广义线性回归极大似然估计的强相合性

设有该文第1节所描述的广义线性回归模型,以$underline{lambda}_n$和$overline{lambda}_n$分别记$sumlimits_{i=1}^{n}Z_iZ_i^{prime}$的最小和最大特征根,$hat{beta}_n$记$beta_0$的极大似然估计.在文献[1]中,当{$Z_i,ige1$}有界时得到$hat{beta}_n$强相合的充分条件,在自然联系和非自然联系下分别为$underline{lambda}_nrightarrowinfty$, $(overline{lambda}_n)^{1/2+delta}=O(underline{lambda}_n)$（对某$delta>0$)以及$underline{lambda}_nrightarrowinfty$, $overline{lambda}_n=O(underline{lambda}_n)$.作者将后一结果改进为只要求$(overline{lambda}_n)^{1/2+delta}=O(underline{lambda}_n)$,从而与自然联系情况下的条件达到一致.

Strong Consistency of the Maximum Likelihood Estimator in Generalized Linear Models

Assuming the generalized linear model as described in S1, let $underline{lambda}_n$and $overline{lambda}_n$ denote the minimum and maximum eigentvalues of$sumlimits_{i=1}^{n}Z_iZ_i^{prime}$ resp., and$hat{beta}_n$ denote the maximum likelihood estimator of $beta_0$. It is shown in [1] that, when {$Z_i,ige1$}is bounded, the sufficient conditions for strong consistency of $hat{beta}_n$ are as follows:$underline{lambda}_nrightarrowinfty$, $(overline{lambda}_n)^{1/2+delta}=O(underline{lambda}_n)$(for some $delta>0$) with natural link function, and $underline{lambda}_nrightarrowinfty$,$overline{lambda}_n=O(underline{lambda}_n)$ with nonnatural link function resp.. In this paper, the authors improvethe latter result by showing that even in the case of nonnatural link function, the condition$(overline{lambda}_n)^{1/2+delta}=O(underline{lambda}_n)$ remains to be sufficient.