Best invariant estimation of a distribution function under the Kolmogorov-Smirnov loss function
Given a random sample of size n from an unknown continuous distribution function F, we consider the problem of estimating F nonparametrically from a decision theoretic approach. In our treatment, we assume the Kolmogorov-Smirnov loss function and the group of all one-to-one monotone transformations of real numbers onto themselves which leave the sample values invariant. Under this setup, we obtain a best invariant estimator of F which is shown to be unique. This estimator is a step function with unequal amounts of jumps at the observations and is an improper distribution function. I Consider a k-within-m-out-of-n system consisting of n independent and identically distributed random lifetimes with a common distribution function q. The author finds, under mild conditions, the limiting distribution of the system life-time as \(n\to \infty\).
scientific article; zbMATH DE number 4060504