Improved Score Tests for One{Parameter Exp Onential

Improved score tests for one{parameter exp onential

family mo dels

Silvia L.P.Ferrari Gauss M. Cordeiro

Departamento de Estatstica Departamento de Estatstica

Universidade de S~ao Paulo, Brazil Universidade Federal de Pernambuco, Brazil

Miguel A. Urib e{Opazo

Francisco Cribari{Neto

Departamento de Matematica

Department of Economics

Universidade Estadual do Oeste do Parana,

Southern Il linois University, USA

Brazil

Abstract: Under suitable regularity conditions, an improved score test was derived by Cordeiro and Ferrari (1991).

The test is based on a corrected score statistic which has a chi-squared distribution to order n under the null

hyp othesis, where n is the sample size. In this pap er we follow their approach and obtain a Bartlett-corrected score

(0)

statistic for testing H : = , where is the scalar parameter of a one-parameter exp onential family mo del. We

apply our main result to a numb er of sp ecial cases and derive approximations for corrections that involveunusual

functions. We also obtain Bartlett-typ e corrections for natural exp onential families.

Keywords: Bartlett-typ e correction; chi-squared distribution; exp onential family; score statistic; variance function.

1 Intro duction

Three commonly used large sample tests are the score (or Lagrange multiplier), likeliho o d ratio (LR) and

Wald (W) tests. These three tests are asymptotically equivalent under the null hyp othesis. It has b een

argued that after mo difying their critical regions to force the three tests to have the same size to order n ,

where n is the numb er of observations, the score test is usually more p owerful than the LR and W tests to

order n ; see Chandra and Joshi (1983), Chandra and Mukerjee (1984, 1985) and Mukerjee (1990a, 1990b).

Another advantage of the score test is that it only requires estimation of the unknown parameters under

the null mo del. All three tests rely on a rst order asymptotic approximation whichmay b e a p o or one in

samples of small to mo derate size. It is p ossible to improve the chi-squared approximation to the LR test by

multiplying the test statistic by a scalar correction factor known as the Bartlett correction; see Lawley (1956),

Hayakawa (1977), Cordeiro (1987, 1993a) and the references therein. Recently, Cordeiro and Ferrari (1991)

obtained a Bartlett-typ e correction to the score statistic. Their correction is de ned as a second degree

p olynomial on the score statistic with co ecients that dep end on cumulants of log-likeliho o d derivatives.

Both unmo di ed statistics havea distribution to rst order under the null hyp othesis H , where q denotes

the numb er of restrictions imp osed byH . On the other hand, the mo di ed statistics havea distribution

Correspondence to: Silvia L.P.Ferrari, Departamento de Estatstica, Universidade de S~ao Paulo, Caixa Postal

66281, 05389-970 S~ao Paulo, SP, Brazil. 1

1 2 2 1

to order n under H . In other words, Pr[S x]=Pr[ x]+o(1) and Pr[S x]=Pr[ x]+o(n ),

q q

where S and S represent the unmo di ed and mo di ed statistics, resp ectively. The nite-sample accuracy

of these corrections has b een extensively evaluated through Monte Carlo simulation; see Cordeiro (1993b),

Cordeiro and Cribari{Neto (1993), Cordeiro, Cribari{Neto, Aubin and Ferrari (1995), Cordeiro, Ferrari and

Paula (1993), Cribari{Neto and Cordeiro (1995), Cribari-Neto and Ferrari (1995a, 1995b), Cribari{Neto and

Zarkos (1995) and Ferrari and Cordeiro (1995). Such exp eriments have shown that Bartlett and Bartlett-

typ e corrections usually lead to improvements in the size b ehaviour of the likeliho o d ratio and the score tests

even in nonnormal and multivariate regression mo dels.

The main purp ose of the present pap er is to obtain a general closed-form expression for the Bartlett-typ e

correction to the score statistic in one-parameter exp onential family mo dels. This expression can b e easily

used to derive Bartlett-typ e corrections for many imp ortant distributions. Our aim here is to givea new

formula for correcting the score test in one-parameter exp onential family mo dels which is algebraically more

app ealing for applications than the general formulas develop ed by Harris (1985). Unlike Harris' general

formulas, our result can b e readily used by applied researchers since it only requires trivial op erations on

suitably de ned functions and their derivatives. It should b e mentioned that the results in the present pap er

can b e extended to more general problems, such as, two-parameter exp onential family mo dels.

Consider a set of n indep endent and identically distributed random variables with density (or probability)

function

(y ; )= expf ( )d(y )+v(y)g; (1)

()

where is a scalar parameter, (); ();d() and v () are known functions and () is p ositivevalued. We also

assume that the supp ort set of this distribution is indep endentof and that () and ()have continuous

rst four derivatives, d ( )=d and d ( )=d b eing di erent from zero for all in the parameter space. Here,

()

( )= ; (2)

() ()

primes denoting derivatives with resp ect to : This quantity will play an imp ortant role in the derivation of

the correction.

In the next section, we derive a simple expression for the Bartlett-typ e correction to the score test in

the exp onential family (1). This new expression involves only the functions () and () and their rst

three derivatives, and can b e easily implemented in a computer algebra system suchasMATHEMATICA

(Wolfram, 1991) or MAPLE V (Ab ell and Baselton, 1994). In Section 3, we presentanumb er of sp ecial cases

thus showing that our main result has a wide range of imp ortant applications. Finally, Section 4 considers

Bartlett-typ e corrections for the class of natural exp onential families assuming di erent forms of variance

functions.

2 Derivation of the correction

Consider n indep endent and identically distributed random variables y ;:::;y having any regular unipara-

1 n

metric distribution with density (or probability) function (y ; ) = expft(y ; )g; where is a scalar param-

0 r (r ) 0

eter, and de ne v = Ef(t (y ; )) g and v = Eft (y ; )g, for r = 1;:::;4; with t (y ; ) = dt(y ; )=d

(r ) 2

(r ) r r

and t (y ; ) = d t(y ; )=d : The v 's satisfy certain regularity relations such as v = 0; v = v ;

1 2

(2)

0 0 00 00 2

v =2v 3v and v = 3v +8v 6v +3v ; where v =Ef(t (y ; )) g (Lawley, 1956).

3 4

(3) (4) 2(2) 2(2)

2 3 2

(0)

The score statistic for testing H : = against a two-sided alternative can be written as S =

0 R

0 (0) 2 (0)

t (y ; )) =(nv ); where v is evaluated at : The evaluation of S do es not require any esti- (

l R

(2) (2)

l=1

mation since it only dep ends on the value of sp eci ed under H : Cordeiro and Ferrari (1991) have shown

that the mo di ed score statistic

n o

S = S 1 (c + bS + aS ) ; (3)

R R

where

2 2 2

3 10 5 3

2 2

1 1 1

; b = ; c = ; (4) a =

36 36 12

3=2

2 1 2 2

has a distribution to order n : Here, = v =v and =(v 3v )=v are the usual measures

1 2

(3) (4)

(2) (2)

(2)

of skewness and kurtosis of the score function for a single observation.

Next, let (y ; ) b e de ned as in (1) so that t(y ; )=log ( ) ( )d(y )+v(y): It then follows that

0 0 0 0 00 0 0 00 000 0 00 00 0 000

Efd(y )g = ,varfd(y )g = = , v = ;v =2 ;v =3( + ) and

2 3 4

002 0 0 02 02 0 2 0 1

v = = + : Hence, the score statistic is given by S = n ( + d) = ; where d = n d(y )

R l

2(2)

0 (0)

and and are evaluated at : Finally, using again the relations among the v 's and plugging them into

the expressions for a; b and c in (4), we get

0 00 0 00 2

( )

a = ; (5)

03 03

02 002 0 0 00 00 02 002 0 02 000 02 0 000

+11 10 3 +3

; (6) b =

03 03

02 002 0 0 00 00 02 002 0 02 000 02 0 000

4 +5 +3 3

c = : (7)

03 03

0 00 000 0 00 000 (0)

Plugging (5)-(7) into (3) and evaluating , , , , and at ; we get Cordeiro and Ferrari's (1991)

correction to the score statistic for testing H in one-parameter exp onential family mo dels. Two features of

equations (5)-(7) are noteworthy. First, c is equal to the co ecient divided by 12 which determines the

correction to the LR statistic (see eq. (4) of Cordeiro, Cribari{Neto, Aubin and Ferrari, 1995), and thus the

present pap er can b e viewed as an extension of their pap er. Second, a; b and c dep end on the mo del (1)

only through the functions and and their rst three derivatives.

3 Sp ecial cases

In this section, we use equations (5)-(7) to derive Bartlett-typ e corrections to the score statistic for a number

of distributions that b elong to the one-parameter exp onential family (1). The calculations were done using

MATHEMATICA and MAPLE V. It should be remarked that it is not p ossible to guarantee that the

mo di ed statistic has a chi-squared distribution to order n when the distribution is discrete. The sp ecial

cases listed b ellow have a wide range practical applications in various elds such as engineering, biology,

medicine, economics, among others (Johnson and Kotz, 1970a, 1970b; Johnson, Kotz and Kemp, 1992).

The following distributions are considered: 3

(i) Binomial (0 < <1, m 2 IN , m known, y =0;1;2;:::;m): ( )=logf=(1 )g, ( )=(1) ,

, d(y)=y, v(y) = log

(2 1) 22 ( 1)+7 (1 ) 1

a = ; b = ; c = :

36m (1 ) 36m (1 ) 6m (1 )

(ii) Negative binomial (0 < < 1, > 0, known, y = 0; 1; 2;:::): ( ) = log , ( ) = (1 ) ,

+y 1

d(y )=y, v(y) = log ,

2 2

( +1) 7 +8 +7 1 (1 )

a = ; b = ; c = :

36 36 6

(iii) Poisson ( > 0, y = 0; 1; 2;:::): ( ) = log( ), ( ) = expf g, d(y ) = y , v (y ) = log(y !), a =

1=(36 ); b = 7=(36 ); c =1=(6 ):

(iv) Truncated Poisson (>0, y =1;2;:::): ( )=log( ), ( )=e (1 e ), d(y )=y,v(y)=log(y !),

2 2 2 2 3

a = f +3 +1+e ( 3 2) + e g =f36e (1 + e ) g;

2 3 4 2 3 4

b = f7+36 +71 +39 +7 +e (28 108 140 9 +8 )

2 2 3 4 3 2 3 4

+ e (42 + 108 +67 33 +7 )+e (28 36 +2 +3 )+7e g=

f36e (1 + e ) g;

2 3 4 2 3 4

c = f2+ 6 +16 +9 +2 +e (818 40 9 2 )

2 2 3 4 3 2 3 4

+ e (12 + 18 +32 3 +2 )+e (86 8 +3 )+2e g=

f12e (1 + e ) g:

(v) Logarithmic series (0 < <1, y =1;2;:::): ( )=log( ), ( )=log(1 ), d(y )=y, v(y)=

log(y ),

2 2 3

a = flog(1 )(3 + log (1 ) + log(1 )) + 2 g =f36 ( + log(1 )) g:

3 2 2 3 2

b = [22 ( + 3 log(1 )) + flog(1 )g (28 +73)+3flog(1 )g (12 +12 )

4 2 3

+ flog(1 )g (7 + 8 +7 )]=f36 ( + log(1 )) g;

3 2 2 3 2

c = [2 ( + 3 log(1 )) + 8 flog(1 )g (1 + )+3flog(1 )g (2 + 2 )

4 2 3

+ 2flog(1 )g ( + 1)]=f12 ( + log(1 )) g:

(>0;a 0;y =0;1;2;:::): = log( );()= a ;d(y)=y; v(y) = log(a ); (vi) Power Series

y y y

y=0

0 2 00 2

(g +3g + g )

; a =

0 3

36 (g + g )

2 0 2 02 00 3 0 00 000 4 002 0 000

7g +36g g + (69g +2gg )+3 (14g g gg )+ (10g 3g g )

b = ;

0 3

36 (g + g )

2 0 2 02 00 3 0 00 000 4 002 000

2g +6g g +8 (3g gg )+3 (g g gg )+ (5g 3gg )

c = ;

0 3

12 (g + g ) 4

where g = g ( )=dlog()=d: Note that cases (ii), (iii), (iv) and (v) can b e obtained from this case

by simple sp eci cation of the function g ().

(vii) Zeta (>0, y =1;2;3;:::): ( )=+1, () = Zeta( + 1), d(y ) = log(y ), v (y )=0,

002 002 0 000 002 0 000

g 10g 3g g 5g 3g g

a= ; b = ; c = ;

03 03 03

36g 36g 12g

( +1)

i (see, e.g., Patterson, where is the Riemann zeta-function, i.e., ( ) = Zeta( +1)=

i=1

1988) and g = g ( ) = d log Zeta( +1)=d:

(viii) Non-central hyp ergeometric (>0, m ;m ;r are known p ositiveintegers, k = maxf0;r m g

1 2 1 2

m m

1 2

yminfm ;rg = k ): ( )=, ()=D (), d(y )=y, v(y) = logf g,

1 2 0

y r y

2 3 2 2 3

a =(D D 3D D D +2D ) =f36(D D D ) g;

3 0 1 2 0 2

0 1 1

4 2 3 2 3 2 2 2 4 6 4

b = (10D D 48D D D D +28D D D +45D D D 66D D D +22D 3D D D

1 2 3 3 0 2 2 4

0 3 0 0 1 0 1 2 1 1 0

3 3 3 2 2 3

+ 9D D +3D D D )=f36(D D D ) g;

4 0 2

0 2 0 1 1

4 2 3 2 3 4 6 4 3 3

c = (5D D +18D D D D 8D D D +6D D D 2D +3D D D 9D D

1 2 3 3 0 2 2 4

0 3 0 0 1 1 1 0 0 2

3 2 2 3

3D D D )=f12(D D D ) g;

4 0 2

0 1 1

m m

1 2

y expfy g;p=0;1;2;3;4: where D = D ( )=

p p

y=k

y r y

2 1 3 2 2

(>0, y>0): ( )=(2 ) , ()= , d(y)=y , v(y) = log(y 2= ), a =2=27; b = (ix) Maxwell

11=27; c =1=9:

(x) Gamma (k>0; > 0, y>0):

(a) k known: ( )=,()= ,d(y)=y,v(y)=(k1) log(y ) logf(k )g, a =1=(9k ); b = 11=(18k );

c =1=(6k ):

(b) known: (k )=1k, (k)= (k ), d(y ) = log(y ), v (y )=y ,

00 2 00 2 0 000 00 2 0 000

(k ) 10 (k ) +3 (k) (k ) 5 (k ) 3 (k ) (k )

a = ; b = ; c = ;

0 3 0 3 0 3

36 (k ) 36 (k ) 12 (k )

where () and () are the gamma and digamma functions, resp ectively.

( > 0, b > 0; b known, y > 0): ( ) = , ( ) = g ( )= , d(y ) = (xi) Burr system of distributions

log G(y );v(y) = logfjd log G(y )=dy jg, a =1=9; b=11=18; c =1=6; where the functions g (:) and

G(:) must be p ositive. Di erent choices for g ( ) and G(y ) lead to di erent distributions; see Burr

(1942). Burr I and Burr IX distributions do not b elong to the exp onential family.

2 2 2

(xii) Rayleigh (>0, y>0): ( )= , ()= , d(y)=y , v(y) = log(2y ), a =1=9; b= 11=18; c =

1=6:

(xiii) Pareto ) , d(y ) = log(y ), v (y ) = 0, ( > 0, k > 0, k known, y > k ): ( ) = +1, () = (k

a =1=9; b=11=18; c =1=6: 5

(xiv) Weibull ( > 0, > 0, known, y > 0): ( ) = , ( ) = , d(y ) = y , v (y ) =

log ()+(1) log(y ), a =1=9; b=11=18; c =1=6:

( > 0, > 0, known, y > ): ( ) = 1 , ( ) = , d(y ) = log(y ), v (y ) = 0, (xv) Power

a =1=9; b=11=18; c =1=6:

(>0, 1 0): ( )= , ()=2, d(y)=jykj, v(y)=0, (xvi) Laplace

a=1=9; b=11=18; c =1=6:

(xvii) Extreme value (1 < < 1, > 0, known, 1 < y < 1): ( ) = expf=g, () =

expf=g, d(y ) = expfy=g, v(y)=y=, a =1=9; b=11=18; c =1=6:

(xviii) Truncated extreme value ( > 0, y > 0): ( ) = , ( ) = , d(y ) = expfy g 1, v (y ) = y ,

a =1=9; b=11=18; c =1=6:

2 2

(xix) Lognormal ( > 0, 1 << 1, known, y > 0): ( )= , ()=, d(y) = (log y ) =2,

v (y )=log y flog(2 )g=2, a =2=9; b=11=9; c =1=3:

(xx) Normal (>0, 1 <<1, 1

1 1=2 2

(a) known: ( )=(2) , ()= , d(y)=(y) , v(y)=flog(2 )g=2, a =2=9; b =11=9; c =

1=3:

2 2

(b) known: () = = , () = expf =(2 )g, d(y ) = y , v (y ) = fy + log(2)g=2, a = 0;

b =0; c=0:

(xxi) Inverse Gaussian (>0, >0, y>0):

1=2 2 2 3

(a) known: ( ) = , ( ) = , d(y ) = (y ) =(2 y ), v (y ) = flog(2y )g=2, a = 2=9;

b = 11=9; c =1=3:

2 3

(b) known: () = =(2 ), () = expf=g, d(y ) = y , v (y ) = =(2y ) + [log f=(2y )g]=2, a =

=(4 ); b = 5=(4 ); c =0:

(xxii) McCullagh (>1=2, 1 1, known, 0

d(y ) = log [y (1 y )=f(1 + ) 4y g], v (y )=[logfy (1 y )g]=2,

00 00 2

( ( +1) ( +0:5))

a = ;

0 0 3

36( ( +0:5) ( + 1))

00 00 2 0 0 000 000

10( ( +1) ( +0:5)) +3( ( +1) ( +0:5))( ( +1) ( +0:5))

b = ;

0 0 3

36( ( +0:5) ( + 1))

00 00 2 0 0 000 000

5( ( +1) ( +0:5)) 3( ( +1) ( +0:5))( ( +1) ( +0:5))

c = ;

0 0 3

12( ( +0:5) ( + 1))

where B(; ) is the b eta function (see McCullagh, 1989).

(>0, 0 < < 2, known, 0

v (y )=0,

00 2 00 2 0 000 00 2 0 000

r ( ) 10r ( ) +3r ()r ( ) 5r ( ) 3r ( )r ( )

a= ; b = ; c = :

0 3 0 3 0 3

36r ( ) 36r ( ) 12r ( )

where I () is the mo di ed Bessel function of rst kind and th order, and r ( )=I ()=I ( ).

0 6

(xxiv) Generalized hyp erb olic secant (=2 =2, r > 0, r known, 0 < y < 1): ( ) = , ( ) =

(sec ) , d(y )=logfy=(1 y )g= , v (y )=logfy (1 y )g=2,

2 2 2

1 (cos ) 11 14(cos ) 1 4(cos )

a = ; b = ; c = :

9r 18r 6r

Among the cases studied here, the following distributions were previously considered by Cordeiro, Ferrari

and Paula (1993): binomial, Poisson, gamma (case a), inverse Gaussian (case b) and normal (case b). The

latter case corresp onds to testing the mean of a normal distribution with known variance and is the only

one for which a = b = c = 0 which is in agreement with the fact that in this case S has an exact

null distribution. An interesting feature of the results obtained here is that several cases corresp ond to

the following constants a =1=9;b=11=18 and c =1=6, thus implying that S = S f1 (3 11S +

R R

2S )=(18n)g. Since a; b and c determine the expansion to the distribution function of the score statistic to

order O (n ) under the null hyp othesis (see Cordeiro and Ferrari, 1991), an interesting conclusion is that the

1 0

score statistics for such examples have the same distribution to order O (n ) and not only to order O (n ): It

is easy to verify that a; b and c equal the constants given ab ove when one of the following conditions holds:

(a) ( ) ( )=c or (b) ( )=c +c ; (c and c are usually equal to 1 or 1) and ( )=c =(c ); where

1 1 2 1 2 3

c ;:::;c are known scalars. These conditions are individually sucient, but not necessary.

1 4

For some of the sp ecial cases considered here, the correction has a very simple form and for some of

them the correction do es not even dep end on . In some cases, however, the correction is avery compli-

cated function of (e.g., McCullagh, von Mises and zeta distributions) requiring the evaluation of Bessel,

p olygamma and zeta functions. In order to simplify the evaluation of the Bartlett-typ e correction in such

cases, we shall derive simple approximations for a; b and c.

We start by considering the corrections that involve p olygamma functions. For large values of k ,

1 1 1 1 1

0 9

(k )= + + + + O (k ):

2 3 5 7

k 2k 6k 30k 42k

Then, for the gamma distribution ( known)

1 1 1 1 1 1 1 1

4 4 4

a = + + O (k ); b = + O (k ); c = + + O (k )

2 2 3 2 3

36k 72k 9k 18k 72k 12k 24k 24k

for large k:

0 0 0 0 2

For the McCullagh distribution, we use the equation (z +1) (z +1=2)=2 (z)4 (2z ) z ;

to obtain

1 11 1 1 3 2

3 3 3

+ O ( ); b = + + O ( ); c = + + O ( ); a =

2 2 2

9 24 9 6 3 8

for large values of : For the von Mises case, the correction involves the function r () and its rst three

derivatives. For large values of (Abramowitz and Stegun, 1970, pp.416-421)

1 1 1 25 13

r ( )=1 +

2 3 4 5

2 8 8 128 32

By making use of this expansion, we get

1 1 3 2 23 11 77 1 19

4 4 4

+ a = + + O ( ); b = + O ( ); c = + O ( )

2 3 2 3 2 3

9 8 48 9 2 48 3 8 8 7

for large . For small ,wehave (Mardia, 1972, p.63)

2 4

1 + r ( )=

2 8 48

and it then follows that

2 2 2

1 19 3

4 4 4

a = + O ( ); b = + O ( ); c = + + O ( ):

32 8 96 8 8

Finally, consider the zeta distribution and let

) (

j +1 j

(log m) (log k )

; = lim

m!1

k j +1

k =1

j =0;1;2;3, b eing Euler's constant, i.e., 0:577. It is p ossible to obtain

0 0

1 1 4 1

2 3 4 2 3 2 2 4 5

a = + ( (2 (21 +2 ) +6 +3 ) + +84 +54 +30 2+ 10 ) + O( );

1 0 1 2 1 0 3

0 0 0 0 1

9 3 9 9

7 31 1 11

2 3 4 2 3 2 2

( (2 (174 +2 ) + +6 +3 ) + 696 + 441 b = + 255

1 0 1 2 1 0 2

0 0 0 0 1

18 3 9 9

4 5

+ 85 ) + O ( );

11 1 1

2 3 4 2 3 2 2 4 5

+2( (2 (69 +2 ) +6 +3 ) + + 276 + 171 c = + 105 +35 ) +O( )

1 0 1 2 1 0 2 3

0 0 0 0 1

6 3 3

for small values of : It is p ossible to use MAPLE V (Ab ell and Baselton, 1994) to numerically evaluate the

's and simplify these expansions as

7099 15043 24134 1

2 3 4 5

+ + + O ( ); a =

9 113556 327410 42607

18068 29835 24407 11

2 3 4 5

+ + O ( ); b =

18 29741 83788 62468

1 7099 137395 17629

2 3 4 5

c = + + + O ( ):

6 18926 362472 39036

The approximations given ab ove are not exp ected to work well in practice for all values of in the

parameter space. Some of them are go o d approximations for \large" and the others, for \small" : Strictly

sp eaking, an approximated corrected statistic obtained through these approximations do es not havea

distribution to order n under H . However, if the approximations are go o d, the approximated corrected

statistic and the exact corrected statistic are approximately equal, any di erence b eing negligible. An

interesting question is then: How large or how small should b e for obtaining go o d approximations to a; b

and c? To shed some light on this issue, we present plots of a; b and c against in Figure 1 for the gamma and

McCullagh distributions. For the gamma distribution, our approximation works well when k 2, whereas

for the McCullagh distribution the approximations b ecome reliable when, say, 4. A similar graphical

analysis can b e p erformed for the other approximations obtained ab ove. This is not done to save space. For

the von Mises distribution, the approximations work well for 1:5 and 5 for small and large values of

, resp ectively. The zeta distribution requires 0:3 for the approximations to work well. 8 0.1 2 4 6 8 10 -0.1 0.08 -0.2 0.06 -0.3 0.04 -0.4 0.02 -0.5

2 4 6 8 10

0.22 2 4 6 8 10 0.21 -0.02

2 4 6 8 10 -0.04 0.19

-0.06 0.18

0.17 -0.08 0.16

0.7

0.65 2 4 6 8 10 0.6 -1.05 0.55

-1.1 0.5

-1.15 0.45

-1.2 2 4 6 8 10

0.35

Figure 1. Aproximations for the gamma ( known) and McCullagh distributions. The top two

panels show the approximations for a and b for the gamma distribution (small k ), the two panels

on the second row show the approximations for c for the gamma distribution (small k ) and for a

for the McCullagh distribution (large ), whereas approximations for b and c for the McCullagh

distribution (large ) are given in the b ottom two panels, resp ectively. Solid lines indicate exact

values and dashed lines indicate approximations. 9

4 Natural exp onential family

In Section 3, we presented the Bartlett-typ e correction for the score test in the one-parameter exp onential

family parameterized in terms of the parameter (see eq. (1)). This parameterization is quite convenient for

(0)

studying how the correction varies with : However, it is also informative to write the correction for the

exp onencial family parameterized in the natural form. By doing this, it is p ossible to write the co ecients

a; b and c in a very ellegant and simple way, and also giveaninterpretation for these co ecients.

The one-parameter natural exp onential family is written as

(y ; )= expf d(y )+v(y)g; (8)

( )

where is the natural parameter and d(y ) is the canonical statistic. Since (1) and (8) de ne a one-to-one

corresp ondence between and , the co ecients a; b and c given in (5)-(7) reduce to the corresp onding

2 002 03 000 02

expressions in (4) with = = and = = ; where = d logf ( )g=d , with primes denoting

derivatives with resp ect to : In order to give an interpretation for the co ecients a; b and c, it should

be noticed that for distributions in (8) one has that ( ) is the cumulant generator of d(y ) with the

r r 2

(r + 1)th cumulant given byd =d : Hence, and are the third and fourth standardized cumulants of

d(y ): Then, a is prop ortional to which is the usual measure of skewness of d(y ) and c is prop ortional to

5 3 and may b e viewed as a measure of nonnormality or noninverse normalityof d(y) (see McCullagh

and Cox, 1986). The co ecient b is a linear combination of a and c:

When the cumulant generating function of d(y ) can b e written as ( ) = expfc + c + c g; where

0 1 2

c ; c and c are arbitrary constants, we have that a = b = c = 0, that is, the Bartlett-typ e correction

0 1 2

vanishes. This is the case for the normal distribution with known variance. On the other hand, if

1=(9k )

( ) = expfc g(k + c ) c ; (9)

0 1 2

where c ;c and c are any real numb ers and k>0, wehave that a = k , b = 11k=2 and c =3k=2. For 13

0 1 2

of the cases considered in the previous section, a, b and c are constant with a>0, and it is p ossible to show

that their cumulant generating functions satisfy (9) and that the ab ove relations hold.

Next, we shall derive simple expressions for the co ecients a; b and c in terms of the mean of the

canonical statistic d(y ) for some sub classes of distributions in the one-parameter exp onential family. First,

it should b e noticed that there is, up to a linear tranformation, only one distribution in (8) with a sp eci ed

variance function (Jrgensen, 1987) and hence uniquely characterizes any distribution in (8). The following

families of variance functions are considered (c ;:::;c 2 IR):

0 3

0 p

(1) Power variance function : = =c ; for p 0 and p 1;c >0:

0 0

2 p2 p2 p2

p p(4p +3) p(3 p)

a = ; b = ; c = :

36c 36c 12c

0 0 0

Here, p =0;1;2;3 for the normal, Poisson, gamma and inverse Gaussian distributions, resp ectively.

For other distributions in this class, see Jrgensen (1987).

0 2

= c + c + c ; where a; b and c are given by the corresp onding : (2) Quadratic variance function

0 1 2

expressions for the cubic variance function (see b elow) with c =0: Morris (1982) showed that there

3 10

are only six distributions in this class, namely: normal (c = ; c = c =0;>0; known), Poisson

0 1 2

(c = c =0;c = 1), binomial (c =0;c =1;c =1=m; m 2 IN , m known), negative binomial

0 2 1 0 1 2

(c = 0; c = 1; c = 1= ; > 0; known), gamma (c = c = 0; c = 1=k ; k > 0 known) and

0 1 2 0 1 2

generalized hyp erb olic secant(c =r;c =0;c =1=r;r>0 known).

0 1 2

0 2 3

(3) Cubic variance function : = c + c + c + c :

0 1 2 3

0 1 2 2 2

a = (36 ) fc 4c c + c (6c +8c +9c )g+4c ;

0 2 3 1 2 3 2

0 1 2 2 3

b = (36 ) f7(c 4c c )+c (18c 24c 38c 45c )g22c ;

0 2 3 0 1 2 3 2

0 1 2 2

c =(6 ) fc 4c c c (9c +3c +c )g+2c :

0 2 3 0 1 2 2

Letac and Mora (1990) showed that there are only six distributions in this class with c 6= 0, namely:

Ab el (c =0;c =1;c =2=p; c =1=p ;p>0 known), Takacs (c =0;c =1;c =(2m+1)=(mp);

0 1 2 3 0 1 2

2 2

c =(m+1)=(mp ); p>0 and m>0 are known), strict arcsine (c = c =0;c =1;c =1=p ;

3 0 2 1 3

2 2

p known), large arcsine (c = 0; c = 1; c = 2=(mp); c = (1 + m )=(mp) ; p > 0 and m > 0

0 1 2 3

are known), Ressel (c = c = 0; c = 1=p; c = 1=p ; p > 0 p known) and inverse Gaussian

0 1 2 3

(c = c = c =0;c =1= ; > 0; known).

0 1 2 3

0 1=2

(4) Bab el class : = c +(c +c ) ; where is a p olynomial of degree smaller than 3 which is not a

0 1 2

p erfect square. For this class a; b and c are easily obtained using MATHEMATICA but the resulting

expressions are very cumb ersome and consequently they are not presented here.

Acknowledgement

We thank an anonymous referee for suggestions that led to improvements in our pap er. The partial nancial

supp ort of CNPq and CAPES/Brazil is also gratefully acknowledged.

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