A simple solution to a nonparametric maximum likelihood estimation problem
type: Item
instance of: scholarly article
title: A simple solution to a nonparametric maximum likelihood estimation problem
zbMATH Open document ID: 0591.62032
author: Julius R. Blum
author: Gilbert G. Walter
published in: The Annals of Statistics
publication date: 1984-01-01T00:00:00Z
review text: The nonparametric maximum likelihood (ML) approach in estimating a density function f encounters, among others, the difficulty that the likelihood \[ L(f)=\prod^{n}_{i=1}f(x_ i)=\prod^{n}_{i=1}<\delta_{x_ i},f>, \] where \(x_ 1,...,x_ n\) are i.i.d. with density function f, has no finite maximum as f is allowed to range over an appropriate space of densities. One way to overcome this difficulty is to modify the linear functional \(<\delta_{x_ i},f>\) in some suitable way. The authors propose three approaches. The first is based on the method of sieves. The second method is to replace the functional by an inner product in a Hilbert space which contains both \(\delta\) and f. The third method is a standard ML method with the additional condition that \(D^ 2f\) belong to a bounded set. The authors consider one specific sieve (a double exponential convolution sieve) and show that the first method includes the other two, and they all lead to the problem of maximizing the same function. The authors then obtain a closed-form expression for the ML estimator in this case and show that it is consistent.
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 3949497
zbMATH Keywords: density estimation
zbMATH Keywords: consistency
zbMATH Keywords: maximum likelihood
zbMATH Keywords: method of sieves
zbMATH Keywords: inner product
zbMATH Keywords: Hilbert space
zbMATH Keywords: double exponential convolution sieve
zbMATH Keywords: closed-form expression
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176346415
OpenAlex ID: W2093039498
DOI: 10.1214/AOS/1176346415
description: scientific article; zbMATH DE number 3949497