Asymptotic Properties of Estimators in a Model of Life Data with Warnings

Abstract

We consider a model where elements of a single type are life tested. All elements are observed up to the time of their failures or censorings. Three types of events are possible to observe during life testing for each element: failure, censoring, and warning, where a warning can only be observed before a failure or before censoring has occurred. It is essential to know if warnings influence subsequent failures. Two subsets of data are simultaneously considered: the first consisting of only the times of the first occurrences of failure, censoring, or warning, and the second consisting of the times for those elements where warnings occurred before failures or censorings. The first subset belongs to the competing risks model, and the second consists of left-truncated data. Estimators of the cumulative hazard function before and after warnings are derived and proved to be consistent, with asymptotic normal distributions. A null hypothesis where the cumulative hazard functions before and after warnings are proportional and a corresponding alternative hypothesis that they are not proportional are defined. Under this null hypothesis an estimator for the constant of proportionality is derived and showed to be strongly consistent. Martingale techniques are used and numerical examples are provided.

Key Words:

• Consistency
• Cumulative hazard function
• Independent competing risks model
• Left truncation
• Nonparametric estimators
• Proportional hazards