Stochastic estimation and testing
type: Item
instance of: scholarly article
title: Stochastic estimation and testing
zbMATH Open document ID: 0644.62028
author: P. W. Millar
author: Rudolf Beran
published in: The Annals of Statistics
publication date: 1987-01-01T00:00:00Z
review text: Several stochastic procedures (tests, estimates and confidence intervals) are introduced and the asymptotic theory of these procedures is developed. For example, let \(\{Q_{\theta}\), \(\theta\in \Theta \}\) be a family of distribution functions on \(R^ d\) with density f(\(\theta\),x), \(\Theta \subset R^ d\), \(X_ n=(x_ 1,...,x_ n)\) be an i.i.d. sample of size n, \(L_ n(\theta,X_ n)=\prod^{n}_{1}f(\theta,x_ i)\) be the likelihood function. Let \(S_ n=(s_ 1,...,s_{j_ n})\) be a certain random sample of \(j_ n\) elements in \(\Theta\). The stochastic maximum likelihood estimate (SMLE) \({\hat \theta}_ n={\hat \theta}_ n(X_ n,S_ n)\) is defined by the requirement \[ L_ n({\hat \theta}_ n,X_ n)=\max (L_ n(s_ j,X_ n),\quad 1\leq j\leq j_ n). \] Under some regularity assumptions, the SMLE is asymptotically (n\(\to \infty)\) normal and asymptotically efficient (like the usual MLE).
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 4050752
zbMATH Keywords: randomized tests
zbMATH Keywords: iterated bootstrap techniques
zbMATH Keywords: random sampling schemes
zbMATH Keywords: stochastic minimum distance tests
zbMATH Keywords: stochastic confidence bands
zbMATH Keywords: critical value
zbMATH Keywords: Monte Carlo
zbMATH Keywords: minimum distance estimate
zbMATH Keywords: goodness-of-fit test
zbMATH Keywords: likelihood ratio test
zbMATH Keywords: stochastic search
zbMATH Keywords: asymptotic normality
zbMATH Keywords: asymptotic efficiency
zbMATH Keywords: likelihood function
zbMATH Keywords: stochastic maximum likelihood estimate
reviewed by: Vsevolod K. Malinovskii
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350497
OpenAlex ID: W2048077627
DOI: 10.1214/AOS/1176350497
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description: scientific article; zbMATH DE number 4050752