Dimension reduction estimation for probability density with data missing at random when covariables are present

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• 作者：Jianqiu Deng ; Qihua Wang ; ^{qhwang@amss.ac.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Kernel density estimation ; Kernel regression ; Dimension reduction ; Missing at random ; Asymptotic normality
• 刊名：Journal of Statistical Planning and Inference
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：181
• 期：Complete
• 页码：11-29
• 全文大小：707 K

文摘

We develop dimension reduction estimating methods for probability density with data missing at random in the presence of covariables. In this paper, we propose two families of sufficient dimension reduction based nonparametric density estimators by modifying the regression calibration estimator and the inverse probability weighted estimator due to Wang (2008). The proposed methods overcome the challenges faced with high dimensional covariates: model specification and curse of dimensionality. The curse of dimensionality is overcome by replacing the covariables X_i$X_i$ in the regression calibration estimator and the inverse probability weighted estimator, respectively, with a root-n$n$ consistent estimator Image" height="18" width="33" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0378375816301021-si13.gif">$\hat{S}\left(X_i\right)$ of a score S(X_i)$S\left(X_i\right)$ for i=1,2,…,n$i=1,2,\dots ,n$. Three different scores S(⋅)$S(\cdot )$ are found by dimension reduction techniques. It is shown that the two families of proposed estimators are asymptotically normal, respectively, by taking three different scores. The asymptotic variances are the same when the same score is taken. With different scores, the asymptotic variances are different. A comparison for the two families of density estimators is made by taking different scores. Simulations are carried out to demonstrate the excellent performances of the proposed methods. A real data analysis is used to illustrate our methods.