论文标题
在广义线性混合模型中M估计器的指数一致性
Exponential Consistency of M-estimators in Generalized Linear Mixed Models
论文作者
论文摘要
广义线性混合模型是用于分析群集数据的强大工具，其中未知参数是通过最大似然和受限的最大似然性过程经典（并且最常见的）估计的。但是，由于已知基于可能性的程序对异常值高度敏感，因此M估计器已成为流行，作为在可能的数据污染下获得可靠估计值的一种手段。在本文中，我们证明，为了足够平滑的一般损耗函数，定义了通用线性混合模型中M估计量的一般损耗函数，则估计回归系数与真实回归系数之间偏差的尾巴概率具有指数结合。这意味着在适当的假设下，这些M估计器的一致性呈指数级，从单变量到多变量响应的现有指数一致性结果。我们已经在线性和逻辑混合模型的设置中进一步说明了最大似然估计器和强大的最小密度差异估计量的特殊示例和强大的最小密度差异估计量的特殊示例，这是一个理论上的示例，并将其进一步说明了这一理论结果。
Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However, since the likelihood based procedures are known to be highly sensitive to outliers, M-estimators have become popular as a means to obtain robust estimates under possible data contamination. In this paper, we prove that, for sufficiently smooth general loss functions defining the M-estimators in generalized linear mixed models, the tail probability of the deviation between the estimated and the true regression coefficients have an exponential bound. This implies an exponential rate of consistency of these M-estimators under appropriate assumptions, generalizing the existing exponential consistency results from univariate to multivariate responses. We have illustrated this theoretical result further for the special examples of the maximum likelihood estimator and the robust minimum density power divergence estimator, a popular example of model-based M-estimators, in the settings of linear and logistic mixed models, comparing it with the empirical rate of convergence through simulation studies.
