Abstract
A class of density estimators based on observed incomplete data are proposed. The method is to use a conditional kernel, defined as the expectation of a given kernel for the complete data conditioning on the observed data, to construct the density estimator. We study such kernel density estimators for several commonly used incomplete data models and establish their basic asymptotic properties. Some characteristics different from the classical kernel estimators are discovered. For instance, the asymptotic results of the proposed estimator do not depend on the choice of the kernel $k(\cdot )$. Simulation study is conducted to evaluate the performance of the estimator and compared with some exising methods.
Highlights
Estimating the density function is one of the fundamental problems in nonparametric statistics
At the end of this section, we summarize the results for general incomplete data models
After studying the conditional kernel density estimation for the above five incomplete data models, we summarize the results for general incomplete data models as in Theorem 6, without proof
Summary
Estimating the density function is one of the fundamental problems in nonparametric statistics. Common models of incomplete/missing data include the left truncated, right censoring, doubly censoring, interval censoring of types I (or the current status data) and II, multiplicative censoring and convolution model For these types of data, estimates of the survival function, distribution function and density function has been extensively explored. Other common methods for density estimation in incomplete data models include the nonparametric maximum likelihood estimator (NPMLE), or taking the left/right derivative of an estimated distribution function. K(·|F, h, Y ) involves the underlying unknown distribution F , we plug in an estimator Fn based on the observed data such as NPMLE This kernel estimator has some features different from the other methods, such as its asymptotic distribution does not depend on the sujectively chosen kernel, this is in contrast to most existing methods using kernel smoothing.
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