Estimating the Survival Curve When New is Better than Used of a Specified Age

Abstract

Suppose that the survival function S of a random variable X is a member of the new-better-than-used-at-age- t 0 (NBU-{ t 0 }) class. As defined by Hollander, Park, and Proschan (1986), S must satisfy the condition S ( x + t 0 ) ≤ S ( x ) S ( t 0 ) for all x ≥ 0. Given a random sample from S , an estimator is sought that itself belongs to the NBU-{ t 0 } class and has the same asymptotic optimality properties as the empirical survival curve S n . The estimator S of S , given implicitly by S ( x ) = min[ S n ( x ), Ŝ( t 0 )Ŝ( x - t 0 )] when x ≥ t 0 , with Ŝ( x ) = S n ( x ) for x ≤ t 0 , is studied. A closed-form representation for Ŝ is derived. Ŝ is shown to be a strong uniformly consistent NBU-{ t 0 } estimator of S that converges to S at the best possible (pointwise and mean squared) rates. The asymptotic distribution theory of Ŝ is discussed and is derived explicitly for functions S for which S ( x + t 0 ) ≤ S ( x ) S ( t 0 ) for all x > 0. Confidence procedures based on Ŝ are discussed. The final section describes the results of a simulation study that lends strong support to the claim that Ŝ estimates an NBU-{ t 0 } survival curve more precisely than does the empirical survival function S n .