Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs. (English) Zbl 0957.62056

Chen, Kani; Hu, Inchi; Ying, Zhiliang

Ann. Stat. 27, No. 4, 1155-1163 (1999).

Summary: Strong consistency for maximum quasi-likelihood estimators of regression parameters in generalized linear regression models is studied. Results parallel to the elegant work of Lai, Robbins and Wei and Lai and Wei on least squares estimation under both fixed and adaptive designs are obtained. Let \(y_1, \dots, y_n\) and \(x_1, \dots, x_n\) be the observed responses and their corresponding design points \((p\times 1\) vectors), respectively. For fixed designs, it is shown that if the minimum eigenvalue of \(\sum x_ix_i'\) goes to infinity, then the maximum quasi-likelihood estimator for the regression parameter vector is strongly consistent. For adaptive designs, it is shown that a sufficient condition for strong consistency to hold is that the ratio of the minimum eigenvalue of \(\sum x_ix_i'\) to the logarithm of the maximum eigenvalues goes to infinity. Use of the results for the adaptive design case in quantal response experiments is also discussed.

MSC:

62J12	Generalized linear models (logistic models)
62F12	Asymptotic properties of parametric estimators
62J05	Linear regression; mixed models

Keywords:

exponential family; strong consistency; martingale difference; maximum quasi-likelihood estimators; fixed designs; adaptive designs

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