Bivariate Density Estimation with Randomly Truncated Data

In this study bivariate kernel density estimators are considered when a component is subject to random truncation. In bivariate truncation models one observes the i.i.d. samples from the triplets (T, Y, X) only if T[less-than-or-equals, slant]Y. In this set-up, Y is said to be left truncated by T and T is right truncated by Y. We consider the estimation of the bivariate density function of (Y, X) via nonparametric kernel methods where Y is the variable of interest and X a covariate. We establish an i.i.d. representation of the bivariate distribution function estimator and show that the remainder term achieves an improved order of O(n-1 ln n), which is desirable for density estimation purposes. Expressions are then provided for the bias and the variance of the estimators. Finally some simulation results are presented.