ON ESTIMATING A SURVIVAL-CURVE SUBJECT TO A UNIFORM STOCHASTIC ORDERING CONSTRAINT

If F and G are cumulative distribution functions on [0, infinity) gove rning the lifetimes of specific systems under study, and if FBAR and G BAR are their corresponding survival functions, then F is said to be u niformly stochastically smaller than G, denoted by F <(+) G, if and on ly if the ratio l(x) = G(x)BAR/F(x)BAR is nondecreasing for x is-an-el ement-of [ 0, sup {t : F(t)BAR > 0}). When F and G are absolutely cont inuous, F <(+) G is equivalent to the assumption that the correspondin g failure rates are ordered. The applicability of the notion of unifor m stochastic ordering in reliability and life testing is discussed. Gi ven that a random sample X1, . . . , X(n) of lifetimes has been obtain ed from F, where F is assumed to satisfy the uniform stochastic orderi ng 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 estimat ion fails to provide consistent estimators in this type of problem. He re, 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 functi on, 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.