Let X be a positive random variable. The distribution F of X is said to be ‘new better than used in expectation,’ or ‘NBUE,’ if E(X)≥E(X-t•X>t) for all t≥0. Suppose X1,...,Xn, is a random sample from a NBUE distribution F. The problem of estimating F by a distribution which is itself NBUE is considered. The estimator Gn, defined as the NBUE distribution supported on the sample which minimizes the (sup norm) distance between the NBUE class and the empirical distribution function, is studied. The strong uniform consistency of Gn is proven, and a numerical algorithm for obtaining Gn is given. The present approach is applied to provide an estimate of the distribution of lifetime following the diagnosis of chronic granulocytic leukemia based on data from a National Cancer Institute study.
