Estimating multivariate density function of mixed measurement error data

Abstract

Density estimation has been a long frontline research area in nonparametric smoothing. However, real applications oftentimes see the data contaminated with different types of measurement errors. Further data analysis, therefore, should take care of these errors to have a reliable statistical inference procedure. In this proposal, nonparametric density estimation for the data contaminated super-smooth, ordinary-smooth, Berkson measurement errors will be thoroughly investigated. Classical kernel and deconvolution kernel smoothing are used as building blocks to construct the estimators. In the first part, we propose a nonparametric mixed kernel estimator for a multivariate density function and its derivatives when the data are contaminated with different sources of measurement errors. The proposed estimator is a mixture of the classical and the deconvolution kernels, accounting for the error-free and error- prone variables, respectively. Large sample properties of the proposed nonparametric estimator, including the order of the mean squares error, the consistency, and the asymptotic normality, are discussed. The optimal convergence rates among all nonparametric estimators for different measurement error structures are derived, and it is shown that the proposed mixed kernel estimators achieve the optimal convergence rate. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. In the second part, we consider the nonparametric estimation for the joint density function of two random variables, when one variable is contaminated with Berkson measurement error, and another variable can be observed directly. Two estimators are proposed with or without applying the kernel smoothing for the data with Berkson measurement error. Mean squared errors are calculated for both estimators. Large sample properties, including weak consistencies, strong consistencies, uniform strong consistencies in probability, and asymptotic normality are derived. In addition, we develop a method for bandwidth selection in the kernel estimate of the probability density using the least squares cross-validation method. The performance of this method is further assessed by a simulation study.