Distribution of Record Values

Recent Developments in Record Values

M. Ahsanullah Rider University Lawrenceville, NJ,USA

In this talk the record values of continuous and discrete random variables will be discussed.

1. 2

Consider the following sequence of observations

6,10,5,6,11, 11,14,4,18,4, 1, …

6,10,11,14,18, … -upper record values 6,5,4,1, … -lower record values 6,10,11,11,14.18, - upper weak records 6,5,4,4.1, … - lower weak records

Suppose we consider 5 more observations. 6,10,5,6,11,11,14,4,18,4,1, ,8,0,21,21,0,…

6,10,11,14,18, 21,.. upper record values 6,5,4,1, 0,.. lower record values 6,10,11,11,14.18,21,21, upper weak records 6,5,4,4.1, 0,0,.. - lower weak records Thus the past records are in the data.

1.0. INTRODUCTION AND EXAMPLES OF RECORD VALUES In 1952 Chandler defined so called Record Times and Record Values and a groundwork for a mathematical Theory of Records. For six decades beginning his pioneering work there appeared about 500 papers and some monographs devoted to different aspects of the theory of records. Records are very popular because they arise naturally in many fields of studies such as climatology, sports, medicine, traffic, industry and so on. The Records as such are memories of their time.

Suppose we consider the weighing of objects on a scale missing its spring. An object is placed on the scale and its weight is measured. The 'needle' indicated the correct weight but does not return to zero when the object is removed. If various objects are placed on the scale, only the weights greater than the previous ones can be recorded. These recorded weights are the upper record value sequence. Let us consider a sequence of products that may fail under stress. We are interested to determine the minimum failure stress of the products sequentially. We test the first product until it fails with stress

less than X1 then we record its failure stress, otherwise we consider the next product. In general we will record stress Xm of th the m product if Xm < min (X1,..., Xm-1), m >1. The recorded failure stresses are the lower record values. One can go from lower records to upper records by replacing the original sequence of random variables {Xj} by {-Xj j >1} or if P(Xj > 0) = 1 by { 1/Xj, i > 1}, j=1,2,…

Chandler (1952) proved the interesting result that for any given distribution of the random variables the expected value of the inter record time is infinite. Feller

(1952) gave some examples of record values with respect to gambling problems.

1.1. DEFINITION OF RECORD VALUES AND RECORD TIMES

Suppose that X1, X2, .... is a sequence of independent and identically distributed random variables with cumulative distribution function F(x). Let Yn = max (min){X1,X2,...,Xn} for n > 1. We say Xj is an upper(lower) record value of {Xn, n >1}, if Yj > (<)Yj-1, j > 1. By definition X1 is an upper as well as a lower record value. One can transform the upper

records to lower records by replacing the original sequence of {Xj} by {-Xj , j > 1} or (if P(Xi > 0)=1 for all i) by {1/Xi, i >1}; the lower record values of this sequence will correspond to the upper record values of the original sequence. The indices at which the upper record values occur are given by the record times {U(n)}, n > 0, where U(n) = min{j|j>U(n-1),Xj >XU(n-1), n >1} and U(1) =1. The record times of the sequence {Xn n >1} are the same as those for the sequence {F(Xn), n > 1}.

Since F(X) has an uniform distribution, it follows that the distribution of U(n) , n > 1 does not depend on F. We will denote L(n) as the indices where the lower record values occur. By our assumption U(1) = L(1) = 1. The distribution of L(n) also does not depend on F.

The kth record values XU(n,k) are a natural extension of records XU(n) . It is interesting that distributions of the kth records can be expressed via distributions of the classical record values. Really, together with a sequence of independent random variables X1,X2,… having a common distribution function F, let us consider one more sequence Y1=min{X1,X2,…,Xk}, Y2= min{Xk+1,Xk+2,…,X2k }, ….

Let now XU(n,k) be the kth record value based on X’s and Y(n) be the classical records based on the sequence Y1, Y2,…. Then for any fixed k=2,3,… and any n=1,2,… the vector (XU(1,k) ,XU(2,k) ,…,XU(n.k) has the same distribution as the vector (YU(1) ,YU(2) ,…,YU(n).

1.2. THE EXACT DISTRIBUTION OF RECORD VALUES (CONTINUOUS RVS) Many properties of the record value sequence can be expressed in terms of R(x), where R(x) = - ln Fx(), 0 < F (x) < 1 and Fx() = 1 - F(x). Here 'ln' is used for the natural logarithm. If we define Fn(x) as the distribution function of XU(n) for n > 1, then we have

F1(x) = P[XU(1) < x] = F(x) (1.2.1)

F2(x) = P[XU(2) < x ]  x y i1 =    (F(u)) dF(u)dF(y) i1 x y dF(u)   dF(y)   1 F(u) x  R(y)dF(y) (1.2.2)

If F(x) has a density f(x), then the probability density function (pdf) of

XU(2) is f2(x) = R(x) f(x) (1.2.3)

It can similarly be shown that the pdf Fn(x) of XU(n) is

Fn(x) = P( XU(n) < x ) x un u2 f (un1) f (u1)  f (un )dun dun1.  du1.   1 F(un1)  1 F(u1) n1 x R (u)   dF(u),  x .  (n) (1.2.4) This can be expressed as n1 R(x) u F (x) =  eu du , n  (n)  x n1 (R(x)) j Fn (x)1Fn (x)F (x)  j0 ( j 1)

n1 j (R(x))  eR( x)  j0 ( j 1) 14

The pdf fn(x) of XU(n) is R n1 (x) fn(x) = f (x),  x . (n) (1.2.5)

The joint pdf f(x1,x2, ... , xn) of the n record values XU(1), XU(2), ..., XU(n)) is given by

f (x1, x2 ,....,xn ) r(x1 )r(x2 )....r(xn1 )f (xn ) for

 x1 x2...xn1xn , (1.2.6) d f (x) where r(x) = R(x) , dx 1 F(x)

0 < F(x) < 1.

The function r(x) is known as hazard rate. The joint pdf of XU(i) and XU(j) is f(xi,xj) (R(x ))i1 (R(x ) R(x )) ji1 = i r(x ) j i f (x ) (i) i ( j  i) j (1.2.7)

for  xi x j  . Thus E(Vk) = k and Var(Vk) = k. The conditional pdf of XU(j)| XU(i) = xi if( xj | XU(i) = xi ) = f (x , x ) ij i j fi (xi ) (R(x )  R(x )) ji1 f (x ) = j i j ( j  i) 1 F(xi )

(1.2.8)

for   xij x   . fik,im (x, y |XU(i) z) 1 1   [R(y)  R(x)]mk1[R(X ) R(z)]k1 (m  k) (k) f (y)r(x) . F(z)

for - < z < x < y < .

The marginal pdf of the nth lower record value can be derived by using the same procedure as that of the nth upper record value. Let H(u) = -ln F(u), 0< F(u) < 1 and d h(u) H (u), then du

P(XL(n) < x ) = n1 x {H (u)}  dF(u)  (n) (1.2.9) and the corresponding the pdf f(n) can be written as (H (x))n1 f(n)(x) = f (x). (n) (1.2.10) The joint pdf of XL(1), XL(2), ... , XL(m) can be written as f (1),(2),...,(m) (x1,x2 ,...,xm )h(x1 )h(x2 )...h(xm1 )f (xm )

 xm xm1  ...x1

= 0, otherwise. (1.2.11) The joint pdf of XL(r) and XL(s) is f(r),(s) (x,y) = (H (x))r 1[H (y)  H (x)]sr 1 h(x)f (y) (r) (s  r)

for s > r and - < y < x <  . (1.2.12)

Proceeding as in the case of upper record values, we can obtain the conditional pdfs of the lower record values.

Example 1.2.1. Let us consider the exponential distribution with pdf f(x) as f(x)=e-x ,0x and the cumulative distribution function (cdf) F(x) as F(x) = 1- e-x, 0 < x <∞. Then R(x) = x and n1 x x fn(x) = e , x 0 (n) =0, otherwise. The joint pdf of XU(m) and XU(n) , n> m is fm,n(x,y) xm1 = (y  x)nm1e y (m)(nm) 0 xy = 0, otherwise.

The conditional pdf of XU(n) | XU(m) = x) is f(y|XU(m)=x) (y  x)nm1 = e( yx) (n  m)

= 0, otherwise. Thus the conditional distribution of XU(n) - XU(m) given XU(m) is the same as the unconditional distribution of XU(n-m) for n > m.

Example 1.2.2. Suppose that the random variable X has the Gumbel distribution with pdf f(x) = x ex e e ,x  . Let F(n) and f(n) be

the cdf and pdf of XL(n). It is easy to see that nu x e eu F(n)(x) =  e du  (n) nx e ex and f(n)(x) = e , x . (n) Let f(m,n)(x,y) be the joint pdf of XL(m) and XL(n) , m< n. Using (1.2.16), we get for the Gumbel distribution

f m,n x, y =  y  x nm1 mx e  e e  y e y e e , n  m m    y  x   Thus the conditional pdf f(n|m)(y|x) of XL(n)| XL(m) = x is given by

f(n|m)(y|x)= y x nm1 (e  e ) y x eye(e e ). (n  m)  yx .

y x nm1 (e  e ) y (ey ex) e e ,yx  . (n  m) E(XL(m)XL(n)) =  

 y xy f(m),(n)(x, y)dxdy =  y x nm1 mx   (e  e ) e y   xy  eyee dxdy  y (n  m) (m) .

= 2 E(X L(n) )+E(T)E(XL(n)) where  E(T) = (n) ( (m)(n-m-1))-1  0 .(1-e- t)n-m-1 e-mt dt

Cov (XL(m)XL(n)) = Var(XL(n)). and

Var(XL(r)) d  2 r1 1 ()r =  =   2 , k > 1 dr 6 k 1 k  2 = for k = 1. 6  Let Var(XL(r)) = Vrr, , r = 1,2, ..., then  2 V *  1,1 6   2 V j, j V j1, j1 ( j 1) , j2

Further n  1 E(XL(m)) = E(XL(n)) +  1 pm p Var(XL(n-1)) - Var(XL(n)) = (n-1)-2

Example 1.2.3.

A random variable is said to have generalized Pareto distribution if its probability density function is of the following form: 1 x  1 f (x, , ,  )  (1( ))(1 ) 0   x , for0,

  x   1, for0 1 1 = e(x) ,x  for   0  = 0, otherwise. It can be shown that for   0 d   n XU (n)    Ui   i1 where U1,U2,...,Un are independently and identically distributed with

1 P(Ui< x)= 1(x) , x 1,  0,

= (x)1/  , 00. For   0 , we have d n X U (n)   Zi i1 where Z1, Z2, ....., Zn are independently and identically distributed with d P(Zi < z) = 1 - e-z , z > 0 , here  denotes the equality in distribution. For   0 , we have  E(X ) =  {(111 ) n }, U(n)  1 Var( X ) 2 2{(1 2 )n  (1  )2n}, U (n) 2 . 2 -2 mn m 2m Cov(XU(m),XU(n))    (1  ) {(1 2 ) (1  ) } Let m,n be the correlation

coefficient between XU(m) and XU(n), then 1 (12 )m (1 )2m  2  = (1  )mn   , m,n  n 2n   (12 ) (1 )   < 1/2. ={(tm 1) /(tn 1)}1/ 2 , where t = (1  )2 and  < 1/2. 12 As   0, m,n   (m/n) which is the correlation coefficient between XU(m) and XU(n) when  = 0 i.e. for the exponential distribution.

Example 1.2.4.

A random variable is said to have Type II extreme value distribution

28 if its cumulative distribution function is of the following form: x ( ) F(x)e  ,x , 0, 0.

Suppose XL(1), XL(2) .... be the sequence of lower record values and f(n)(x) is the pdf of XL(n), n = 1,2,... We can write (H (x))n1 f (x) f (x) (n) (n)

n x (n 1) x  ( ) ( )   = e  (n)

We can write 1 X L(n)   d (W W ....W )  , where  1 2 n

WW12, ,..., Wn are independent and

identically distributed as exponential with unit mean.

1.3 Moments Theorem 1.3.1.

 If | x |r  dF(x), for some  > 0,  then E(XU(n))r is finite for all n > 1.

Theorem 1.3.2. If E(X) = 0 and Var(X) =1, then 2n |E(XU(n+1))| <  n 1 .

Theorem 1.3.3.

Suppose the random variable X is symmetric about zero and has variance 1,

30 then | E(XU(n+1))| 1 2 1  2n 1  n  < n  2  0 [ln(1u)lnu] du 2 [(n1)]  .

Table 1.3.1 Values of h(n), g(n) and b(n)

N h(n) g(n) b(n) 1 0.906896 1 1.102662 2 1.726929 2.236068 1.294824 3 3.162147 4.358899 1.378462 4 5.916078 8.306624 1.404076 5 11.224972 15.84298 1.411405 6 21.494185 30.380915 1.413448 7 41.424630 58.574739 1.414008 8 80.218452 113.441615 1.414159 9 155.916644 220.497166 1.414199 10 303.937494 429.831362 1.414210

Theorem 1.3.4.

Let { Xi , i=1,2,...} be a sequence of independent and identically distributed random variables and suppose that for 1 < 2 n1 m < n, E(X1 (ln(1 F(X1)))   . Then

(XU (m) , XU (n) )  m / n , where (X , Y) is the correlation between X and Y. The equality holds if and only if X1 is distributed as exponential. We will call F is "new better than used " (NBU) if for x, y > 0, F (xy) < F (x)F (y), and F is " new worse than used" (NWU) if for x, y > 0, F (xy) F (x)F( y) . We will say F belongs to the class C1 if either F is NBU

32 or NWU. The following Theorem is based on NBU(NWU) properties.

Theorem 1.3.5.

Let { Xn , n > 1} be a sequence of independent and identically distributed random variables with distribution function F(x) and the corresponding density function f(x). If E(Xn) , n > 1 is finite and F belongs to the class C1, then E {XU(m+1) - XU(m) } < (> ) E(Xn) , for any fixed m and n according as F is NBU(NWU).

Theorem 1.3.6.

Let {Xi, i=1,2,…} be ac sequence of i.i.d. continuous non-negative rv’s with common cdf F(x) and pdf f(x).Suppose that XU(1), XU(2), … are the upper record values of this sequence and Zn+1,n = XU(n+1) – XU(n), n = 1,2,… with XU(0)=0 . If E(Dn+1) exists and F belongs to class C2, then E(Zn+1 ) > (<) E(Zn) according as F is IFR or DFR.

1.4. DISTRIBUTION OF INTER- RECORD TIMES

Let r = U(r+1) - U(r) and (r) = L(r+1) - L(r), r = 1, 2, ...We will call r and  (r) as the upper and lower inter record times respectively. Since U(1) = 1 = L(1), we have U(r+1) = 1 + 1 + 2 + ... + r . Similarly L(r+1) = 1 +  (1) +  (2)+... +  (r) .

Lemma 1.4.1

For any n > 1, P(n <  ) = 1

= P(  ()n   ). ( Similarly it can be shown that

P(  k| X  x ) n U (n) n k1 = (F(xn )) (1F(xn ))

p(n  k) n1  k1 (ln(1F(x))  (F(x)) (1F(x)) f (x)dx  (n)

k1 k1 i 1 = .  (i ) (1) n i0 (2i)

(1.4.3)  n1  j1 {ln(1 F(x))} P(n  k) 0  (F(x)) (1 F(x)) f (x)dx jk (n)

k1 k1 i 1 =  (1) i n i0 (1 i) (1.4.4) Proceeding similarly, the following theorem can be proved.

Theorem 1.4.2. P((n)=k)

k1 k1 i 1  ( )(1) = i n I0 (2i) For all n > 1, n and (n) are identically distributed. Their pdfs are independent of the parent distribution F(x).

Theorem 1.4.3.

E(n-1) = n(Sn+1 - 1), where  Sn =  ()j n j  1 -1 Table 1.4.1. Values of E(n ) n -1 E(n ) 1 0.644934 2 0.404114 3 0.246970 4 0.147711 5 0.086715 6 0.050096 7 0.028541 8 0.016067 9 0.008951 10 0.004942 15 0.000229

Using the relation between S2n and the Bernoulli numbers, we can express E(n-1) for odd n in terms of the Bernoulli numbers.

Theorem 1.4.4. 1 s P(n > s ) =  P(n1 j) s j1 In particular 1 P( > s ) = , 1 s  1 and 1 s P(2 > s) =  (1 j)1. s1 j1

Let Pg(n(s)) be the probability generating function of n ( |s| < 1), then

  x 1 x Pg(1(s)) =  P(1  x)s =  s = x1 x1 x(x1) 1s 1 + ln(1s), s

 n k k1 E[ s |XU (n)  xn ]  s (F(xn )) (1F(xn ))= k1

1F(xn ) s 1sF(xn ) and Pg(n(s)) n1  1F(x) (ln(1F(x))  s  f (x)dx  1tF(x) (n) 1 s(1u) (ln(1u))n1 =   du 0 1su (n)

 j j s 1 = 1- .  (1) j  n , | s |1 j0 (1 s) (1 j) (1.4.10) The following table gives the values of P(n < k) for some selected values of n. We have

m1 m1 j 1 (1) P(n < m) =1 -   j  n j0 (1 j)

Table 1.4.2. Values of P(n

M \ n 1 2 3 4 5 6 2 0.5000 0.2500 0.1250 0.0625 0.0313 0.0156 5 0.8000 0.5433 0.3323 0.1890 0.1039 0.0552 10 0.9000 0.7071 0.4936 0.3143 0.1870 0.1058 15 0.9333 0.7788 0.5803 0.3926 0.2459 0.1451 20 0.9500 0.8201 0.6365 0.4484 0.2912 0.1773 30 0.9667 0.8668 0.7071 0.5251 0.3586 0.2283 50 0.9800 0.9100 0.7813 0.6153 0.4460 0.3002 100 0.9900 0.9481 0.8573 0.7209 0.5615 0.4062 200 0.9950 0.9706 0.9095 0.8047 0.6657 0.5137 500 0.9980 0.9864 0.9522 0.8836 0.7780 0.6453 1000 0.9950 0.9925 0.9712 0.9235 0.8425 0.5985 5000 0.9998 0.9982 0.9916 0.9734 0.9353 0.7798

Theorem 1.4.5. α - α E(Δn )   and E(Δn )   for α ≥ 1 and n ≥ 1. OPEN PROBLEM  E(n ) ? for α>1

1.5. RELATION BETWEEN OCCURRENCE OF RECORDS AND POISSON PROCESS

In this section we restrict the discussion to upper record values. The corresponding results for lower record values are similar.

Let {Xn, n > 1}be a sequence of i.i.d. random variable with continuous cdf F such that F(0) = 0. We define the process N(0.t] for 0 < t <  as N(0.t] = #{m: XU(m) < t }. It is easy to see that Fm(t) = P[XU(m) < t ] = P(N(0,t] > m ).

Hence P(N(0,t]>m)=. m1 t {R(u)}  f (u)du 0 (m) Thus assuming E(N(0.t] ) < , we get  EN()[,]0 t  PN ( [,]0 t m ) m1  t {R(u)}m1 = f (u)du  0 m 1 (m) Now interchanging the summation and the integral and simplifying, we obtain

t f (u) E(N(0,t] ) dulnF(t) 0 F(u) Since t {R(u)}m1 P(N > m ) = f (u)du , (0,t] 0 (m)

P(N[0,t] = k) = P(N(0,t] > k) - P(N(0,t] > k+1) k 1 t (R(u)) = f (u)du - 0 (k 1) k t (R(u))  f (u)du 0 (k)

= Fk (t)Fk 1(t) . Using the relation n 1 (())Rx j Fx()1 Fx ()  Fx () nn  j! j  0 n 1 (())Rx j  e Rx() j  0 j! we have (())R t k 1 e  R()t P(N[0,t] = k) = ()!k  1 , k = 1,2,... Thus the process N(0,t] is a Poisson Process with mean = R(t) = -ln Ft().

Example 1.5.1

Suppose { Xn, n > 1} is a sequence of i.i.d. random variables with F(x) = 1- e-x , x > 0 and  > 0. Then n1 x u Fn(x) =  eu du and N(0,t] is 0 (n 1)! a homogeneous Poisson Process with E(N(0,t] ) =  t and intensity function  ( = - Ft() ) = . t t Ft()

1.6. NUMBER OF RECORDS IN A SEQUENCE OF OBSERVATIONS

We will consider here the number of upper records among the sequence of observations X1, X2, ..., Xn, the result for the lower records are identical. Let Mn be the number of upper records among the sequence X1, ...., Xn. P[Mn< 2] = P[U(2) > n] = P[ 1 > n-1] = 1 . n P[ Mn < 3] = P[ U(3) > 3] n1   p[1  n  m,2  m 1] m1

1 n1 1   n m1 n  m In general P[ Mn < k+ 1 ] = P[ U(k+1) > n]

1 1 1 1 =     . n 1m1....mk n n mk n mk1 m1 Theorem 1.6.1.

n E(Mn) =  1 i1 i n i1 and Var(Mn) = 2 , n > 1. i1 i

For n  , E(Mn) - Var(Mn) 2 / 6.

M n (n  2) E(2 ) PM (2) = = n+1 (n 1)(2) and

Var( (2M n )  E( (2M n )2 --( E(2M n ))2 2 = PM(4)-(PM(2)) 2 (n  4)  (n  2)  =   (n 1)(4)  (n 1)(2)  Thus 2Mn -1 is an unbiased estimate of n.

The following table gives the exact and asymptotic values of E(Mn) and Var (Mn)..

Table 1.6.1 Exact and Asymptotic Values of E(Mn) and Var(Mn). N E(Mn) Var(Mn) E(Mn) Var(Mn)

10 2.929 1.379 2.880 1.235 100 5.187 3.552 5.182 3.537 500 6.793 5.150 6.792 5.147 1000 7.485 5.842 7.485 5.840 (The last two columns are asymptotic values)

OPEN PROBLEM

E(Mn)k =?

1.7. ENTROPIES OF RECORD VALUES

Let X be a continuous random variable with the pdf f(x), then the entropy H(x) of X is defined as  f (x)lnf (x)dx H(x) = - 

48 where f(x) ln f(x) is integrable. Let Hn(x) be the entropy of XU(n) for a continuous random variable, then  Hn (x)  fn (x)lnfn (x) dx

Suppose that the sequence of i.i.d. random variables Xn has the Weibull pdf , f(x) where c c f(x) = x c1ex /a , 0 < x, a, c <. a In this case, we have Hn(x) 1 1 = ln(n)(n ) (n) lnc lnan. c c If we take c=1, the we get the entropy of the nh upper record value of the exponential distribution as Hn(x) = ln(n) (n 1)(n)  n.

1.8. REPRESENTATION OF RECORDS

Let Y1, Y2,…, Yn,… be a sequence of independent and identically distributed random variable with the cdf as F0(x) = 1-exp(-x), x>0. Further suppose that X1, X2,….be a sequence i.i.d. r.v.’s with continuous cdf F. Then one has d  ln(1 F(x))  Y1. The following theorem gives the representation of the nth record as sum of functions n independent random variables.

Theorem 1.8.1. XU(n) d 1  gF (gF (X1)  gF (X2) ...  gF (Xn)) , where gF(x) = -ln(1-F(x) and -1 -1 -x gF (x)= F (1-e ).

Exponential distribution For the two parameter exponential distribution, E(, ), with cdf F(x) as F(x) = 1-e-(x-) /  , x> 0, we obtain

d X U(n)  X1  X 2 ,,,  X n  (n 1)

Weibull distribution

For the Webull Distribution, W(α,β) with  F(x) = 1-e x , x > 0, α > 0, β > 0, = 0, for x<0. Using Theorem 1.8.1, we obtain 1 d     X U(n)  (X1  X 2 ,,,  X 2 )

Example 1.8.3

Consider the Power Function Distribution, POW (δ,α,β) with cdf F(x) as F(x)   0, x  0       x  1   ,  x  ,  0,     ,      1, x  . 

For this distribution, the following representation is true

1 d  XÚ(n)    ( ) (  X1)(  X 2 )...(  X n ).

Example 1.8.4.

Consider the Pareto distribution, P(,α, β) with F(x) as         1   , x   F(x)    x        0, x  . It can easily be shown the following representation of XU(n) 1 X d (X   )(X   )...(X   ). Ú(n)  n1 1 2 n -. (   )

The following Theorem can be proved following the same

procedure as given in Theorem 3.4.1 (for details see Ahsanullah and Nevzorov (2001 p. 242).

Theorem 1.8.2

Suppose that {X1,X2, …… } and {Y1, Y2,…, } are to sequence of i.i.d. r.v.’s with continuous distribution functions F and H respectively. Then d {XU(1,XU(2), … }  {G(H((YU(1), YU(2),….} where G is the inverse function of F.

RECORDS OF DISCRETE DISTRIBUTIONS 2.0. INTRODUCTION

Let X1, X2,… , Xn,… be a sequence independent and identically distributed random variables taking values on 0,1,2,…such that F(n) <1 for all n, n=0,1,2,…We define the upper record times, U(n) as U(1) =1, U(n+1) = min {j >U(n), Xj > XU(n) }, n=1,2,…. The nth upper record value is defined as XU(n). Let pk = P(X1=k), k P(k)   p( j), P(k)  1 P(k) and P()  1. j0

The joint probability mass function (pmf) of the XU(1), XU(2),…XU(n) is defined as P1,2,…,n(x1,x2,..xn) = P( XU(1)=x1, XU(2)=x2,…., XU(n)=xn) = p(x1) p(x2) p(xn1) ... p(xn) P(x1) P(x2) P(xn1) 0< x1

= 0, otherwise. (2.0.1)

The marginal pmf’s of the upper record values are given as p1(x1) = P(XU(1)= x1) = p(x1) , x1 = 0,1,2,…, p2(x2) = P(XU(2)= x2) = R1(x2) p(x2), where p(x) R1(x2) =  B(x1), B(x)  P(x) 0x1x2 x2 =1,2,…., pn(xn) = P(XU(n)=xn) = Rn-1(xn) p(xn) ,

Rn-1(xn) =  B(x1)B(x2 )....B(xn1), xn  n 1,n,... 0x1x2...xn

(2.0.2) The joint pmf of XU(m) and XU(n), m

Pm,n(xm,xn) = P(XU(m) =xm, XU(n) =xm) = Rm-1(xm)Rm+1,n(xm,xn) p(xn) ,m< xm < xn –n + m <, (2.0.3) where Rm+1,n(x,y) =

 B(xm1)...... B(xn1) , m xmxm1xm2...xn

2.1 GEOMETRIC DISTRIBUTION

A discrete random variable X is said to have geometric distribution if its probability mass function (pmf) is of the following form: p(k) = P(X = k) = pqk -1, 0 < p < 1, q = 1-p , k  Ao = 0, otherwise, (2.1.1) where An = is the set of integers n+1, n+2, ..., and n > 0. We say X  GE(p), if the pmf of X is as given in (2.0.1). For k > 0, we define r(k) = P[X = k| X > k ] . We choose to distinguish between GE(p) and the larger class of distributions having geometric

tail (GET). We write X  GET(s, p) if the pmf of X is as follows: p(k) = P[X = k] = cqk-1, q = 1-p, k  As , = 0, otherwise, (2.1.2)  where c is such that  p(k)  1.. k s1 If s = 0, then GET(s, p) = GE(p) with c = p.The geometric distribution like the exponential distribution possesses the memory less property i.e. P(r  s)  P(r)P(s)

(2.1.3) where r and s are positive integers and

 P( j)   p(k). k  j1 .

Geometric distribution is said to a discrete analogue of the exponential distribution.

DISTRIBUTION OF RECORD VALUES

If X  GE(p), then Px() qx and p(x) = pqx-1, for x Ao.

Substituting the values of Px()i and p(xi) in (2.0.1), we get m x m p (x1,x2,.., xm) = p q m , 1

= 0, otherwise. Thus XU(n) –XU(n-1) is independent of XU(n-1) and XU(n) –XU(n-1)  GE(p), n=2,3,….

Let V1 = XU(1) V2 = XU(2)- XU(1) Vn = XU(n) – XU(n-1). Then Vi’s are independent and Vi  GE(p). We have XU(n) = V1 +V2 +…+ Vn. It is known that if X. G E(p),then  ps E(s x )   s x pq x1  x1 1 qs

(2.1.5) Using (2.1.5), We obtain E(s XU (n) )  E(sV1V2 ...Vn )

ps =( )n 1 qs n X U ( n ) V1 V 2 ...Vn  ps  E ( s )  E (s )   1 qs  (2.1.6) n x  ps    The coefficient of s in   is 1 qs  x1 n xn   p q , x  n.  n1 Thus the marginal pmf of XU(m) can be written as

 x1 m xm pm (x) p[XU (m) x]  p q ,xAm1 ,m  1 m1 = 0, otherwise. (2.1.7) We see that XU(m) has a negative binomial distribution with parameters m and p. We can write

XU(n)|XU(m) =xm d  Um1  ... Un  xm , n>m. and nm XU (n) xm  ps  E(s | XU (m)  x)  s   . 1 qs  The coefficient of sy in nm xm  ps  s   is 1 qs  yx 1  m  nm yxmnm  p q ,  nm1  Thus we obtain the conditional pmf of XU(n) given XU(m) as P(XU(n)=xn|XU(m)= xm)= x x 1  n m  nm xnxmnm  p q ,  nm1  0

< ,xm < xn-n+m<.

But we know that the marginal pmf of XU(m) is

 x1 m xm pm (x) p[XU (m) x]  p q ,xAm1 ,m  1 m1 Thus the joint pmf of XU(m) and XU(n) is  x1 yx1  n yn pm,n (x,y)P[XU (m) xm ,XU (n) xn )   p q  m1n  m1 m < x < y -n+m <  = 0, otherwise.

Let Zmn,()() X Un X Um , 0 < m < n <, then P( Zm.n = z | XU(m) = x) = P( XU(n) = z + x| XU(m) = x )

z1 nm zn m = nm1 p (1 p) , for z  An-m- 1,

= 0, otherwise. (2.1.8) Thus Zn,m and XU(m) are independent. Further Zn,m and XU(n-m) are identically distributed.

ESTIMATION OF PARAMETERS

d Since XU(m)  V1 +V2 + .... + Vm, where V1, ..., Vm are independent and identically distributed as GE(p). Using this property, we get

E(XU(n)) = np-1, Var(XU(n)) = np-2 q. (2.1.9) Cov( XU(n), XU(m)) = Var(XU(m)) = (n -m ) p-2 q (2.1.10) Suppose we have observation the first m record value r12,r ,...,rm and we wish to estimate a function of the unknown parameter p. It is evident from (2.1.3) that XU(m) is a sufficient statistic for p . Further XU(m) is a complete sufficient statistic.

Table 2.1.1 UMVUE for some selected functions of p

Function UMVUE of p

1/p rm / m 1/ p2 rm (rm 1) / m(m

CHARACTERIZATIONS

There are several characterizations of the

geometric distributions based on the (i) independence of XU(n) - XU(m) and XU(m) (ii) conditional distribution of XU(n ) | XU(m) and (iii) moment properties of some functions of XU(n) for n >m . The following theorem (originally proved by Srivastava (1979)) is based on the probability of XU(2) - XU(1) | XU(1).

Theorem 2.1.1.

Suppose F(x) is the distribution function of the sequence of i.i.d. random variables { Xn, n > 1} with positive mass function only at 1,2,... Then P[XU(2) - XU(1) = 1 | XU(1) = i] = P[ XU(2) - XU(1) = 1] for i = 1,2,...,if and only if Xn has the geometric distribution with pmf as given by

Pj = P[ X= j] = c p (1-p)j- 2, j=2,3,.... (2.1.9) and

 p1 = 11 pcj , 0 < p < 1, j  2 0 < c < 1.

A generalization of the Theorem 6.1.9 is the following theorem.

Theorem 2.1.2

Let { Xn, n > 1} be a sequence of independent and identically distributed discrete random variables with common distribution function F. Suppose X is concentrated on the positive

72 integers and a = sup { x |F(x) < 1} = . Then Xn  GET(n,p) for some fixed n, n > 1, if and only XU(n+1) - XU(n) and XU(n) are independent.

Theorem 2.1.3.

Let { Xn, n > 1} be a sequence of independent and identically distributed random variables with common distribution function F. Suppose X is concentrated on the positive integers with

a = sup { x| F(x) < 1} = . Further if P[ XU(n+ 1) - XU(n) = u | XU(n) = y) = P [ X1 = u] for two fixed y  A n-1, yy12, relatively prime and all u  Ao, then X GET(n,p)

Srivastava (1979) gave a characterization of the geometric distribution using the condition E( XU(2)| XU(1) = y) = + y . Ahsanullah and Holland (1984) proved the following

74 theorem which is a generalization of Srivastava's result.

Theorem 2.1.4.

Let { Xn, n > 1} be a sequence of independent and identically distributed discrete random variables with common distribution function F. Suppose X is concentrated on the positive integers with a = sup{ x| F(x) < 1} = . Further suppose E[XU(n+1)]2 < . If

E(XU(n+1) | XU(n) = y ) = y + p -1 for all y An-1, then X1  GET(n,p)and 0 < p < 1.

If fact it can be shown that k E[(XU (n1)  XU (n) ) | XU (n)  y], for any positive integer k, is independent of y. It will be interesting to know whether this is a characteristic property of the geometric distribution. As a partial solution to this question, we have the following theorem..

Theorem 2.1.5.

Further suppose E[XU(2)]2 < . Then E([XU(2) - XU(1)]2| XU(1) = y ) = c, where c is a constant independent of y, if

and only if Xr  GE(p), r > 1 and 0 < p < 1. If in the Theorem 6.1.5, we replace the condition E([ XU(2) - XU(1)]2| XU(1) = y ) = c, by E([ XU(n) - XU(n)]2| XU(n) = y ) = c, y Ar,, r > n, then we can obtain a characterization of GET(p) with 0 < p < 1.

Dembinska and Wesolowski (2000) proved the following Theorem.

THEOREM 2.1.6.

< 1} = . If E(XU(n+2)) < and E( XU(n+2) - XU(n) | X-

U(n)) = b, then Xn  GET(n, b/2).

PREDICTION OF RECORD VALUES

Given observed values of XU(1), XU(2), ....XU(m), we are interested to predict XU(s) for s > m. It is easy to verify that E (xU(s)=s |XU(m) = rm, X U(m-1)=rm-1, ...., XU(1) = r1) = E(XU(s) | XU(m)= rm ) = (s-m) / p + rm. If p is known, then (s-m) / p + rm is the minimum variance unbiased predictor of XU(s).

(2.1.29) If p is unknown, then substituting the unbiased estimator of p-1, we get X Us() as a predictor of XU(s), where

X Us()= srm/m (2.1.30) which is unbiased for XU(s). The non stochastic multiple of X Us() with smallest mean square error for predicting XU(s) is  1 srm p X Us()  1 ()mp11 (2.1.31)

Since P-1 is unknown, substituting MVLUE estimator for p-1 in (6.1.31), we get

2 srm X Us() . ()mrm1 m

2.2. WEAK RECORDS

Vervaat ( 1973) introduced the concept of weak records of discrete distribution. Let X1, X2, …be a sequence of independent and identically distributed random variables taking values on 0,1,….with

82 distribution function F such that F(n) <1 for any n. The weak record times Uw(n) and weak upper record values

XUw (n) are defined as follows: Uw(1)=1 Uw(n+1) min {j> Lw(n) ,Xj > max (X1, X2, …Xj-1)} and the corresponding weak upper record value is defined as XUw (n1) . If in the above expression if we replace > by >, then we obtain record times and record values instead of weak record times and weak record values.

The joint pmf of

XU (1), XU (2),...,X is given by w w Uw(n), n1 p(xi ) Pw,1,2,..,n (x1, x2,..., xn )   p(xn ) i1 P(xi 1) (2.2.1) for 0 < x1 < x2 <…xn <.

For any m>1 and n>m, we can write P(X ..., X ) Uw(n), Uw(m1)|.XU (m)xm,;;;,XU (1) x1 n1 p(x ) p(x )   i n im1 P(xi 1) P(xm 1) (2.2.2)

It follows easily from (2.2.1) and (2.2.2) that the weak

X X X records, Uw(1), Uw(2),... Uw (1) , form a Markov chain. The marginal pmf’s of the upper weak records are given by

P(XUw (1)  x1) = Pw,1(x1) = p(x1), x1 = 0,1,2,…,….

P(XUw (2)  x2) = Pw,2(x2) = Rw,1 (x2) p(x2), x2= 0,1,2,…. where Rw,1 (x2) = p(x )  1 0x1x2 P(x1 1)

P(XUw (n)  xn) = Pw,n(xn)= Rw,n-1(xn) p(xn), where Rw,n-1(xn)

n1 p(xi )    p(xn ) 0x1x2...xn1xn i1 P(xi )

The joint pmf of

XUw(m) and XUw(n) , m

where Rw,m,n(x,y) =

n1    Aw (xm1)....Aw (xn1) xmxm1...xn i1 for m

Pw,n|m(=xXUw (n) n|XUw (m) = xm) p(xn  Rw,m1,n (xm, xn ) P(xm 1)

for m < xm < xn<. .

. Example 2.3..1. Geometric distribution

Suppose X1, X=2, … be a sequence of independent and identically random variables k  with p(k) = pq and P(k 1)  qk , k=0.1,2,…

p(x1) Here R x ) =   = w,1 ( 2 P(x 1) 1x1 x2 1 x2 p. Thus

Pw,2(k) = Rw,1 (k) 2 k p(k)= k p q , k= 0,12,….

Since Pw, n| n-1(=(xn| XUw(n1)  xm)

p(xn ) x  x  = pq n m , P(xn1 1) xn> xn-1, we obtain x 3 2 x x x Pw,3(x3) =  x2 p q 2 pq 3 2 x20 x 3 3 x = x2 p q 3 x2 0 x (x 1) 3 3 p3qx3 =2 , x3= 0,1,2, ,… By induction it can be proved that Pw,n(xn) = x (x 1)...(x  n  2) n n n p xn q xn1 (n 1)!

 x n2  n x =  n  p q n  n1  n > 2 and xn =0,1,2,…

E( XUw(n) | XUw(n1)  xm )

 x x = xn pq n n1 = xn xn1 q  xn-1 p .

The conditional pmf of

XUw(n) given XUw(n2) , is given by

Pw, n|,n-2 (|XXUw(n) U(n-2)= xn-2) =

p(xn) Rw,n-2,n (xn, xn2)  , P(xn2 1) For 0 < xn-2 < xn <, where

Rw,n-2,n(x,y) =

 Aw(xn1)  (x  xn2 1) . xn2xn1x

The conditional expectation of XUw(n) | XUw(n2)  x is

E(XUw(n) | XUw(n2)  xn2) =  2 xx  x(x  xn2 1)]p q n2 xxn2 =  2 xx xn2  [x  xn2)(x  xn2 1)]p q n2 xxn2 2q = xn-2.+ p The following theorem due to Wesolowski and Ahsanullah ( 2001).

THEOREM 2.3.1.

Let X1, X2, …be a sequence of independent and identically distributed random variables taking values on 0,1,…, N .with distribution function F such that F(n) <1 for any n 2, then Xi’s GE(b/(2+b))

3.1. LIMITING DISTRIBUTION OF INTER-RECORD TIMES

We will discuss here the inter-record times corresponding to the upper record values. The corresponding results related to inter-record values of the lower records are similar.

We know ( see Theorem 1.4.2) that 1 s P( s) P(  j) n s n1 j1 (3.1.1)

Let Gn(s) = P(  n > s) , then P( n = s) = Gn(s) - Gn (s + 1) and 1 s Gn (s)  Gn1( j). s j1 Now P(n = s) = 1 s 1 s1  Gn1( j)  Gn1( j) s j1 s 1 j1 1 s Gn1(s 1) =  Gn1( j) s(s 1) j1 s1

1  s1 1 s s    Gn1 ( j) Gn1 ( j) Gn1 ( j)Gn1(s)Gn1 (s1) s1  s    j1 j1 j1  1 (s 1)P( s1)P( (s) = s 1 n n1 (3.1.2)

The relation (3.1.2) can be used to calculate P(n = s) by the method of iteration.

Example 3.1.1.

Using Theorem 1.4.1, we have 4 4 1 P(  5)  (1)i 0.0484 2   i  2 i0   (2i) 3 3 1 P( 4)  (1)i 0.0528 3   i  3 i0   (2  i) P(3 = 5) = 1 4(0.0528) 0.0484 [4P(  4)P(  5)]  0.0433 6 3 2 6 4 4 1 P(  5)  (1)i 0.0433 3   i  3 . i0   (2i)

We know thatP( > n ) = 1 1 n 1 n 1 n 1  P( i) and P(2 > n ) n 1 n i . i1 i1 Since for large n,

n 1  ln n  j , where  is the j1 Euler's constant, we can write 1 P( > n) = (lnn ) . 2 n (3.1.3) Using (3.1.3) ,Theorem 1.4.2 and approximation for large n, we obtain

1 1 . (lnn)2 P(3 > n) = n 2! and

1 1 . (ln n)m1 P(m > n ) = n (m1)! , m > 1.

| s(m1,n) |  n! , (3.1.4) where |(smn , )| is the absolute value of s(m,n) and s(m,n) is the Stirling's number of the first kind.

Example 3.1.2.

For m = 4, P( 4 > 5000 ) = 1 (ln5000)3 .0206 5000 3! . Using Table 1.4.2, we get P( 4 > 5000) = 1 - .9734 =.0266.

The following Theorem is due to Tata (1969).

Theorem 3.1.1.

lim ln n  n  d N(0,1) , n where N(0,1) is the standard normal distribution.

ln  n   1 x2 lim n 1 2 n P[ k] e dx . n k 2

3.2. LIMITING DISTRIBUTION OF RECORD VALUES

We will discuss here the limiting distribution of the upper record values. The distribution of R(Xn) is exponential, E(0,1). It can be shown that

dW W  W R(XU(n)) 12 n (3.2.1)

where WW12,,..., Wn are i.i.d. E(0,1). Hence 2 R(XU (n) )n x 1  1u P[ x] e 2 du . n  2 (3.2.2) From (3.2.1), it follows that

R(X ) U (n) 1 n with probability one. (3.2.3) From the relation (3.2.1), we have the following result of the law of iterated logarithm for R(XU(n)).

Result 3.2.1. With probability one

limsup R(XU (n) )n n 1  1 (2nln ln n) 2

liminf R(XU (n) )n n 1 1. (2nlnln n) 2 (3.2.4) Tata (1969) proved the following theorem.

100

Theorem 3.2.1.

Let R(x) be is convex and twice differentiable with R(x) lim  x x lim R(x) x 0. x2 Then there exists constants an > 0 and bn such that  X b  x  1t2 lim U (n) n 1 2 n P x e dt .  an   2 . The following Theorem related to inter- record time is similar. Theorem 3.2.3.

As n , P[

k1 U (n  k)U n1 1 (ln(1x)) j x] x  j! , k > 1. U (n1)U (n) j0

3.3. LIMITING DISTRIBUTION OF RECORD TIMES

We will consider here the limiting distribution of U(n) . The corresponding results for L(n) are similar. We have seen in section 1.4 that {U(n), n > 1} forms a

102

Markov chain with the probabilities 1 P(U (2)k) k(k1) , k > 1.

j P(U (n1)  k|U (n) j) k(k1) , k > j > 1 , n > 1 and  j  P(U (n1) k|U (n) j)min ,1 , k > j > 1 ,  k  n > 1.  U (n) x  P  |U (n)xP(U (n1) y|U (n)x) U (n1) y  = x . y Since any real x in the interval can be approximated by the rational m/n, we have the following Theorem. The

Theorem was originally proved by Tata (1969).

Theorem 3.3.1.

U (n) The distribution of U (n1) is asymptotically uniform over the unit interval, i.e.

lim  U (n)  n P  xx,0x1. U (n1) 

Theorem 3.3.2.

As n  , P [ Un() < x ] = Un() k k1 x (lnw)k1 ( lnx)i dwx . 0 (k)  ( j1) j0

104

3.5. LIMITING DISTRIBUTIONS OF NUMBER OF RECORDS

We have seen in Theorem 1.5.1 that n 1 E(M n ) j1 j and n i1 Var(M ) n  2 , i1 i n where Mn = Z i . Considering the i1 joint and the marginal densities, it can be shown easily that the random variables Zi's are independent. Considering one-sided Chebyshev inequality P[Mn > t] < var(M n ) 2 , var(M n ) {(tE(M n )} we get

P[Mn > 20] < 0.036. As n , E(Mn)   + ln n Var(M )  + ln n - 2 ,  is n 6 the Euler's constant = .5772... The table 8.4.1 gives the exact and approximate values of E(Mn) and Var(Mn) for some selected values of n.

Table 3.5.1 Exact and approximate values of E(Mn) and Var(Mn)

N E(Mn)  + ln n Var(Mn) 2  + ln n - 6 5 2.28 2.19 0.82 0.54 10 2.93 2.88 1.38 1.23 20 3.60 3.57 2.00 1.93 50 4.50 4.48 2.87 2.84 100 5.19 5.18 3.55 3.54

106

200 5.88 5.87 4.24 4.23 500 6.79 6.79 5.15 5.15 1000 7.49 7.48 5.84 5.84

E(Mn)   as n   and       var(M )   n  < . We have   n  n1  E(Z )    i   i 1  M n  1 with probability one as n  lnn . It can be shown that the random variables Zi' s satisfy the Liapunov condition, i.e. 1 n E |Z  E(Z ) |2  2  j j 0, as n   . 2 j1 {var(M n )} Thus M lnn n d N(0,1) as n   . lnn

Thus as n , ln()s is the logarithm of a Poisson generating

function with mean ln b . Hence as a n , b (ln ) j a P[ M - M a . . bn an = j ] = j! b

108

4.0 EXPONENTIAL DISTRIBUTION

A continuous random variable X is said to be exponentially distributed with parameters  and , >0, if its pdf is of the following form f x   1 exp 1x  , x    0, otherwise . (4.0.1) The corresponding distribution function F(x) and the hazard rate r(x) of the rv X are respectively Fx  1 exp 1 x  , x   and

110

rx  f x/1 Fx   1. (4. 0.2) We will denote the exponential distribution with the pdf as given in (4.0.1) as E( ,). The exponential distribution possesses the memory less property i.e. an item whose lifetime is exponentially distributed, the residual life does not depend on the past life. In terms of probability, we can write P[X> s+t | X> t ] = P[X>s] (4.0.3) In terms of the distribution function we can write (2.0.3) as 1-F(s+t) = [ 1-F(s)] [ 1-F(t)] . (4.0.4)

This property is utilized in many characterization problems of the exponential distribution.

112

4.1. DISTRIBUTION OF RECORD VALUES

Using the relation (1.2.7) and noting R(x)=-1(x- ), we have n  n1 1 f n x  x   exp x   , x   n  0, otherwise. (4.1.1) The corresponding cdf F is k x n1  1  x   Fn (x)1   e . k0 k!   Xnmand X ,  The joint pdf of Unm( U(n) ( by using 1.2.9) is   n x   m 1 f x, y   y  x n  m 1 exp   1 x   , m ,n  m n  m xy    ,  0, otherwise . (4.1.2)

It is easy to see that X Un() X U(n-1) and

XXUm() Um (1 ) are identically distributed

for 1

XdXUmUm() Um (1 ),(1 ) where U is independent of XUm()and XU(m-1) and is identically distributed as Xi’s iff 1exp(-x 1 ). For>0, it is easy to verify that n P[X U (n) wX U (n1) ]w ,n 1.

The conditional pdf of XXUn()| Um ( )  xis  mn fyX(| x ) ()yxnm11exp (())  yx Um() ()nm   xy   ,  0, otherwise . (4.1.3)

Thus PX U n  X U m  y | X U m  x does not depend on x. It can be shown that if  =0., then X U n  X U m  is identically distributed as X , n  m. U n  m If we take  = 0 and  = 1 and Wn = XU(1)+XU(2) +...+ XU(n),

114 the characteristic function of Wn can be written as 1 1 1  ()t   . n 1 it 12 it 1 nit (4.1.4) Inverting (2.1.4), we obtain the pdf fW(w) of Wn as fW(w) = n 11() nj   ejwj/ n2 . (4.1.5) j  1 ()jnj()1

Theorem 4.1.1.

Let Xj , i = 1,2, ... be independently and exponentially distributed with  = 0 and  = 1 . X Ui() Suppose i  i =1,2, m-1 then X U ()i  1

i 's are independent..

Corollary 4.2.1. k Let Wk = (k ) , k = 1,2,..., m-1, then W1, W2 , ...., Wm-1 are independent and identically distributed uniform (over the interval (0,1)) random variable.

4.2. MOMENTS OF RECORD VALUES

Without any loss of generality we will consider in this section the standard exponential population, E(0, 1), with pdf f(x) = exp(-x), 0 x , in which case we have f(x) =1-F(x). From (2.1.1) it is obvious that XU(n) can be written as the sum of n i.e. random variables V1,

116

V2, ..., Vn each of which is distributed as E(0, 1). We have already seen that EXU n   n

VarXU n  n, and CovXU n , XU m   m, m  n. (4.2.1) For m < n , E( p q XU (n) XU (m))  u 1 1 q x m p1 n  m1   . u e v (u v) dvdu 0 0 (m) (nm)

Substituting t u = v and simplifying we get

p q E(XU (n) XU (m) )

  1 1   un pq1ext m p1(1 t)nm1dtdu 0 0 (m)(n  m) . = (m  p)(n  p  q) (m)(n  p) It can be shown that if Wn = XU(1)+ XU(2)+ ... + XU(n), then E(W ) = nn() 1 and Var(W ) = n 2 n nn()()12 n 1. 6 Some simple recurrence relations satisfied by single and product moments of record values are given by the following theorems.

Theorem 4.2.1

For n > 1 and r = 0, 1, 2, ...

118

r 1 r 1 r E X U n  E X U n 1  r  1E X U n  (4.2.2) and consequently, for 0 < m < n-1 we can write n r1 r1 r EX U n  EX U m  r 1  EX U p  pm1 (4.2.3) r1 0 with EX U 0  0 and EX U n 1.

Remark 4.2.1. The recurrence relation in (2.2.2) can be used in a simple way to compute all the simple moments of all record values. Once again, using the property that f(y)=1-F(y), we can derive some simple recurrence relations for the product moments of record values.

Theorem 4.2.2.

For m > 1 and r, s = 0, 1, 2, ... r s1 EX U m X U m1   rs1 r s (4.2.4)  EX U m  s 1 EX U m X U m1  , and for 1 < m < n-2, r,s = 0, 1, 2, ... r s1 EX U m X U n  r s1 r s  EX U m X U n1   s 1 EX U m X U n . (5.2.5)

Remark 4.2.2. By repeated application of the recurrence relation in (5.2.5), with the help of the relation in (5.2.4), we obtain for n>m+1 that r s1 EX U m X U n  n rs1 r s  EX U m  s 1  EX U m X U p . pm1 (4.2.8) For n > m+1,

120

CovX U m , X U n   VarX U m .

Remark 4.2.3. The recurrence relations in equations (4.2.4) and (4.2.5) can be used in a simple way to compute all the product moments of all records values.

Theorem 4.2.3

For m > 2 and r,s = 0,1,2..., r1 s rs1 r s E(XU (m1) XU (m) )E(XU (m) ) (r 1)E(XU (m) XU (m1) ) (4.2.11) and for 2 < m < n-2 and r,s = 0,1,2..,

r1 s r1 s r s E(X U (m1) X U (n1) )E(X U (m) X U (n1) ) (r 1)E(X U (m) X U (m1) )

(4.2.12) Corollary 4.2.3. By repeated application of the recurrence relation in (4.2.12), with the help of the relation in (4.2.1), we obtain for 2 < m < n-1 and r,s = 0,1,2.. n1 r1 s rs1 r s EX U (m1) X U (n1) E X U (n1) (r 1) E X U ( p) X U (n) pm Corollary 4.2.4. By repeated application of the recurrence relations in (4.2.11) and (4.2.12), we also obtain for m > 2 r1 E X r1 X s  (1) p (r 1)( p) E X rs1 p U (m1) U (m)  U (m p) p0 and for 2 < m < n-2 r1 E X r1 X s  (1) p (r 1)( p) E X r1 p X s U (m1) U (n1)  U (m p) U (n1 p) p0 where (r+1)(i) is as defined earlier. It is also important to mention here that this approach can easily be adopted to derive recurrence

122 relations for product moments involving more than two record values.

4.3 ESTIMATION OF PARAMETERS We shall consider here the linear estimation of  and .

(a) Minimum Variance Linear Unbiased Estimates (MVLUE)

Suppose X U 1 , X U 2 ,..., X U m  are the m record values from an i.i.d.

1 sequence E, . Let Yi   XU i  , im 12,,...,, then

EYi  i  VarYi , i,1,2,...,m, and CovYi ,Yj  mini, j.

Let X  X U 1 , X U 2 ,..., X U m , then EX  L  

VarX   2V , where L  1,1,...,1 ',  1,2,...m'

V  Vij ,Vij  mini, j, i, j  1,2,..., m. The inverse V 1 V ij  can be expressed as Vifijmij  2121   ,,...,  1 if i j m 1112if | i j | , i , j , ,..., m  0 otherwise.

The minimum variance linear unbiased estimates (MVLUE)  , of  and  respectively are ˆ   'V 1L 'L' V 1 X /  ˆ  L'V 1 L 'L' V 1 X / , where 2   L'V 1L  'V 1  L'V 1  and Varˆ  2 L'V 1 / 

124

Varˆ   2 L'V 1L /  Covˆ,ˆ   2 L'V 1 /  . It can be shown that L'V 1  1,0,0,...,0 ,  'V 1  0,0,0,...,1, '.V1  m and  m 1 On simplification we get

ˆ  mX U 1  X U m /m 1

ˆ  X U m  X U 1 /m 1 (4.3.1) with Varˆ  m 2 /m 1,Varˆ    2 /m 1 and Covˆ,ˆ   2 /m 1. (5.3.2)

(b) Best Linear Invariant Estimators The best linear invariant (in the sense of minimum mean squared error and invariance with respect to

the location parameter ) estimators (BLIE)   of  and  are

~  E12     1 E22  and ~   ˆ /1 E22  , where  and  are MVLUE of  and  and

Varˆ Covˆ,ˆ  2  E11 E12        Covˆ,ˆ Var ˆ   E12 E22  The mean squared errors of these estimators are

~ 2 2 1 MSE()(!) E11 E 12 E 22  and

~ 2 1 MSE   E22 1 E22  We have ~ ~ 2 1 E        E12 1 E22  . Using the values of E11, E12 and E22 from (2.3.2), we obtain

126

~   m 1 X U 1  X U m / m ,

ˆ  X U m  X U 1 / m m 1 Var~   2 and m m 1 Varˆ   2 m 2 4.4 CHARACTERIZATIONS

We will give several characterization theorems of the exponential distribution under various assumptions.

(a) Under the Assumption of Independence. We have already seen that XU(n) - XU(m) and XU(m) , n> m > 1, are independent. This is a characteristic property of the exponential distribution. For n = 2

Tata (1969) proved the following characterization theorem.

Theorem 4.4.1

Let { Xn, n> 1} be an i.i.d. sequence of non- negative random variables with cdf F(x) and pdf f(x). Then for Xn to belong to E(, ), it is necessary and sufficient that XU(2) - XU(1) and XU(1) are independent. The following theorem is a generalization of theorem 5.4.1.

Theorem 4.4.2

Let { Xn, n> 1} be a sequence of i.i.d. random variables with

128 common distribution function F which is absolutely continuous with pdf f . Assume F(0) =0. Then for Xn  E(0,  ) it is necessary and sufficient that XU(n) and XU(n+1) - XU(n) , n > 1, are independent.

Proof. The following theorem is a generalization of the theorem 4.4.2.

Theorem 5.4.3.

Let { Xn , n > 1} be independent and identically distributed with common distribution function F which is absolutely continuous and F(0) = 0 and F(x) < 1 for all x> 0.

Then for Xn E(0,), it is necessary and sufficient that Zn,m and XU(m) ( n > m > 0) are independent. Here Zn,m = XU(n) - XU(m).

(b) Under the Assumptions of Identical Distribution.

We have seen that if the sequence { Xn, n > 1} of i.i.d. rvs are from E(0,), then XU(n) d Z12 Z ....Zn , where Z1, Z2,... , Zn are i.i.d. E(0, ).

130

The following theorem gives a characterization of the exponential distribution using the above property. If F is the distribution function of a non- negative random variable, we will call F is "new better than used " (NBU) if for x, y > 0, F (xy) < F (x)F (y), and F is " new worse than used" (NWU) if for x, y > 0, F (xy) F (x)F( y) . We will say F belongs to the class C1 if either F is NBU or NWU.

Theorem 4.4.4.

Let Xn, n> 1 be a sequence of i.i.d. random variables which has

absolutely continuous distribution function F with pdf f and F(0) = 0, Assume that F(x) < 1 for all x>0. If Xn belongs to the class C1 and Zn+1,n ( = XU(n+1) - X U(n)) has an identical distribution with Xk , k > 1, then Xk E(0,) , k > 1.

The following theorem is proved under the assumption of monotone hazard rate. We will say F belongs to the class C2 if r(x) is either monotone increasing or decreasing.

Theorem 4.4.5.

If Xk, k > 1 has an absolutely continuous distribution function F

132 with pdf f and F(0) = 0. If Zn+1,n and Zn,n-1 , n > 1, are identically distributed and F belongs to C2, then Xk E(0,), k> 1.

Theorem 4.4.6.

Let Xn, n> 1 be a sequence of independent and identically distributed non-negative random variables with absolutely continuous distribution function F(x) with f(x) as the corresponding density function. If F  C2 and for some fixed n,m, 1< m < n < ,

Zn,m d X Un() m , then Xk  E(0,), k> 1.

Theorem 4.4.7.

Let { Xn, n > 1} be a sequence of independent and identically distributed non-negative random variables with absolutely continuous distribution function F(x) and the corresponding density function f(x). If F belongs to C2 and for some m , m > 1, XU(m) = XU(m-1) + U, where U is independent of XU(m) and XU(m-1) and is distributed as Xn's ,then Xn E(0, ) , for some  >0.

134

Theorem 4.4.8.

Let X1,...,Xm,... be independent and identically distributed random variables with probability density function f(x), x> 0 and m is an integer valued random variable independent of X's and P(m = k) = p(1-p)k-1 , k = 1, 2, ... and 0 < p <1. Then the following two properties are equivalent: (a) X's are distributed as E(0,), where  is a positive real number, m (b) p  XdXjUnUn() X (1 ) , for some j1 fixed n, n> 2, Xj  c2 and Xj  c2 and E( Xj ) < .

(c) Under the Assumption of Finite Moments. We will proof the following characterization theorem under the assumption of the finite first moment.

Theorem 4.4.9.

Let Xn , n > 1 be a sequence of independent and identically distributed non-negative random variables with absolutely continuous distribution function F(x) and the corresponding density function f(x). Let a = inf{x|F(x) >0} =0 , F(x) < 1 for all x > 0. If F belongs to the class C1 and E(Xk) , k > 1 is finite., then Xk E(0,), if

136 and only if for some fixed n, n > 1, E(Zn)= E(Xk).

The following theorem uses the property of homocedasticity but does not use NBU or NWU property.

Theorem 4.4.10.

Let xn, n > 1 be a sequence of independent and identically distributed random variables with common distribution function F which is absolutely continuous and inf{ x| F(x) > 0} =0 and E(Xn2) < . Then Xk, k > 1 has the exponential distribution if and only if var(Zn| XU(n ) = x) = b for all x,

where b is a positive constant independent of x and Zn = XU(n+1) - XU(n). The following Theorem gives a characterization of the exponential distribution using the hazard rate.

Theorem4.4.11.

Let { Xn, n > 1 } be a sequence of independent and identically distributed non negative random variables with continuous distribution function F(x) and the corresponding density function f(x). Let a = inf{x|F(x) = 0} = 0, F(x) < 1 for all x > 0 and F belongs to class C2. Then Xk  E(0,), if and only if for some fixed n, n > 1,

138 the hazard rate r1 of Zn+1,n = the hazard rate r of Xk , where Zn+1,n = XU(n+1) - XU(n).

The exponential distribution can also be characterized using lower record values. Ahsanullah and Kirmani (1991) characterized the exponential distribution using lower record values. The following result is due to Ahsanullah and Kirmani (1991). Suppose {Xn, n > 1} be a sequence of i.i.d. random variables with cdf F and F(0) = 0. Let N is the rv defined as N = min { i > 1: Xi < X1}. It can easily be shown that P(N = n) = 1 , n =2,3,... n(n 1)

Theorem 4.4.12.

Basak( 1996) give a similar characterization based on k- records. The result is given in the following theorem.

Theorem 4.4.13.

Suppose {Xn, n > 1} be a sequence of i.i.d. random variables with cdf F(x) F with F(0) = 0.and lim   , λ x x0 >0, . If (L(n,k) –k+1) XL(n,k) and X1,n , k > 1, are identically distributed, then X has the exponential distribution with F(x) = exp (-λx)..

140

. A distribution function F with F(0) = 0 and μ=E(X) <, is said to be harmonic new better ( worse) than used in expectation , HBBUE or  t /  HNWUE,) if t F (x)dx  ()  e ,t>0,

We say F  C3 if F is either HNBUE or HNWUE,). The following Theorem replaces the equality of distribution in Theorem (2.4.13) by the equality of expectation.

Theorem 4.4.14.

Suppose {Xn, n > 1} be a sequence of i.i.d. random variables with cdf F with F(0) = 0. .If F  C3, then

E( (L(n,k) –k+1) XL(n,k) ) = E( X1,k ) if and only if F is exponential.

142

Goodness of Fit Consider the 100 Meter Freestyle –Women Olympic Swim Records

Year Time Record Holder minute. Seconds (Country) 1912 1 22.2 Fanny Durack (AUS) 1920 1 13.6 Etheida Bleitrey (USA) 1924 1 12.4 Ethel lackie (USA) 1928 1 11.0 Albino Osipowich (USA) 1932 1 06.8 Helene Madison (USA) 1936 1 05.9 Rio Masterbroek (Ned) 1948 1 05.3 Greta Anderson (DEN ) 1952 1 04.8 Katalin Szoke (HUN) 1956 1 02.8 Dawn Fraser (AUS) 1960 1 01.2 Dawn Fraser (AUS) 1964 0 59.5 Dawn Fraser (AUS) 1972 0 58.5 Sandra Neilson (USA) 1976 0 55.65 kormelia Enderr (E GER) 1980 0 54.79 Barbara Krause (E GER) 1992 0 54.65 Zuang Yong (CHN) 1996 0 54.50 Le Jingyi (CHN) 2000 0 53.83 Inge de Burjin (NED) 2004 0 53.52 Jodie Henry (AUS)

Generalized extreme value was fitted to the data.Linear invariant predictors of the next two records are

53.42 and 53.28

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