Estimating the Survival Curve When New is Better Than Used of a Specified Age
针对新比指定年龄的旧更好（NBU-{t0}）类生存函数，提出一个保持该性质且渐近最优的估计量，推导其闭式表达式，证明其强一致性和最优收敛速度，并通过模拟验证其优于经验生存函数。
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 .