Kernel density estimation with doubly truncated data

Carla Moreira,Jacobo De Uña-Álvarez

doi:10.1214/12-ejs683

Abstract

In some applications with astronomical and survival data, doubly truncated data are sometimes encountered. In this work we introduce kernel-type density estimation for a random variable which is sampled under random double truncation. Two different estimators are considered. As usual, the estimators are defined as a convolution between a kernel function and an estimator of the cumulative distribution function, which may be the NPMLE [2] or a semiparametric estimator [9]. Asymptotic properties of the introduced estimators are explored. Their finite sample behaviour is investigated through simulations. Real data illustration is included.

Highlights

Truncated data play an important role in the statistical analysis of survival times as well as in other fields like astronomy or economy
Because we are only aware of individuals with event times in the observational window, the inference with truncated data is based on sampling information from a conditional distribution
In this paper we have introduced kernel density estimation for a variable which is observed under random double truncation

Summary

Introduction

Truncated data play an important role in the statistical analysis of survival times as well as in other fields like astronomy or economy. Woodroofe [19] investigated the properties of the nonparametric maximum-likelihood estimator (NPMLE) of the distribution function (df) with left-truncated data, see [3]. Let X∗ be the random variable of ultimate interest, with df F , and assume that it is doubly truncated by the random pair (U ∗, V ∗) with joint df T , where U ∗ and V ∗ (U ∗ ≤ V ∗) are the left and right truncation variables respectively This means that the triplet (U ∗, X∗, V ∗) is observed if and only if U ∗ ≤ X∗ ≤ V ∗, while no information is available when X∗ < U ∗ or X∗ > V ∗. In Subsection 2.3 we consider the problem of estimating the density function on the basis of these two cumulative estimators

The NPMLE of the cumulative df

The semiparametric estimator of the cumulative df

The density estimators

Simulation study

Real data illustration

Conclusions and final discussion