Testing whether a survival distribution is new better than used of an unknown specified age

A survival variable is a nonnegative random variable X with distribution function F and a survival function (F) over bar = 1 - F. This variable is said to be new better than used of specified age t(0) if (F) over bar(x + t(0)) less than or equal to (F) over bar(x)(F) over bar(t(0)) for all x greater than or equal to 0 and a fixed t(0). Testing H-0: (F) over bar(x + t(0)) = (F) over bar(x)(F) over bar(t(0)) for all x greater than or equal to 0 against H-1: (F) over bar(x + t(0)) less than or equal to (F) over bar(x)(F) over bar(t(0)) for x greater than or equal to 0 when the point t(0) is unknown but can be estimated from the data is proposed when t(0) = mu, the mean of F, and also when t(0) = xi(p), the pth percentile of F. It is shown that, while the known tests for known t(0) (Hollander, Park & Proschan, 1986; Ebrahimi & Habibullah, 1990) continue to hold when t(0) = mu and is estimated by (X) over bar, a simpler and distribution-free test is possible when t(0) = xi(p) and is estimated by X-([np]). The performance of this test is presented.