### Influence Functions for Censored Data

Stanford University, 1979

Abstract

It is frequently necessary in practice to analyze data sets that have some censored observations. For example, in reliability studies some items may not break when subjected to the maximum available stress. In medical studies it is often necessary to analyze the data before all the subjects in the study have failed. In addition, some subjects in a medical study are lost to follow-up. Methods of estimation have been developed to use the information available in these censored data points. In particular, the Kaplan-Meier estimate is the censored data analogue of the empirical cumulative distribution function. In this thesis we consider influence curves of censored data estimators. The influence curve is a function used in the theory of robust estimation both as a qualitative description of the robustness of an estimator and as a tool for calculating its asymptotic variance.

First the influence curve of the Kaplan-Meier estimate is calculated. It is shown that the influence curve provides an alternative derivation of the asymptotic variance of this estimate. The regularity conditions required to make this derivation rigorous are verified in the appendix. Secondly a chain rule for influence curves is established and is used to calculate the influence curves of censored data estimators that are functions of the Kaplan-Meier estimate. The robustness properties and asymptotic variances of these estimates follow directly. Finally, some examples of this unified approach to calculating variances are given. In particular, the theory developed for M-estimation and L-estimation is extended to the censored data case.