Method of medians for lifetime data with Weibull models

Abstract

The Weibull family of distributions is frequently used in failure time models. The maximum likelihood estimator is very sensitive to occurrence of upper and lower outliers, especially when the hazard function is increasing. We consider the method of medians estimator for the two-parameter Weibull model. As an M-estimator, it has a bounded influence function and is highly robust against outliers. It is easy to compute as it requires solving only one equation instead of a pair of equations as for most other M-estimators. Furthermore, no assumptions or adjustments are needed for the estimator when there are some possibly censored observations at either end of the sample. About 16 per cent of the largest observations and 34 per cent of the smallest observations may be censored without affecting the calculations. We also present a simple criterion to choose between the maximum likelihood estimator and the method of medians estimator to improve on the finite-sample efficiency of the Weibull model. Robust inference on the shape parameter is also considered. The usefulness with contaminated or censored samples is illustrated by examples on three lifetime data sets. A simulation study was carried out to assess the performance of the proposed estimator and the confidence intervals of a variety of contaminated Weibull models.