ON IMPROVING THE χ2 APPROXIMATION OF SCORE TESTS IN LOCATION-SCALE NONLINEAR MODELS

Abstract

ABSTRACT This paper considers the issue of testing parameters based on score tests in location-scale nonlinear models assuming known scale parameter, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. Significance levels derived from the score statistic can be misleading, particularly in small samples. We obtain, in matrix notation, a Bartlett-type correction formula to improve score tests in this class of models, thus generalizing results by Ferrari and Cordeiro (Ferrari, S.L.P.; Cordeiro, G.M. Corrected Score Tests for Exponential Family Nonlinear Models. Statist. Probab. Lett. 1996, 26, 7–12) and Ferrari and Arellano-Valle (Ferrari, S.L.P.; Arellano-Valle, R.B. Modified Likelihood Ratio and Score Tests in Linear Regression Models Using the t Distribution. Braz. J. Probab. Statist. 1996, 10, 15–33.). Our results are used to obtain a corrected score statistic for testing that a subset of the nonlinear regression coefficients equals a given vector of constants. The corrected score statistic is distributed as chi-squared with an error of order , n being the sample size, whereas the original score statistic has a chi-squared distribution with error of order . We show that the formulae derived for the Bartlett-type corrections generalize a number of previously published results. We present simulation results comparing the sizes of the usual score tests and their modified versions for linear and nonlinear regression models when the scale parameter is known or it is replaced by a consistent estimate. The paper also provides a numerical comparison of the sizes of analytical corrections for score and likelihood ratio tests and bootstrap tests.