Bandwidth selection for kernel density estimation with doubly truncated data

Abstract

In this work we introduce and compare several bandwidth selection procedures for kernel density estimation of a random variable that is sampled under random double truncation. The work is motivated by the fact that this type of incomplete data is often encountered in studies in astronomy and medicine. The bandwidth selection procedures we study are appropriate modifications of the normal reference rule, the least squares cross-validation procedure, two types of plug-in procedures, and a bootstrap based method. The methods are first shown to work from a theoretical point of view. A simulation study is then carried out to assess the finite sample behavior of these five bandwidth selectors. We also illustrate the use of the various practical bandwidth selectors by means of data regarding the luminosity of quasars in astronomy.