Diao Guoqing, Ning Jing, Qin Jing
George Mason University, USA.
Int J Biostat. 2012 Jun 27;8(1):/j/ijb.2012.8.issue-1/1557-4679.1372/1557-4679.1372.xml. doi: 10.1515/1557-4679.1372.
In the statistical literature, the conditional density model specification is commonly used to study regression effects. One attractive model is the semiparametric density ratio model, under which the conditional density function is the product of an unknown baseline density function and a known parametric function containing the covariate information. This model has a natural connection with generalized linear models and is closely related to biased sampling problems. Despite the attractive features and importance of this model, most existing methods are too restrictive since they are based on multi-sample data or conditional likelihood functions. The conditional likelihood approach can eliminate the unknown baseline density but cannot estimate it. We propose efficient estimation procedures based on the nonparametric likelihood. The nonparametric likelihood approach allows for general forms of covariates and estimates the regression parameters and the baseline density simultaneously. Therefore, the nonparametric likelihood approach is more versatile than the conditional likelihood approach especially when estimation of the conditional mean or other quantities of the outcome is of interest. We show that the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in practical settings. A real example is used for illustration.
在统计文献中，条件密度模型设定常用于研究回归效应。一种有吸引力的模型是半参数密度比模型，在此模型下，条件密度函数是一个未知基线密度函数与一个包含协变量信息的已知参数函数的乘积。该模型与广义线性模型有着自然的联系，并且与有偏抽样问题密切相关。尽管此模型具有吸引人的特征和重要性，但大多数现有方法限制过多，因为它们基于多样本数据或条件似然函数。条件似然方法可以消除未知基线密度，但无法对其进行估计。我们提出基于非参数似然的有效估计程序。非参数似然方法允许协变量采用一般形式，并同时估计回归参数和基线密度。因此，非参数似然方法比条件似然方法更具通用性，尤其是在对条件均值或结果的其他量进行估计时。我们证明了非参数最大似然估计量是一致的、渐近正态的且渐近有效的。模拟研究表明，所提出的方法在实际情况下表现良好。文中用一个实际例子进行说明。
