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$\sum\limits_{i=1}^{n}Z_iZ_i^{\prime}$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\{$Z_i,i\ge1$\}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$\underline{\lambda}_n\rightarrow\infty$, $(\overline{\lambda}_n)^{1/2+\delta}=O(\underline{\lambda}_n)$（对某$\delta>0$)以及$\underline{\lambda}_n\rightarrow\infty$, $\overline{\lambda}_n=O(\underline{\lambda}_n)$.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$\sum\limits_{i=1}^{n}Z_iZ_i^{\prime}$ resp., and$\hat{\beta}_n$ denote the maximum likelihood estimator of $\beta_0$. It is shown in 1] that, when \{$Z_i,i\ge1$\}is bounded, the sufficient conditions for strong consistency of $\hat{\beta}_n$ are as follows:$\underline{\lambda}_n\rightarrow\infty$, $(\overline{\lambda}_n)^{1/2+\delta}=O(\underline{\lambda}_n)$(for some $\delta>0$) with natural link function, and $\underline{\lambda}_n\rightarrow\infty$,$\overline{\lambda}_n=O(\underline{\lambda}_n)$ 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.