Skovgaard’s adjustment to likelihood ratio tests in exponential family nonlinear models

Abstract

Likelihood ratio tests can be substantially size distorted in small- and moderate-sized samples. In this paper, we apply Skovgaard’s [Skovgaard, I.M., 2001. Likelihood asymptotics. Scandinavian Journal of Statistics 28, 3–32] adjusted likelihood ratio statistic to exponential family nonlinear models. We show that the adjustment term has a simple compact form that can be easily implemented from standard statistical software. The adjusted statistic is approximately distributed as χ 2 with high degree of accuracy. It is applicable in wide generality since it allows both the parameter of interest and the nuisance parameter to be vector-valued. Unlike the modified profile likelihood ratio statistic obtained from Cox and Reid [Cox, D.R., Reid, N., 1987. Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B 49, 1–39], the adjusted statistic proposed here does not require an orthogonal parameterization. Numerical comparison of likelihood-based tests of varying dispersion favors the test we propose and a Bartlett-corrected version of the modified profile likelihood ratio test recently obtained by Cysneiros and Ferrari [Cysneiros, A.H.M.A., Ferrari, S.L.P., 2006. An improved likelihood ratio test for varying dispersion in exponential family nonlinear models. Statistics and Probability Letters 76 (3), 255–265].