Nonnull Asymptotic Distributions of Three Classic Criteria in Generalised Linear Models

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY This paper develops simple formulae for the asymptotic expansions up to order n-of the distributions of the likelihood ratio, Wald and score statistics in generalised linear models, under a sequence of Pitman alternatives. The powers of all three criteria, which are equivalent to first order, are compared under specific conditions based on second order approximations. The asymptotic distributions of all three statistics are obtained for testing a subset of linear parameters when the dispersion parameter is known, for testing the dispersion parameter and for testing the whole set of linear parameters assuming that the dispersion parameter is unknown. The formulae derived are simple enough to be used analytically to obtain closed-form expressions for these expansions in special models with a Fisher information matrix in closed form.

Local power of some tests in exponential family nonlinear models

“…It should be noticed that the above expressions depend on the model only through f and the rank of the matrix X Ã ; they do not involve the unknown parameter b. The coefficients b ik (i= 1,2,3 and k= 0,1,2,3) are exactly the same given by Cordeiro et al (1994) for GLMs. Therefore, the nonnull asymptotic expansions of the statistics S 1 , S 2 and S 3 for testing H 0 : f ¼ f 0 are the same for any nonlinear regression structure with the same p. The coefficients b 4k (k =0,1,2,3), which are also valid for GLMs, seem to be a new result.…”

“…The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the dispersion parameter, thus generalising the results given in Cordeiro et al (1994) andFerrari et al (1997). We also present Monte Carlo simulations in order to compare the finite-sample performance of these tests.…”

“…The coefficients b ik 's (i=1,2,3,4 and k=0,1,2,3) can be written, after extensive algebra, as b 11 =b 11 L +b 11 and k= 0,1,2,3) become equal to the corresponding coefficients developed for GLMs (Cordeiro et al, 1994) by replacing Z Ã and Z Ã 1 by Z and Z 1 , respectively. On the other hand, the coefficients b ik NL (i=1,2,3 and k= 0,1,2,3) may be regarded as the amount of nonlinearity in the likelihood ratio, Wald and Rao score statistics induced by the systematic component of the model, since they vanish for linear models.…”

“…According to Cordeiro et al (1994), the GLMs for which this equality holds have the link function defined by Z l ¼ R V ¼ 1, . .…”

“…Also, the equality P 1 ¼ P 2 ¼ P 3 ¼ P 4 holds only for normal models with identity link function. Power comparisons among the likelihood ratio, Wald and score tests in GLMs are not presented here and can be found in Cordeiro et al (1994). Now, we present an analytical comparison among the local powers of the four tests for testing the null hypothesis…”