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Locally minimum variance unbiased estimator in a discontinuous density function. (English) Zbl 0608.62030
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 $$n^{-2}$$.
Reviewer: J.Melamed

MSC:
 62F10 Point estimation
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References:
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