Nonparametric Likelihood Ratio Estimation of Probabilities for Truncated Data

Abstract

We consider the problem of interval estimation of probabilities for randomly truncated data. The best-known method is based on the normal approximation of the product-limit estimator and has a drawback that it may produce intervals containing impossible values outside the range [0, 1]. Its small-sample performance is also not satisfactory. In this article we investigate an alternative approach that derives confidence intervals directly from a conditional nonparametric likelihood ratio. The method is an exact nonparametric analog of the classical parametric likelihood ratio theory, with the parameter space now being the family of all distributions. It has an appealing property that the resulting confidence intervals are always subintervals of [0, 1]. It also demonstrated a better small-sample performance in our simulation studies. This approach is generalized to obtain confidence intervals for the ratio of two probabilities, make joint inferences on any finite number of probabilities, and test goodness of fit of a given distribution function. The small-sample performances of three different methods are investigated in a Monte Carlo study. An illustration is also given using the Centers for Disease Control's transfusion related acquired immune deficiency syndrome data.

Key Words:

• Acquired immune deficiency syndrome
• Empirical likelihood
• Goodness of fit
• Human immunodeficiency virus
• Incubation period
• Interval estimation
• Product limit estimate
• Truncation