LIMIT THEOREMS FOR ASYMPTOTICALLY MINIMAX ESTIMATION OF A DISTRIBUTION WITH INCREASING FAILURE RATE UNDER A RANDOM MIXED CENSORSHIP/TRUNCATION MODEL

ABSTRACT

The search for optimal non-parametric estimates of the cumulative distribution and hazard functions under order constraints inspired at least two earlier classic papers in mathematical statistics: those of Kiefer and Wolfowitz[1] and Grenander[2] respectively. In both cases, either the greatest convex minorant or the least concave majorant played a fundamental role. Based on Kiefer and Wolfowitz's work, Wang3-4 found asymptotically minimax estimates of the distribution function F and its cumulative hazard function Λ in the class of all increasing failure rate (IFR) and all increasing failure rate average (IFRA) distributions. In this paper, we will prove limit theorems which extend Wang's asymptotic results to the mixed censorship/truncation model as well as provide some other relevant results. The methods are illustrated on the Channing House data, originally analysed by Hyde.5-6

Keywords:

• Asymptotic approximation
• Cumulative hazard and distribution functions
• Greatest convex minorant
• Increasing failure rate
• Kaplan–Meier estimator
• Nelson–Aalen estimator

Acknowledgments