The dependence of uncorrelated statistics
type: Item
instance of: scholarly article
title: The dependence of uncorrelated statistics
zbMATH Open document ID: 0808.62053
DOI: 10.1016/0893-9659(94)90067-1
author: V. Pereyra
published in: Applied Mathematics Letters
publication date: 1995-03-16T00:00:00Z
review text: Let \(f(x,\theta)\) be the density function of the independent sample elements \(X_ 1, X_ 2, \dots, X_ n\), \(\theta= (\theta_ 1, \theta_ 2,\dots, \theta_ k)\) is a parameter vector. Under some regularity conditions the maximum likelihood estimator (MLE) \(\widehat{\theta} (n)\) of the parameter vector \(\theta\) is asymptotically normal, more precisely \[ \sqrt{n} (\widehat {\theta}(n)- \theta)\to N(0,\Sigma) \quad \text{as } n\to\infty. \] Thus, if the covariance matrix \(\Sigma\) is diagonal, then the coordinates of \(\widehat{\theta}(n)\) are asymptotically independent. Note, however, that if only the marginals of the limit distributions are known to be normal and \(\Sigma\) is diagonal, then the components of \(\widehat{\theta}(n)\) are not necessarily independent asymptotically; in fact, their maximal correlation can be any given number. A typical example where the above mentioned regularity conditions do not hold is the uniform distribution \(U(a,b)\). Here, the MLE of the location and scale parameters \(\theta_ 1= (a+b)/2\) and \(\theta_ 2= b-a\) are \(\widehat{\theta}_ 1= (\widehat{a}+ \widehat{b}) /2\), \(\widehat{\theta}_ 2= \widehat{b}- \widehat{a}\) where \(\widehat{a}= \min X_ i\), \(\widehat{b}= \max X_ i\). Although \(\text{cov} (\widehat{\theta}_ 1, \widehat {\theta}_ 2)=0\), one can easily see that \(\widehat{\theta}_ 1\) and \(\widehat{\theta}_ 2\) are not independent. We study the dependence of these types of uncorrelated estimators.
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62H12
zbMATH DE Number: 713769
zbMATH Keywords: dependence
zbMATH Keywords: uncorrelated \(L\)-statistics
zbMATH Keywords: bivariate distribution
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: asymptotically normal
zbMATH Keywords: asymptotically independent
zbMATH Keywords: regularity conditions
zbMATH Keywords: uniform distribution
zbMATH Keywords: location
zbMATH Keywords: scale
zbMATH Keywords: uncorrelated estimators
MaRDI profile type: Publication
full work available at URL: 0893-9659(94)90067-1
OpenAlex ID: W1972654397
cites work: Q3672893
cites work: Linear Forms in the Order Statistics from an Exponential Distribution
cites work: Q5665994
cites work: How to transform correlated random variables into uncorrelated ones
cites work: On measures of dependence
cites work: Ordinal Measures of Association
cites work: Some Concepts of Dependence
cites work: The asymptotic distribution of the rènyi maximal correlation
cites work: Certain Uncorrelated and Independent Rank Statistics
cites work: Q5606305
description: scientific article; zbMATH DE number 713769