Let \(f_{Y|X,Z}(y|x,z)\) be the conditional probability function of Y given (X, Z), where Y is the scalar response variable, while (X, Z) is the covariable vector. This paper proposes a robust model selection criterion for \(f_{Y|X,Z}(y|x,z)\) with X missing at random. The proposed method is developed based on a set of assumed models for the selection probability function. However, the consistency of model selection by our proposal does not require these models to be correctly specified, while it only requires that the selection probability function is a function of these assumed selective probability functions. Under some conditions, it is proved that the model selection by the proposed method is consistent and the estimator for population parameter vector is consistent and asymptotically normal. A Monte Carlo study was conducted to evaluate the finite-sample performance of our proposal. A real data analysis was used to illustrate the practical application of our proposal.

设\(f_{Y|X,Z}(y|x,z)\)是给定 ( X ,  Z ) Y的条件概率函数，其中Y是标量响应变量，而 ( X ,  Z ) 是协变量向量。本文针对\(f_{Y|X,Z}(y|x,z)\)提出了一个鲁棒的模型选择准则，其中X随机丢失。所提出的方法是基于一组用于选择概率函数的假设模型而开发的。然而，我们提议的模型选择的一致性并不要求正确指定这些模型，而只要求选择概率函数是这些假设的选择性概率函数的函数。在一定条件下，证明了所提方法的模型选择是一致的，总体参数向量的估计量是一致且渐近正态的。进行了蒙特卡罗研究以评估我们提议的有限样本性能。一个真实的数据分析被用来说明我们建议的实际应用。