独立与相依情形下加权广义线性模型选择的重对数律和强一致性

The Law of Iterated Logarithm and Strong Consistency for the Selection of Weighted Generalized Linear Models in Independent and Dependent Cases

在温和条件下,考虑相依响应情形(ρ-混合或m-相依),针对一般广义线性模型(GLM)任意改变其参数权重构造出加权GLM,给出了参数的最大似然估计(MLE),并推导了加权后模型的重对数律(LIL)。应用独立情形下强极限理论证明了任意修正模型与全模型的对数似然之差的渐近结果。借助惩罚加权对数似然函数技术,基于相依LIL证明了模型选择准则的强一致性,并推导出若惩罚项的阶数介于O(log log n)与O(n)之间时,则该准则选取最简单修正模型几乎是必然的。

Under mild conditions,considering the dependent response situation(ρ-mixing or m-dependent),a weighted GLM is constructed for the general generalized linear model(GLM)by arbitrarily changing its parameter weights.The maximum likelihood estimation(MLE)of the parameters is given.The law of logarithm(LIL)of the weighted model is derived.The asymptotic result of the difference between the log-likelihood of the arbitrarily modified model and the full model is proved by the strong limit theory in the independent case.With the help of penalty weighted log-likelihood function technology,the strong consistency of the model selection criteria is proved based on the dependent LIL,and if the order of the penalty term is between O(log log n)and O(n),It is almost inevitable that the criterion chooses the simplest modification model.