Best invariant estimation of a distribution function under the Kolmogorov-Smirnov loss function
type: Item
instance of: scholarly article
title: Best invariant estimation of a distribution function under the Kolmogorov-Smirnov loss function
zbMATH Open document ID: 0649.62005
author: Yaakov Friedman
author: Alexander Gelman
author: Eswar G. Phadia
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: 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 \(n\to \infty\).
Mathematics Subject Classification ID: 62C99
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 4060504
zbMATH Keywords: Kolmogorov-Smirnov loss function
zbMATH Keywords: one-to-one monotone transformations
zbMATH Keywords: best invariant estimator
zbMATH Keywords: improper distribution
zbMATH Keywords: k-within-m-out-of-n system
zbMATH Keywords: limiting distribution
reviewed by: Moshe Shaked
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350959
OpenAlex ID: W2035812387
DOI: 10.1214/AOS/1176350959
description: scientific article; zbMATH DE number 4060504