Nonparametric estimation of nonincreasing densities and use of data from renewal processes
To estimate a density h, vanishing on (-∞,0) and nonincreasing on (0,∞), we use a random sample of size n from h to construct a histogram hn, and then let be the monotonization of hn. (The monotonization of a step function is obtained by replacing its values by their isotonic regression.) In terms of the sup-norm is at least as good as hn; hence some known strong consistency results for histograms translate directly to monotonized histograms. Under appropriate conditions is strongly consistent at every t∊ (0, ∞) where h is continuous and is a strongly consistent estimator of h(0+). We consider life testing questions related to a situation where n objects from a population with life distribution F (with finite mean μ) were initially put in use and, whenever an object failed, it was promptly replaced. At some time θ (long after the first n objects were put in use) a statistician observes the n objects in use at θ. We first consider the situation where the objects are not observed after time θ, so that only ...