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Improved Score Tests for One{Parameter Exp Onential

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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). 1 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 0 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 1 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 1 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. 2 Both unmo di ed statistics havea distribution to rst order under the null hyp othesis H , where q denotes 0 q 2 the numb er of restrictions imp osed byH . On the other hand, the mo di ed statistics havea distribution 0 q 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 ), 0 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 1 (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, 0 () ( )= ; (2) 0 () () 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 (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 P n 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 0 that the mo di ed score statistic n o 1 2 S = S 1 (c + bS + aS ) ; (3) R R R R n 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) 1 (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 P 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 36 02 002 0 0 00 00 02 002 0 02 000 02 0 000 +11 10 3 +3 ; (6) b = 03 03 36 02 002 0 0 00 00 02 002 0 02 000 02 0 000 4 +5 +3 3 c = : (7) 03 03 12 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 0 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).
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