Open Access
February 1999 Nonparametric quasi-likelihood
Jeng-Min Chiou, Hans-Georg Müller
Ann. Statist. 27(1): 36-64 (February 1999). DOI: 10.1214/aos/1018031100

Abstract

The quasi-likelihood function proposed by Wedderburn broadened the scope of generalized linear models by specifying the variance function in-stead of the entire distribution. However, complete specification of variance functions in the quasi-likelihood approach may not be realistic. We define a nonparametric quasi-likelihood by replacing the specified variance function in the conventional quasi-likelihood with a nonparametric variance function estimate. This nonparametric variance function estimate is based on squared residuals from an initial model fit. The rate of convergence of the nonparametric variance function estimator is derived. It is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi-likelihood estimates obtained under correct specification of the variance function, thus establishing the asymptotic efficiency of the nonparametric quasi-likelihood estimates. We propose bandwidth selection strategies based on deviance and Pearson’s chi-square statistic. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method can improve upon extended quasi-likelihood or pseudo-likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters. We illustrate the proposed methods with applications to dental data and cherry tree data.

Citation

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Jeng-Min Chiou. Hans-Georg Müller. "Nonparametric quasi-likelihood." Ann. Statist. 27 (1) 36 - 64, February 1999. https://doi.org/10.1214/aos/1018031100

Information

Published: February 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0978.62056
MathSciNet: MR1701100
Digital Object Identifier: 10.1214/aos/1018031100

Subjects:
Primary: 62G07 , 62G10 , 62J12

Keywords: Bandwidth selection , deviance , efficiency , extended quasi-likelihood , pseudo-likelihood , semiparametric modeling , smoothing , variance function

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 1999
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