An alternative derivation of weak convergence concerning quasi-likelihood estimation with a small-sample correction for simultaneous testing

Abstract

AbstractOften arises in counting data analysis that both violation of distributional assumption and large-scale over-dispersion substantially impair the validity of the methods for multiple comparisons. For over-dispersed data fitting to the generalized linear models, we describe the simultaneous inference method in assessing a sequence of estimable functions based on the root using the quasi-likelihood estimation of the regression coefficients. A new method for deriving the limiting distributions of the score function and the root under a list of mild regularity conditions is presented. This approach has a close connection to the asymptotic normality of the root in general linear models that it provides a heuristic analogy for classroom presentation. Hence, researchers can routinely estimate quantiles based on the limiting distribution of the root for simultaneous inference. We apply the percentile bootstrap method to estimate the quantiles as a resampling-based alternative. As will be shown, the simultaneous test based on both the approximation methods above is anti-conservative in designs with small sample sizes. We propose the simultaneous testing method using Efron's bias-corrected percentile bootstrapping procedure as an improvement. In small-sample designs, we demonstrate through the simulation study that the proposed method provides a viable alternative to the large-sample and the percentile bootstrap approximation methods. Moreover, the proposed method persists in controlling the familywise error rate in simultaneous testing for highly over-dispersed data from substantially small-sample designs, where the percentile-t bootstrap method provides a liberal test.Keywords: Simultaneous inferencequasi-likelihood functionspercentile bootstrapbias-corrected percentile bootstraprobustness of validityover-dispersion AcknowledgmentsThe author would like to thank two anonymous referees for providing insightful comments, which have helped the author improve the article. The author would like to thank Dr. Mei-Qin Chen at The Citadel for a discussion helpful to the proof of Theorem 2.2.Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 The subindex i in Sections 2 and 3 is in association to the subindices i1i2i3i4 with i1=1,2,ı2=1,2,ı3=1,2,3,4, and i4=1,2,3 in Section 8.1 in order.2 The subindex i in Sections 2 and 3 is in association to the subindices i1i2 with i1=1,…,4 and i2=1,…,ni1 in Section 8.2 in order.