Locally minimum variance unbiased estimator in a discontinuous density function
Let \(X_ 1,...,X_ n\) be iid random variables with the following density function w.r.t. the Lebesgue measure \[ f(x;\theta) = \begin{cases} p \qquad &\text{for \(0\leq x\leq \theta\) and \(\theta +1\leq x\leq 2,\)} \\ q &\text{for \(\theta <x<\theta +1,\)} \\ 0 &\text{otherwise,} \end{cases} \] where \(\theta \in [0,1]\) and p and q with \(0<p<q\) and \(p+q=1\) are fixed constants. The exact forms of the locally minimum variance unbiased estimators \({\hat \theta}_ n={\hat \theta}_ n(x_ 1,...,x_ n)\) of \(\theta =\theta_ 0\) are obtained in the cases when \(n=1\) and \(n>1\). Exact expressions for variances of \({\hat\theta}_ 1\) and \({\hat\theta}_ n\) are presented. It is shown that \(var_{\theta}{\hat \theta}_ 1\geq var_{\theta_ 0} {\hat\theta}_ 1\) for all \(\theta\in [0,1]\). It is also proved that \(var_{\theta_ 0} {\hat\theta}_ n\) is of the order \(n^{-2}\).
locally minimum variance unbiased estimators
scientific article; zbMATH DE number 3984318