On Estimating a Survival Curve Subject to a Uniform Stochastic Ordering Constraint

Abstract

If F and G are cumulative distribution functions on [0, ∞) governing the lifetimes of specific systems under study, and if F and G are their corresponding survival functions, then F is said to be uniformly stochastically smaller than G, denoted by F <₍₊₎ G, if and only if the ratio l(x) = G (x)/ F (x) is nondecreasing for x ∈ [0, sup{t: [0, sup {(t: F (t) > 0}). When F and G are absolutely continuous, F ₍₊₎ G is equivalent to the assumption that the corresponding failure rates are ordered. The applicability of the notion of uniform stochastic ordering in reliability and life testing is discussed. Given that a random sample X ₁, …, X _n of lifetimes has been obtained from F, where F is assumed to satisfy the uniform stochastic ordering constraint F <₍₊₎ G (or alternatively, F > ₍₊₎ G), where G is fixed and known, the problem of estimating F is addressed. It has been shown elsewhere that the method of nonparametric maximum likelihood estimation fails to provide consistent estimators in this type of problem. Here, a recursive approach is shown to provide estimators that converge uniformly to F with probability 1 and are as close or closer to F, in the sup norm, than is the empirical distribution function. This leads to a proof of the inadmissibility of the empirical distribution function, relative to the sup norm loss criterion, when estimating F <₍₊₎ G (or F >₍₊₎ < G) with G continuous. The two-sample estimation problem is also discussed briefly.

Key Words:

• Consistency
• Estimation
• Inadmissibility
• Nonparametric
• Recursive
• Reliability