The dependence of uncorrelated statistics (Q1343523)

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.