, H. A., 168 Shaffer, J. P., 109, 124, 311, 360, 365, Stone, C. J., 11, 539 371, 373, 374, 390, 391, 525 Stone, M., 334 Shaikh, A. M., 374 Strassen, V., 328 Shao, J., 27, 648, 690 Strasser, H., 27, 264, 616, 622 Shapiro, S. S., 629 Strawderman, W. E., 69, 239 Sheather, S., 287 Stuart, A., 27, 316, 439, 597, 629, 630 Sherfey, B., 318 Student (W. S. Gosset), 448 Shewart, W., 480 Sugiura, N., 245 Shorack, G., 200, 512, 588, 590, 622, Sun, J., 391 629 Suslina, I., 622, 624 Shorrock, G., 201 Sutton, C., 443 Shrader, R. M., 286-287 Swed, F. S., 146 Shriever, B. F., 630 Shuster, J., 100 Takeuchi, K., 93 Sid´ak, Z., 33, 245, 256, 286, 525, 582, Tallis, G. M., 629 588, 590 Tamhane, A., 353, 354, 368, 373, 374, Siegel, A. F., 96, 143, 470 391 Siegmund, D., 9, 588 Tan, W. Y., 445 Sierpinski, W., 323 Tang, D., 287 Silvapulle, M., 513 Taniguchi, M., 582 Silvapulle, P., 513 Tanur, J., 210 Silvey, S. D., 293 Tapia, J., 127 Simon, R., 374 Tate, R. F., 165, 201, 252 Simpson, D., 582 Taylor,H.,5,22 Singh, K., 108, 665, 687, 690 Thomas, D. L., 210 Singh, M., 448 Thombs, L. A., 451 Sinha, B., 300, 314 Tiao, G. C., 304, 421 Skillings, J. H., 290 Tibshirani, R., 318, 439, 668, 686, 690 Author Index 765

Tienari, J, 582 Wellner, J., 488, 512, 571, 574, 582, Tiku, M. L., 281, 306, 307, 446, 448 585, 588, 590, 612, 622, 629, Tiwari, R., 630 658 Tong, Y. L., 145 Wells, M., 108, 630 Tritchler, D., 190 Welsh, A. H., 629 Troendle, J., 374, 391 Westfall, P. H., 109, 293, 300, 366, Tseng, Y., 336 375, 386, 391, 690 Tu, D., 648, 690 Westlake, W., 561 Tukey, J. W., 20, 77, 291, 374, 391, Wijsman, R., 218, 220, 331, 378, 391 421, 474, 480, 691 Wilk, M. B., 293, 629 Turnbull, H., 39 Wilks, S. S., 526 Tweedie, M. C. K., 197 Williams, D., 55 Wilson, E. B., 108, 435 Unni, K., 398 Winters, F., 480 Uthoff, V. A., 259, 260 Witting, H., 86 Wolf, M., 391, 469, 582, 658, 674, Vadiveloo, J., 190 676, 679, 680, 690, 691 van Beek, P., 428 Wolfe, D. A., 251, 286, 539 van der Laan, M., 690 Wolfowitz, J., 76, 95, 442 van der Vaart, A., 90, 109, 518, 526, Wolpert, R., 108, 400, 414, 415, 526 573, 574, 582, 585, 612, 629, Working, H., 108, 391 658 Wright,A.L.,248 van Zwet, W. R., 534, 582, 677, 679, Wright, F., 287 690 Wu, C. F., 293, 677, 679, 691 Venable, T. C., 135 Wu, L., 157, 233, 287, 561 Ventura, V., 109 Wu, Y., 11 Vermeire, L., 339 Wynn, H. P., 378, 384 von Mises, R., 629 von Randow, R., 334 Yamada, S., 21, 46 Vu, H., 526 Yanagimoto, T., 145 Yandell, B. S., 629 Wacholder, S., 149 Yang, G., 487, 488, 525, 533, 582 Wald, A., 18, 27, 78, 95, 108, 347, Yang, Z., 507 506, 526, 543, 582, 597 Yeh, H. C., 447 Wallace, D., 231, 415 Yekutieli, D., 354 Wand, M. P., 318 Young, S., 109, 293, 375, 386, 391, Wang, H., 414 690 Wang, J., 507 Yuen, K. K., 448 Wang, Q., 145 Wang, Y., 175 Zabell, S., 175 Wang, Y. Y., 231, 448 Zemroch, P. J., 159 Webster, J. T., 231 Zhang, C., 526 Wedel, H., 179 Zhang, J., 481, 526, 630 Wefelmeyer, W., 581, 582 Zhou, S., 526 Weinberg, C. R., 149 Zinn, J., 655 Weisberg, S., 169, 318 Zucchini, W., 248 Welch, B. L., 380, 413 Welch, W., 210, 448 Wellek, S., 582 Subject Index

Absolute continuity (of one measure to maximin tests, 329; relation with respect to another), 33, to unbiasedness, 329. See also 492. See also Equivalence, of Invariance two measures; Radon-Nikodym Almost sure convergence, 440 derivative Almost sure representation theorem, Accelerated, bias-corrected, percentile 443 method, 685, 686 Aloaglu’s theorem, 700 Action problem, 6 Alpha-admissibility, 233 Adaptive test, 539 Alternatives (to a hypothesis), 56 Additive linear models, 318 Amenable group, 334 Additivity of effects, 287, 290; in model II, 298; test for, 290, Analysis of covariance, 297 291 Analysis of variance, 286, 292, 318; Admissibility, 17; and invariance, 26; different models for, 297; Bayes method for proving, for one-way layout, 286; for 236; of confidence sets, 239, two-way layout, 288; history 336; of multiple comparison of, 317; robustness of F -tests, procedures, 369, 370; of 446. See also Components of UMP invariant tests, 332; of variance; Linear hypothesis; UMP unbiased tests, 139, Linear model 232. See also α-admissibility; Ancillary statistic, 152, 395, 400; and d-admissibility; Inadmissibility invariance, 395, 397, 401; and Affinity, 530 sufficiency, 397; history of, Aligned ranks, 290 414; in the presence of missing Almost everywhere (a.e.), 33, 115 observations, 410; maximal, Almost invariance, 23, 263; of 397; paradox for, 414. See also likelihood ratio, 263; of tests, S-ancillary 225, 241; relation to invariance, Anderson Darling statistic, 589, 612 230; relation to invariance of Anderson’s nonparametric confidence power function, 230; relation interval for a mean, 468-469 768 Subject Index

Approximate hypotheses: extended Bayesian confidence sets, see Credible Neyman Pearson lemma for, regions 326, 328 Bayesian inference, 15, 172, 173, 175, Arcsine transformation for binomial 304 variables, 474 Bayes risk, 14 Association, 132; spurious, 133; Yule’s Bayes solution, 14, 23; and complete measure of, 129. See also class of decision procedures, Dependence, positive 18; restricted, 15; to maximize Asymptotically linear statistic, 500 minimum power, 320; to Asymptotically maximin tests: for prove admissibility, 236. See multi-sided hypotheses, 564– also Credible region; Prior 567, for nonparametric mean, distribution 567–570, for Chi-squared test, Bayes sufficiency, 21 593, 594 Bayes test, 94, 264, 309 Asymptotically most powerful test Behrens-Fisher distribution, 202 sequence, 541 Behrens-Fisher problem, 159, 231, Asymptotically normal experiments, 408, 415, 420; bootstrap 549–553 solution, 671, 672; LAUMP Asymptotically perfect test, 432 tests for, 558, 559; many Asymptotically uniformly most sample, 448; nonparametric, powerful (AUMP) tests: in 245; permutation test for, 642; univariate models, 540–549, under nonnormality, 447. See in multiparameter models, also Welch-Aspin test 553–567, in nonparametric Berry-Esseen theorem, 428; models, 567–574 multivariate, 604 Asymptotic equivalence of test Beta distribution, 159, 280; as sequences, 577 distribution of order statistics, Asymptotic pivot, 646 266; noncentral, 280, 307; Asymptotic relative efficiency, relation to F -distribution, 534-540, 582; and goodness 159; relation to gamma of fit, 621; of randomization distribution, 196 tests, 639 Bimeasurable transformation, 214 Asymptotic normality: of functions Binomial distribution b(p, n), 4; of asymptotically normal in comparing two Poisson variables, 436; of sample mean, distributions, 153; as loglinear 426; or sample median, 429. model, 134; completeness of, See also Central limit theorem 116; in comparing two Poisson Asymptotic optimality, vii, 527 distributions, 125, 398; in sign Attributes: paired comparisons by, test, 85; variance stabilizing 169, 291, 510, 526; sample transformation for, 474. See inspection by, 80, 293 also Contingency tables; Automatic percentile method, 686 Multinomial distribution; Autoregressive process, 450 Negative binomial distribution; Average power, maximum, 96, 308, Two by two table 627 Binomial probabilities: comparison of two, 106, 126, 145, 149, Bahadur efficiency, 539 399; confidence bounds Bahadur Savage theorem, 466, 467 for, 75; confidence intervals Banach space, 696–698 for, 167, 434, 435; credible Bartlett correction, 517; relationship region for, 172; one-sided test to bootstrap, 671, 691 for, 67; two-sided test for, Bartlett’s test for variances, 481 113. See also Contingency Basu’s theorem, 152, 210 tables; Independence, test Subject Index 769

for; Matched pairs; Median; combinations, 452; for sample Paired comparisons; Sample median, 429; Lindeberg, 427; inspection; Sign test Lyapounov, 427; multivariate, Binomial trials, 8. 18. 134; minimal 427; uniform, 463, 465 sufficient statistic for, 26. See Characteristic function, 426 also Inverse sampling Chebyshev inequality, 472 Bioassay, 147 Chi-squared distribution, 47; for Bivariate distribution (general): testing linear hypothesis with one-parametric family for, 191; known variance, 310; in testing testing for independence in, normal variance, 114, 155; 192, 271. See also Dependence limit for likelihood ratio, 515, Bivariate normal correlation 516; non-central, 306, 308, coefficient: asymptotic test for, 311; relation to exponential 512; confidence intervals for, distribution, 54; relation to 201; test for, 201, 231, 261, F -distribution, 158; relation to 397; confidence bounds for, t-distribution, 156. See also 273 Gamma distribution; Normal Bivariate normal distribution, 190, one-sample problem, variance; 207; ancillary statistics in, 397; Wishart distribution joint distribution of second Chi-squared test: as a Neyman smooth moments in, 208; test for test, 601; asymptotically independence in, 190 maximin property, 593, Bonferroni procedure, 350, 385 594; for simple hypotheses, Bootstrap, vii, 648; consistency of, 420, 514, 515, 590–597; 650; higher order properties, for composite hypotheses, 664; hypothesis testing, 668; in 597–599; in contingency tables, multiple testing, 658 626; for testing uniformity, Bootstrap calibration, 667 594–597 Bootstrap-t: consistency of, 654; Closure method for multiple testing, higher order properties, 385 665–667 Cluster sampling, 449 Bounded-Lipschitz metric, 471 Cochran-Mantel-Haenszel test, 135 Borel set, 29 Coefficient of variation: asymptotic Bounded completeness, 118, confidence interval for, 509; 228; example of, without confidence bounds for, 273; completeness, 141. See also tests for, 157, 222, 230 Completeness of family of Comparison of experiments, 136, 204 distributions Complement of a set E, denoted Ec, Brownian Bridge process, 585, 588 28 Completeness of a class of decision Calibration, 667 procedures, 17, 18, 108; for Cauchy distribution, 71, 99, 324, 339 one-parameter exponential Cauchy location model: AUMP and family, 141; of classes of LAUMP tests for, 547, 548; one-sided tests, 69; of class of q.m.d. property, 487 two-sided tests, 140; relation Causal influence, 132 to sufficiency, 21. See also CDF, see Cumulative distribution Admissibility function Completeness of family of Center of symmetry: confidence distributions, 115; of intervals for, 203, 206. See also binomial distributions, 116; Symmetric distribution of exponential families, 117; Central limit theorems: for dependent of nonparametric family, variables, 448, 449; for linear 118; of normal distributions, 770 Subject Index

116; of order statistics, 118, 165. See also Simultaneous 143; relations to bounded confidence intervals completeness, 118, 141; of Confidence level, 72 uniform distributions, 116 Confidence sets, 72; admissibility of, Completion of measure, 29 239, 335; average smallest, Complexity: of multiple comparison 251; based on multiple tests, procedure, 373 391; derived from a pivotal Components of variance, 303. See quantity, 254; equivariant, 248, also Random effects model 336; example of inadmissible, Composite hypothesis, 59; vs. simple 336; minimax, 335,336; of alternative, 84 smallest expected Lebesgue Conditional distribution, 40, 41; measure, 200; relation to tests, example of nonexistence, 40 171; unbiased, 164; which are Conditional expectation, 37, 42; not intervals, 225. See also properties of, 39 Credible region; Equivariant Conditional independence: test for, confidence sets; Relevant 133 and semirelevant subsets; Conditional inference, 393, 394, 408; Simultaneous confidence sets optimal, 400 Conjugate distribution, 173 Conditional power, 123, 138, 188, Conservative test, 127 398, 400 Consumer preferences, 135 Conditional probability, 39, 40 Contiguity, 492–494; and limiting Conditional test, 549–553 distribution of a statistic; 499, Confidence bands: for cumulative 500; characterizations of, 496, distribution function, 255, 276; 497; examples of, 498–503 for linear models, 375; for Contingency tables: loglinear models × regression line, 384, 391. See for, 134; r c tables, 127; × × also Simultaneous confidence three factor, 132; 2 2 K, × × intervals 138, 148; 2 2 2, 139; × × × Confidence bounds, 72; equivariant, 2 2 2 L, 148. See also 272; impossible, 300, 408; Two by two tables in presence of nuisance Continuity correction, 127 parameters, 162; most Continuity point, 425 accurate, 72; relation to Continuity theorem, 426 median unbiased estimates, Continuous Mapping theorem, 435, 162; relation to one-sided 436 tests, 163; standard, 76; with Consistent estimator, 432 minimum risk, 102 Contrasts, 382, 472 Confidence coefficient, 72 162; Convergence in distribution (or in conditional, 408 law), 425 Confidence intervals, 6, 76, 162; after Convergence in probability, 431 rejection of a hypothesis, 140, Convergence of moments, 443, 444 408; distribution-free, 189, Convergence theorem: for densities, 203, 251; empty, 300; expected 696; dominated, 32; monotone, length of, 170; history of, 32. See also Central limit 108, 211; in randomization theorem; Continuity theorem; models, 188; interpretation of, Continuous mapping theorem; 162; logarithmically shortest, Cram´er-Wold theorem; Delta 252; loss functions for, 76; method ; Glivenko-Cantelli of bounded length, 197, 198; theorem; Prohorov’s theorem randomized, 166; relation to Cornish-Fisher expansion, 460, 663 two-sided tests, 163; uniformly Correlation coefficient: in bivariate most accurate unbiased, normal distribution, 190, Subject Index 771

548, 549, 557; confidence Design of experiments, 8, 9, 130, bounds for, 273; intraclass, 204, 293. See also Random 313; multiple tests of, 661; assignment; Sample size nonparametric bootstrap test Directional error, 139, 140, 373 of, 670; testing value of, 190, Direct product (of two sets), 33 231, 261. See also Bivariate Dirichlet distribution, 202 distribution; Dependence, Distribution, see the following positive; Multiple correlation families of distributions: coefficient; Rank correlation Beta, Binomial, Bivariate coefficient; Sample correlation normal, Cauchy, Chi-squared, coefficient Dirichlet, Double exponential, Countable additivity, 28 Exponential, F ,Gamma, Countable generators of σ-field, 699 Hypergeometric, Inverse Counting measure, 29 Gaussian, Logistic, Lognormal, Covariance matrix, 89, 305 Multinomial, Multivariate Coverage error, 662–668 normal, Negative binomial, Cram´er’s condition, 459 Noncentral, Normal, Pareto, Cram´er-von Mises statistic 459; Poisson, Polya, Power series, limiting distribution, 616; as t, Hotelling’s T 2, Triangular, a weighted quadratic statistic, Uniform, Weibull, Wishart. 611, 612 See also Exponential family; Cram´er-Wold device, 426 Monotone likelihood ratio; Credible region, 172, 173; highest Total positivity; Variation probability density, 173, 175, diminishing 202 Dominated convergence theorem, 32 Critical function, 58 Dominated family of distributions, Critical region, 56 45, 698, 699 Cross product ratio, see Odds ratio Domination: of one procedure Cumulative distribution function over another, 17. See also (cdf), 30, 52, 424; confidence Admissibility; Inadmissibility bands for, 255, 276; empirical, Double exponential distribution, 259, 245, 255; inverse of, 266. 323, 342; AUMP and LAUMP See also Kolmogorov test for property, 546, 547; locally goodness of fit; Probability most powerful test in, 342; integral transformation q.m.d. property, 487; UMP conditional test in, 402 d-admissibility, 233, 264. See also Duncan multiple comparison Admissibility procedure, 368 Data Snooping, 378 Dunnett’s multiple comparison Decision problem: specification of, 4 method, 390 Decision space, 4, 5 Dvoretzky, Kiefer, Wolfowitz Decision theory, 27, 28; and inference, inequality, 442 6 Deficiency, 157 Delta method, 436–439 EDF, see Empirical distribution Density point, 185 function Dependence: measure of, 129; Edgeworth expansions, 459–462, 481, mo;dels for, 448–451 positive, 662 145; positive quadrant, 145; Efficacy, 536 regression, 191, 240. See Efficient likelihood estimation, 504 also Correlation coefficient; Elliptically symmetric distribution, Independence 314 772 Subject Index

Empirical cumulative distribution 68; sufficient statistics for, function, 245, 255, 441; 27; testing against gamma statistics, 589 distribution, 200; testing Empirical likelihood, 673, 690, 691 against normal or uniform Empirical measure, 475, 589 distribution, 260; tests for Empirical process, 585, 588, 658 parameters of, 93, 195; two- Envelope power function, 262, 337. sample problem for, 259. See See also Most stringent test also Chi-squared distribution; Equi-tailed confidence interval, 649 Gamma distribution; Life Equivalence: of family of distributions testing or measures, 45; of statistics, Exponential family, 46, 55; 26; of two measures, 51 admissibility of tests in, Equivalence classes, 69 234; completeness of, 117; Equivalence hypotheses 81, 90–92; differentiability of, 49; LAUMP tests for, 559–564 equivalent forms of, 123; Equivalence relation, 692 expansion of loglikelihood, Equivariance, 13, 396. See also 483, 484; median unbiased Invariance estimators in, 162; moments Equivariant confidence bands, 255, of sufficient statistics, 55; 376, 384, 390 monotone likelihood ratio of, Equivariant confidence bounds, 272 67; natural parameter space of, Equivariant confidence sets, 248, 48, 55, 119; q.m.d. property, 251, 252, 272, 273, 276; and 488; regression models for, pivotal quantities, 274. See 210; testing in multiparameter, also Uniformly most accurate 119, 121, 123, 234; total confidence sets positivity of, 104. See also One-parameter exponential Error control: strong, 350; weak, 350 family Error of first and second kind, 57, 66; of type 3, 139; familywise error Exponential waiting times, 22, 54, rate, 349; directional, 373 74. See also Exponential distribution Essentially complete class of decision procedures, 17, 54, 69, 96. See Extreme order statistic, 678, 679 also Completeness of a class of decision procedures Factorization criterion for sufficient Estimation, see Confidence bands; statistics, 19, 45, 46 Confidence bounds; Confidence False discovery rate, 354, 386 intervals; Confidence sets; Family of hypotheses, 349, 374 Equivariance; Maximum Familywise error rate (FWER), 349, likelihood; Median: Point 354, 355, 372, 386; control estimation; Unbiasedness based on bootstrap, 658–661 Euclidean sample space, 41 Fatou’s Lemma, 32 Exchangeable, 355 F -distribution, 158; for simultaneous Expectation (of a random available), confidence intervals 381; in 33, 39; conditional, 37, 39, 42 Hotelling’s T 2-test, 306; in Expected order statistics, 243 tests and confidence intervals Experimental design, see Design of for ratio of variances, 166, 299; experiments noncentral, 307; relation to Exponential distribution, 22, 68, beta distribution, 159. See also 74; confidence bounds and F -test for linear hypothesis; intervals in, 74; order statistics F -test for ratio of variances from, 54; relation to Pareto Fiducial, 108; distribution 175; distribution, 94; relation probability, 108, 175 to Poisson process, 54, Fieller’s problem, 197 Subject Index 773

Finite decision problem, 54 Ghosh-Pratt identity, 200 First-order accurate, 666 Glivenko-Cantelli theorem, 441 Fisher’s exact test, 127, 149. See also Goodness of fit test, vii, 256 583; Two by two tables bootstrap tests of, 673; Fisher Information, 485, 486 in multinomial models, Fisher’s least significant difference 514–516; See also Chi-squared method, 368 tests; Kolmogorov-Smirnov; Fisher linkage model, 598 Neyman’s smooth tests; Fisher’s z-transformation, 439 Separate families; Weighted Fixed effects model, 297. See also quadratic tests Linear model; Model I and II Group: amenable, 334; free, 25; Free Group, 25 generated by subgroups, Frequentist point of view, 175 217; linear, 216, 227, 334; of Friedman’s rank test, 290 monotone transformations, F -test for linear hypothesis, 280; 215; orthogonal, 215, 217, admissibility of, 281; as 330;; permutation, 215; scale, Bayes test, 309; for nested 215; transformation, 212, classification, 302; has best 213; transitive, 215, 220; average power, 308; in Fisher’s translation, 215, 219, 333. See least significant difference also Equivariance; Invariance method, 368; in Gabriel’s Group, 692, 693; family, 395, 401 simultaneous test procedure, Guaranteed power: achieved through 368; in mixed models, 426; in sequential procedure, 124, 126, model II analysis of variance, 198, 199 299; power of, 281; robustness of, 445, 446, 448, 480, 491 See Haar measure, 227, 331 also F -distribution Hazard ordering, 101 F -test for ratio of variances, 106, 107, Hellinger distance, 530–534, 582 220, 238; admissibility of, 239; Hierarchical classification. see Nested nonrobustness of, 446. See classification also F -distribution; Normal Higher order asymptotics, 661–668 two-sample problem, ratio of Highest probability density (HPD) variances credible region, 173, 175, 202 F -test in multiple comparison Hilbert space, 696–698 procedures, 366 Hodges-Lehmann efficiency, 539 Fubini’s theorem, 34 Hodges’ superefficient estimator, 525 Fully informative statistics, 96 Holm procedure for multiple testing, Functionals, 571 350, 351, 363, 385 Fundamental lemma, see Neyman- Homogeneity of means: tests of, 285; Pearson fundamental against ordered alternatives, lemma 287; multiple comparisons for, 364, 366; for normal means, Gabriel’s simultaneous test procedure, 285, 287; nonparametric, 286, 368 290, 458. See also Multiple Gamma distribution Γ(g, b), 99, 196; comparisons relation to Beta distribution, Homomorphism, 12 196; scale parameter of, 201; Hotelling’s T 2-test, 306; admissibility shape parameter of, 196. of, 317; as Bayes solution, 317; See also Beta distribution; minimaxity of, 335 Chi-squared distribution; HPD region. see Highest probability Exponential distribution density Gaussian curvature, 341 Huber condition 455 Generalized linear models, 318 Hunt-Stein theorem, 331 774 Subject Index

Hypergeometric distribution, 66, and symmetry, 212; history 134; in testing equality of two of, 276; of likelihood ratio, binomials, 127; in testing for 262; of measure, 227; of power independence in a 2 × 2 table, functions, 227–229; of tests, 131; relation to distribution 214, 276; principle of, 214; of runs, 146. See also Fisher’s relation to equivariance, 13; exact test; Two by two tables relation to minimax principle, Hypergeometric function, 209 25, 329; relation to sufficiency, Hypothesis testing, 5, 56; history of, 220; relation to unbiasedness, 107; loss functions for, 59, 69, 23, 229, 230; warning against 222; without stochastic basis, inappropriate use of, 286. 131, 132 See also Almost invariance; Equivariance Improper prior distribution, 172 Invariant measure, 227, 230; over Inadmissibility, 17; of confidence orthogonal group, 330; over sets for vector means, 335; of translation group, 333 likelihood ratio test, 263; of Inverse Gaussian distribution, 100, UMP invariant test, 306. See 197 also Admissibility Inverse sampling: for binomial trials, Independence: conditional, 133; of 67; for Poisson variables, 68, sample mean from function 98. See also Negative binomial of differences in normal distribution; Poisson process; samples, 152; of statistic from Waiting times a complete sufficient statistic, 152; of sum and ratio of Jackknife, 648, 674 independent χ2 variables, 153; Joint confidence rectangles 657. See of two random variables, 34 also Simultaneous confidence Independence, test for: in bivariate sets normal distribution, 191; in nonparametric models, 241, Kendall’s statistic, 272 271; in r × c contingency k-FWER, 374, 386 tables, 127; in two by two Kolmogorov-Smirnov: and bootstrap tables, 127–130 confidence bands, 658; Indicator function of a set, 33 asymptotic behavior of, 441, Indifference zone, 320 442, 584–589; based on a pivot, Inference, statistical. see Statistical 645; extensions of, 589–590; inference statistic, 256; test for goodness Information matrix, 485, 486 of fit, 256. See also Goodness Integrable function, 31 of fit Integration, 31 Kolmogorov-Smirnov distance, 441 Interaction, 291, 292, 311; as main Kruskal-Wallis test, 286 effects, 311; in random effects Kullback-Leibler information (or and mixed models, 313, 314; divergence) 432; backward, 672 test for absence of, 291 Kurtosis, 459 Interval estimation, see Confidence intervals Large-sample theory, vii, 417 Into, see Transformation Latin squares design, 293, 312 Intraclass correlation coefficient, 313 Lattice distribution, 459 Invariance: of decision procedure, Laws of large numbers: Weak, 431; 12, 13; of likelihood ratio, Strong, 441; Uniform, 463, 464 341; of measure, 299, 518, Least favorable distribution, 18, 84, 519; and admissibility, 26; 85, 86, 321, 361 and ancillarity, 395, 397, 401; Least squares estimates, 281 Subject Index 775

Lebesgue convergence theorems, 39 models, 572; in univariate Lebesgue integral, 31 models, 544–549 Lebesgue measure, 29 Locally most powerful rank test, 244, Legendre polynomials, 599, 600 275 Locally optimal tests, 322, 339, 340, Level of significance, see Significance 403, 511 level Locally unbiased, 340 L´evy distance, 430 Local power 433; of t-test 465, 466 Life testing, 54. See also Exponential Location families (or models), 70, distribution; Poisson process 100, 396; are stochastically Likelihood, 16; function, 503, 504 See increasing 70; comparing two, also Maximum likelihood 219; conditional inference for, Likelihood ratio, 15, 101, 494; 414; condition for monotone censored, 326; invariance likelihood ratio, 323, 401; of, 262; large-sample theory example lacking monotone of, 494, 503; monotone, 65; likelihood ratio, 71; LAUMP preference order based on, 60, tests for, 546–548; strongly 66; sufficiency of, 53. See also unimodal, 401 Monotone likelihood ratio Location-scale families, 12; confidence Likelihood ratio test, 16; example of intervals based on pivot, inadmissible, 263; large-sample 645; comparing two, 258; theory of, 513–517; using LAUMP tests in, 557. See also bootstrap critical values, 670, Normality, testing for 671 Log convexity, 323, 412 Lindley’s Paradox, 95 Logistic distribution, 134, 323, 402 Linear functionals 571; LAUMP Logistic response model, 134 property, 572–574 Loglikelihood ratio, 483; expansion Linear hypothesis, 277, 333; due to Le Cam, 489–491 admissibility of test for 281; Loglinear model, 134, 318 Bayes test for, 309; canonical Lognormal family, 488 form for, 278, 317; F -test Loss function, 3, 7; in confidence for, 200; inhomogeneous, interval estimation, 23, 72, 76; 283; more efficient tests for, in hypothesis testing, 69, 141, 287; parametric form of, 284, 222; monotone, 76 p 309; power of test for, 280; L -space, 697, 698 properties of test for, 280, 308, 333, 338, 341; reduction Main effects, 287, 292; as interactions, of, through invariance, 279; 311; confidence sets for, 289; robustness of tests for, tests for, 287, 291. 451–458. See also Analysis Mallow’s metric, 654 of variance; Additive linear Mantel-Haenszel test, 135 model, Generalized linear Markov chain, 145 model Markov property, 145 Linear model, 277, 318; confidence Markov’s inequality, 472 intervals in, 309; history of, Matched pairs, 138, 183, 221, 239, 317; simultaneous confidence 324; comparison with complete intervals in, 380 randomization, 149; confidence Locally asymptotically uniformly intervals for, 189; rank tests most powerful (LAUMP): for for, 242, 246 equivalence hypotheses, 559– Maximal invariant, 214; ancillarity 564; for one-sided hypotheses of, 395; distribution of, 218; in multiparameter models, method for determining, 216; 553–559; in nonparametric obtained in steps, 217 776 Subject Index

Maximin multiple tests, 354, 357, Model selection, 11 358, 360 Monotone class of sets, 50 Maximin test, 320; by Hunt-Stein Monotone convergence theorem, 32 theorem 333; existence of, Monotone decision rule, 355, 357, 387 338; local, 322; relation to Monotone likelihood ratio, 65, invariance, 329. See also 69, 101, 104; mixtures of Least favorable distribution; distributions with, 341, Minimax principle; Most 401, 403; necessary and stringent test sufficient condition for, 98; of Maximum likelihood, 16, 17, 504–508; differences, 402; of distribution in normal model, 504, 505; in of correlation coefficient, 261; exponential family models, of exponential family, 67; of 505. See also Likelihood ratio location families, 323, 401, test 402; of noncentreal χ2 and F , Maximum modulus confidence 307; of noncentral t, 224; of intervals, 379 scale families, 324; relation to McNemar’s test, 138, 149 total positivity, 103; tests and Measurable: function, 30; set, 29; confidence procedures in the space, 29; transformation, 30, presence of, 65, 69, 73. See also 34 Stochastic increasing Measure, 29 Monotone loss function, 76 Median, confidence bounds for, 105 Monte Carlo simulation 442, 443; Median unbiasedness, 22; relation to for bootstrap, 649; for confidence bounds, 162 subsampling, 679 Meta-analysis, 109 Mortality. see Hazard ordering Metric space, 527, 571, 694. See Most stringent test, 276, 337; also Hellinger; Kolmogorov- existence of, 346 Smirnov; Kullback-Leibler; Moving average process, 450 L´evy; Mallows; Prohorov, Moving blocks bootstrap, 687 Total variation Multinomial distribution, 47, 202; as Minimal complete class of decision conditional distribution, 54; procedures, 17. See also Dirichlet prior for, 202; for Completeness of a class of entries of 2 × 2 table, 128 distributions; Essentially Multinomial model: maximum complete class of decision likelihood estimation in, 514, procedures 515; testing a composite Minimal sufficient statistic, 21 hypothesis in, 597, 598; Minimum Chi-squared estimator, 597 testing a simple hypothesis Minimax principle, 15, 347; and in, 514–516, 590–597; for least favorable distribution, 2 × 2 table, 128, 130; for 18; in confidence estimation, three-factor contingency table, 335; relation to invariance, 133. See also Chi-squared test; 25; relation to unbiasedness, Contingency tables 24. See also Maximin test; Multiple comparison procedures, Restricted Bayes solution iii, 293, 343; complexity Minkowski’s inequality, 697 of, 373; history of, 391; Missing observations, 410 interpretability of, 372; Mixed model, 297, 304, 314, 315 significance levels for, 368, Mixtures of experiments, 392, 394, 370, 371. See also Duncan and 395, 410, 414 Dunnett multiple comparison MLR, see Monotone likelihood ratio methods; Newman-Keuls Model I and II, 297. See also Mixed multiple comparison model; Random effects model procedure; Simultaneous Subject Index 777

confidence intervals; t-distribution, 156, 161, 193, Stepdown procedures; Stepup 224 procedures; Tukey levels; Noninformative prior, 172 Tukey’s T -method Nonparametric: independence Multiple decision procedures, problem, 191, 240, 242; 5. See also Multiple many-sample problem, 286; comparisons; Multiple testing; methods for linear hypotheses, Three-decision problems 290; one-sample problem, 118; Multiple testing, iii, 293, 348; history test in two-way layout, 290. of, 391, maximin procedures, See also Permutation test; 354 Rank tests; Sign test Multiplicity problem, 349 Nonparametric mean 420, 459; and Multivariate cumulative distribution the Bahadur-Savage result; function, 424 466–468; and the bootstrap, Multivariate linear hypothesis, 306, 653, 655; and Edgeworth 318 See also Linear hypothesis expansions, 459–462; and the Multivariate mean: nonparametric t-test, 462–466; asymptotic confidence regions based on maximin and LAUMP bootstrap, 655, 656; multiple property, 567–574; confidence testing for, 661 intervals for based on a root, Multivariate normal distribution, 646, 647; resampling-based 89, 304, 426; testing linear tests for 672, 673. See also combination of means 90, tests Multivariate mean for, 345, 513, 514. See also Nonparametric two-sample problem, Bivariate normal distribution 130, 176, 242; confidence Multivariate normal one-sample intervals in, 188, 203, 268; problem, the mean: confidence omnibus alternatives, 245; intervals for, 415; tests universally unbiased test in, for, 305, 335, 353. See 269. See also Normal scores 2 also Hotelling’s T -test; test; Wilcoxon test Simultaneous confidence sets Nonparametric test, 85 Multivariate t-distribution, 275 Nonparametric variance, LAUMP property, 574 Natural parameter space of an Normal approximation, order of error, exponential family, 48, 55, 119 663, 664 Negative binomial distribution 22, 68, Normal distribution N(ξ,σ2), 5, 86; 144 loglikelihood for, 483; testing Neighborhood model, 326, 328 against Cauchy or double Nested classification, 301, 313 exponential, 259; testing Nested rejection regions, 63, 96, 105 against uniform or exponential, Newman-Keuls multiple comparison 260. See also Bivariate normal procedure, 368, 370 distribution; Multivariate Newton’s identities, 39 normal distribution Neyman-Pearson fundamental lemma, Normality, testing for, 260, 589. See 60, 108; approximate version also Normal distribution of, 326; generalized, 77, 108 Normal many-sample problem: Neyman-Pearson statistic, 503 confidence sets for vector Neyman’s smooth tests, 599–601; means, 252, 336, 366, 375, 378; large sample behavior 601–607 tests for means, 285, 399. See Neyman structure, 115, 118 also Homogeneity of means, Noncentral: beta distribution, 280, tests of 307; χ2-distribution, 306, Normal one-sample problem, the 311; F -distribution, 307; coefficient of variation: 778 Subject Index

confidence intervals for, 273; confidence intervals for, 166, test for, 157, 224, 294, 303 254, 272; credible region for, Normal one-sample problem, the 202; nonrobustness of test for, mean: admissibility of test 446; test for, 107, 157, 259. for, 235; AUMP test for, 555, See also F -test for ratio of 556; confidence intervals for, variances; Ratio of variances 163, 250, 405; credible region Nuisance parameters, 318, 402 for, 172m 174; Edgeworth Null set, 40 expansion for t-statistic, 517; LAUMP test of equivalence Odds ratio, 126, 399; most accurate with unknown variance, 563, unbiased confidence intervals 564; likelihood ratio test for, for, 200. See also Binomial 87; median unbiased estimate probabilities; Contingency of, 164; nonexistence of test table; Two by two tables with controlled power, 157; One parameter exponential family, nonexistence of UMP test 67, 81, 111; complete class for, for, 89; optimum test for, 92, 141; most stringent test in, 155, 156, 260, 283, 401; test 338. for, based on random sample One-sided hypotheses, 65, 124 size, 95; two-stage confidence One-way layout, 285, 353; Bayesian intervals for, of fixed length, inference for, 304; model II 198, 199; two-stage test for, 297; nonparametric, 286. for, with controlled power, See also Homogeneity, tests of; 199; two-sided test for, 260; Normal many-sample problem sequential confidence intervals Onto, see Transformation for, 163, 199. See also Matched Optimality,9,10 pairs; t-test Orbit of transformation group, 214 Normal one-sample problem, the Ordered alternatives, 287 variance: admissibility of test Order notation OP (1), oP (1), 433; for, 238; conditional confidence an  bn, 498; an ∼ bn, 535 intervals for, 415; confidence Order statistics, 37, 38; as intervals for, 165, 201; credible maximal invariants, 215; region for, 174; likelihood ratio as sufficient statistics, 53, test for, 87; optimum test for, 176; completeness of, 118, 87, 92, 154, 220, 325 141; distribution of, 266; Normal response model, 134 equivalent to sums of powers, Normal scores statistic, 269 38; expected values of, 243; in Normal scores test, 243; optimality permutation tests, 176 of, 243, 244 Orthogonal group, 215, 217, 330 Normal subgroup, 257 Orthogonal: transformations, 194, Normal two-sample problem, 215; vector, 697 difference of means: Orthonormal: system, 697; vector, comparison with matched 697 pairs, 204; confidence intervals for, 165; credible region for, Paired comparisons, see Matched 202; optimal tests for for pairs (with variances equal), 107, Pairwise sufficiency, 53 160, 195, 225, 260, 284. See Parameter space, 3 also Behrens-Fisher problem; Parameters, unrelated, see Variation Homogeneity of means, tests independent parameters of; t-distribution; t-test Parametric bootstrap, 651–653; in Normal two-sample problem, ratio of Behrens-Fisher problem, 671, variances, 107, 157, 220, 238; 672 Subject Index 779

Pareto distribution, 94, 196 one-sided test for, 68, 98; Parseval’s identity, 697, 698 one-sided test for sum of, 105 Partial ancillarity, 398, 399 Poisson process, 4, 68, 98; and Partial sufficiency, 106 2 × 2 tables, 130; confidence Pearson’s Chi-squared test. see bounds for scale parameter, Chi-squared test 74; distribution of waiting Percentile method, 685 times in, 22; test for scale Permutation group, 215 parameter in, 68, 98. See also Permutation test, 130, 177, Exponential distribution 187; approximated by Poly´a’s theorem, 429 standard t-test, 180, 447; Poly´a frequency function, 323 as randomization test, 242, Population models, 132 635, 641–643; complete class, Portmanteau theorem, 425 186; computational methods Positive dependence, see Dependence, for, 180; confidence intervals positive based on, 189, 203, 206; for Positive part of a function, 31 testing independence, 192; Posterior distribution, 172; percentiles history of, 210, 690; most of, 175. See also Bayesian powerful, 178; robustness of, inference 447, 638–643; most stringent, Posterior probability, 94 346. See also Nonparametric; Power function, 57; of invariant test, Randomization model 228; of one-sided test, 68; of Pillai-Bartlett trace test, 463; two-sided test, 82 robustness of, 465 Power of a test, 57, 98; conditional, Pitman asymptotic relative efficiency. 124, 399; unbiased estimation see Asymptotic relative of, 123 efficiency Power series distribution, 142 Pivotal: method, 644–646, quantity, Preference ordering of decision 253, 274 procedures, 10, 14 Plug-in estimate, 648 Prepivoting, 657, 668 Point estimation, viii, 5, 7; Prior distribution, 14, 172; improper, equivariant, 13; history of, 27; 172; noninformative, 172. unbiased, 14 See also Bayesian inference; Pointwise asymptotically level α: for Least favorable distribution; confidence sets, 423; for tests, Posterior distribution 422 Probability density (with respect to Pointwise consistent in power, 423 µ), 33; convergence theorem Poisson distribution, 4, 6, 54; for, 696 comparison of two, 125, Probability distribution of a 398; relation to exponential random variable, 30. See distribution, 27, 68, 98; also Cumulative distribution square root transformation function (cdf) for, 474; sufficient statistics Probability integral transformation, for, 19; sums of, 54. See also 97, 266 Exponential distribution; Probability measure, 39, 30 Poisson parameters; Poisson Product measure, 34 process Prohorov’s theorem, 440 Poisson model: for 2 × 2 table, 130, Projection, as maximal invariant, 132; for 2 × 2 × K table, 133, 216, 284 148 Pseudometric space, 694 Poisson parameters: comparing two, P-value, 57, 63, 97, 98, 108; 125, 398; confidence intervals combination of, from for the ratio of two, 168; independent experiments, 97, 780 Subject Index

109; for randomization test, alternative, 265, 266; null 636; for randomized tests, 64; distribution of, 242. See also in multiple testing, 350, 364; Signed ranks in stepdown procedures, 360; Rank-sum test, 147. See also properties of, 64, 139; versus Wilcoxon test fixed levels, 65 Rank tests, 241; as special case of permutation tests, Quadrant dependence, 145, 210, 371, 635, 636; in multivariate 372. See also Dependence, problems, 318; surveys of, positive 286. See also Nonparametric; Quadratic mean derivative, 484 Nonparametric two-sample Quadratic mean differentiable problem; Symmetry; Trend (q.m.d.) families, 484; Ratio of variances: confidence examples of: 486, 488; intervals for, 166, 254, 272, loglikelihood expansion for, 299, 558; in model II, 299; 489; properties of, 485–487 tests for, 157, 220, 259, 298, Quadrinomial distribution, 133 412. See also F -test for ratio of Quality control, 85, 223 variances; Homogeneity, tests Quantiles, 430, 649 of; Random effects model Recognizable subsets, see Relevant Rao’s score tests. see Score tests subsets Radon-Nikodym derivative, 33, 51 Rectangular distribution, see Uniform Radon-Nikodym theorem, 33 distribution Random assignment, 131, 182, 247, Regression, 169, 318, 395; as linear 293 model, 278, 293; comparing Random effects model, 297; for several lines, 295, 312; nested classifications, 301, 313; confidence band for, 384, for one-way layout, 297; for 391; confidence intervals two-way layout, 313. See also for coefficients, 223, 295; Ratio of variances intercepts and ordinates Randomization, 8, 293; as basis for of line, 170; polynomial, inference, 182; possibility of 278; robustness of tests for, dispensing with, 95; relation to 451–458; tests for coefficients, permutation test, 184; tests, 169, 293; with both variables 632–643. See also Random subject to error, 312. See also assignment; Randomized Trend procedure Regression dependence, 191, 240. See Randomization distribution, 637 also Dependence, positive Randomization hypothesis, 633 Regular (estimator sequence), 508, Randomization models, 132, 187; 526 confidence intervals in, 188; Relative efficiency, 539. See also history of, 210 Asymptotic relative efficiency Randomized procedure, 8; confidence Relevant and semirelevant subsets, intervals, 167; in conditioning, 175, 405, 406, 413; history of, 414 414, 415; randomized version Randomized test, 58; representation of, 414; relation to Bayesian as nonrandomized test, 74 inference, 415 Randomness, hypothesis of, 270 Restricted Bayes solution, 15 Random sample size, 95, 142, 210 Riemann integral, 31 Random variable, 30 Riskfunction,4,9,10 Rank correlation coefficient, 272 Robustness, 11, 347; against Ranks, 216; as maximal invariants, dependence, 448–451, 680; 216, 241; distribution under against F -test of means, 445, Subject Index 781

446, 448, 480; of efficiency, 421; of, 536 AUMP and LAUMP of general linear models tests, property, 545; counterexample 451–458 ; of validity, 421; lack to AUMP property, 547 of, for F -test of variances, 446; Score vector (or function), 489, 511 lack of, for Chi-squared test Second-order accurate, 666 of a normal variance, 445; of Selection procedures, 102 test of independence or lack of Separable: family of distributions, correlation, 476; for tests in 698; space, 694 two-way layout, 455; of t-test, Separate families of hypotheses, 220, 444, 445. See also Adaptive 258 test; Behrens-Fisher problem; Sequential procedures, 8, 9, 145, 157, Permutation test; Rank tests 163 Root, 644 Shift, confidence intervals for: based Runs test, for testing independence on permutation tests, 203; in a Markov chain, 145, 146 based on rank tests, 251, 268. See also Behrens-Fisher Sample, 5; haphazard, 181; stratified, problem; Exponential 176, 182, 188 distribution; Nonparametric Sample correlation coefficient, 190, two-sample problem; Normal 207; distribution of, 209; two-sample problem, difference limiting distribution of, 438; of means monotone likelihood ratio of Shift model, 134, 250, 578, 579 distribution, 261; variance σ-field, 29; with countable generators, stabilizing transformation for, 699 438, 439. See also Bivariate σ-finite, 29 normal distribution; Rank Signed ranks, 242; distribution correlation coefficient under alternatives, 270; null Sample covariance matrix, 305, 316; distribution of, 246 distribution of, 208 Significance level, 57; for multiple Sample distribution function, comparisons, 368, 370; for see Empirical cumulative stepdown procedures, 351, 361; distribution function nominal, 387. See also P-value Sample inspection: by attributes, 66, Significance probability, see P-value 223; by variables, 85, 223; for Sign test, 85; asymptotic relative comparing two products, 135, efficiency of, 537, 538; for 225 matched pairs, 138; for testing Sample size, 8; required to achieve consumer preferences, 135; for specified power, 57. 125. 199. testing symmetry with respect 320 to a given point, 137; history Sample median, 429 of, 149; in double exponential Sample space, 30 distribution, 342; limiting Sample standard deviation, 434 behavior, 501, 502; treatment S-ancillary, 398, 399 of ties in, 167, 186. See Scale families, 324; comparing also Binomial probabilities; two, 259, 412; conditional Median; Sample inspection inference for, 414; condition Similar test, 110, 115; relation to for monotone likelihood ratio, unbiased test, 111; history of, 323 149. Scheff´e’s S-method, 375, 380, 384, Simple: class of distributions, 59; 388; alternatives to, 384 hypothesis, 59 Score tests, 511–513; asymptotically Simple function, 31 maximin property, 566, 567; Simple hypothesis vs. simple asymptotical relative efficiency alternative, 60, 415; with 782 Subject Index

large samples, 503. See also Student’s t-test, see t-test Neyman-Pearson lemma Subfield, 34 Simpson’s paradox, 132 Sufficient statistic, 19, 44, 54, Simultaneous confidence intervals, 55; Bayes definition of, 21; 375, 391; bootstrap, 657; for factorization criterion for, 19, all contrasts, 382. See also 45; for exponential families, Confidence bands; Dunnett’s 47; in presence of nuisance multiple comparison method; parameters, 96; likelihood ratio Scheff´e’s S-method; Tukey’s as, 53; minimal, 21; pairwise, T -method 53; relation to ancillarity, 397; Simultaneous confidence sets for a relation to fully informative family of linear functions, 375, statistic, 96; relation to 381; smallest, 378; taut, 378 invariance, 220; statistics Simultaneous testing, 349. See also independent of, 151, 152. See Multiple comparisons also Partial sufficiency Single step procedure for multiple Subsampling, 673–676; comparisons testing, 351 with bootstrap, 677–680; for Singly truncated normal distribution hypothesis testing, 680, 681 (STN), 144 Superefficient estimator, 525; Skewness, 459, 662 bootstrap of, 679 Slutsky’s theorem, 433 Symmetric: confidence interval, 649 Small-sample theory, iii distribution, 53 Smirnov test, 245 Symmetry, 11, 13; and invariance, 12, Smooth function of means, 656 212; sufficient statistics for Spherically symmetric distributions, distributions with, 53; testing 194, 314 for, 241, 246, 270; testing, with Stagewise tests, 367 respect to given point, 137, Standard confidence bounds, 77, 175 246, 248, 270 Starshaped, 101 Stationarity, 145 Tautness, 378 Statistic, 30, 34; and random t-distribution, 156, 161, 286; variables, 31; equivalent approximation to permutation representations of, 36; fully distribution, 180; as informative, 96; subfield distribution of function of induced by, 34 sample correlation coefficient, Statistical inference, 3; and decision 207; as posterior distribution, theory, 6; history of, 27 174; Edgeworth expansion for, Stein’s two-stage procedure, 198 517; in two-stage sampling, Stepdown procedures, 351, 352, 391; 198; monotone likelihood ratio canonical form for, 360; large of, 224; multivariate, 275; sample bootstrap, 658–661 noncentral, 156, 161, 193, 224 Stepup procedures, 351, 356 Test (of a hypothesis), 5, 56; Stochastically increasing, 70, 135 almost invariant, 225, 241; Stochastically larger, 70, 101, 240, conditional, 394, 400, 403; 354 invariant, 214, 276; locally Stratified sampling, 176, 182, 188 maximin, 322; locally most Strictly unbiased, 112 powerful 339; maximin, 322; Strongly unimodal, 323, 401, 412, most stringent, 337; of type D 546, 547 and E, 340, 341; randomized, Studentization, 286, 445 58, 127; strictly unbiased, 112; Studentized range, 367, 390 unbiased, 110; uniformly most Student’s t-distribution, see powerful (UMP), 58 t-distribution Three-decision problems, 81, 124 Subject Index 783

Three factor contingency table, 132 Two by two by two table, 135 Ties, 136 Two-sample problem, see Behrens- Tight sequence, 439 Fisher problem; Binomial Time series models, 450, 451 probabilities; Exponential Total positivity, 71, 103, 115, 308, 323 distribution; Matched pairs; Total variation distance, 529 Nonparametric two-sample Transformation: into, 30; of integrals, problem; Normal two-sample 34; onto, 30; probability problem; Permutation test; integral, 97; variance Poisson parameters; Shift, stabilizing, 439 confidence intervals for; Transformation group, 12, 212, 213. Two-by-two tables See also Invariance; Group Two-sided alternatives, 81 Transitive: binary relation, 569; Two-way contingency tables, see transformation group, 285 Contingency tables; Two by Trend: test for absence of, 271 two tables Triangular distribution, 259 Two-way layout, 287, 290, 304; mixed Trimmed mean, 647, 648 models for, 314, 315; multiple t-test: admissibility of, 235, 237, testing in, 374; rank tests 281; as Bayes solution, 237; for, 290; reorganization of as likelihood ratio test, 25, variables in, 311; robustness in, 87; comparison with Wilcoxon 455; simultaneous confidence and sign tests, 537, 538; for intervals in, 383; with one matched pairs, 183, 204; for observation per cell, 287; regression coefficients, 169, with m observations per cell, 294; in linear hypothesis with 290. See also Contingency one constraint, 281; local tables; Interactions; Nested power of, 465, 466; one-sample, classifications; Two by two 89, 156, 192, 260; optimality tables in nonparametric model, 567–574, permutation version UMP invariant test, 150, 218, 219; of, 180, 635, 638, 639; power admissibility, 232; conditional, of, 156, 192, 193; relevant 404; conditions to be UMP subsets for, 408; robustness almost invariant, 227; example of, 445, 446; two-sample, 161, of inadmissibility, 232; 176; two-stage, 199; under examples of nonuniqueness, local alternatives, 501; uniform 231, 232; relation with UMP asymptotic behavior, 465, 466. unbiased test, 230; trivial, 232. See also Normal one- and See also Invariance; Linear two-sample problem hypothesis Tukey levels for multiple comparisons, Uniformly most powerful (UMP) 368, 387 test, 58, 108; conditional, 394, Tukey’s T -method, 367, 374, 388, 401, 403; examples involving 389, 390 two parameters, 93, 95; for Two by two by K tables, 138, 148 exponential distributions, 93; Two by two tables: alternative models for monotone likelihood ratio for, 128, 130, 132; comparison families, 65; for one-parameter of experiments for, 130; exponential families; for Fisher’s exact test for, 127, uniform distribution, 92, 149; for matched pairs, 138, 99; in inverse Gaussian 149; McNemar’s test for, 138, distribution, 100; in normal 149; multinomial model for, one-sample problem, 87, 88; 128, 130; S-ancillaries for, 399. in Weibell distribution, 99; See also Contingency tables nonparametric example of, 85 784 Subject Index

UMP unbiased test, 111; admissibility intervals; Simultaneous of, 139; example of confidence intervals and sets nonexistence of, 140; for Unimodal, 412. See also Strongly multiparameter exponential unimodel families, 119, 121, 150; for Unrelated parameters, 398 one-parameter exponential U-statistic, 678 families, 111; for strictly totally positive families, 115; Variance components, see relation to UMP almost Components of variance invariant tests, 230; via Variance stabilizing transformation, invariance, 150, 230. See also 439 Unbiasedness Variation diminishing, 71. See also Unbiasedness, 13, 27, 110; and Total positivity admissibility, 26; and Variation independent parameters, invariance, 23, 229, 230; and 398 minimax, 24; and similarity, Vector space, 696–698 111; for confidence intervals, Vitali’s theorem, 32 23, 131; for point estimation, 14, 22, 27; for two-decision Waiting times, 22, 98 procedures, 13; of tests, 110, Wald tests and confidence regions, strict, 112. See also UMP 508–510, 646; efficiency of, 536; unbiased test; Uniformly most AUMP and LAUMP property, accurate confidence sets 548, 549 Undetermined multipliers, 80 Weak compactness theorem, 700, 701 Weak convergence, 425, 694 Uniform confidence bands, 442 Weak conditionality principle, 400 Uniform distribution U(a, b), 9, 22; Weibull distribution, 99 as distribution of probability Weighted quadratic test statistics, integral transformation, 97; 607, 608; examples of, 611, completeness of, 116, 141; 612; local power calculations, discrete, 142; distribution of 614, 615 order statistics from, 267; not Welch approximate t-test, 231, 447, q.m.d., 488, 533; of p-values, 448 64, 65; one-sample problem Welch-Aspin test, 231, 408 for, 92, 99, 413; relation to Wilcoxon one-sample test, 246 exponential distribution, 93; Wilcoxon signed-rank statistic, 269, sufficient statistics for, 26; 493, 502, 503 testing against exponential or Wilcoxon signed-rank test. see triangular distribution, 260; Wilcoxon one-sample test other tests for, 480, 482 Wilcoxon statistic, 268, 269; Uniformly asymptotically level α: for expectation and variance of, confidence sets, 423, 424; for 265 tests, 422 Wilcoxon two-sample test, 243, Uniformly integrable, 472 245; alternative form of 265; comparison with T -test, 537. Uniformly most accurate confidence 538; confidence intervals sets, 72, 73; equivariant, 249; based on, 251; history of, 276; minimize expected Lebesgue optimality of, 243, 244, 267, measure, 251; relation to 268 UMP tests, 73; unbiased, 164. Wilson confidence interval for See also Confidence bands; binomial, 435, 647 Confidence bounds; Confidence intervals; Confidence sets; Simultaneous confidence Yule’s measure of association, 129 Springer Texts in Statistics (continued from page ii)

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