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Suppose $X_1,\ldots,X_n$ are a random sample of uniform distribution on interval $(0,1)$. If $$ Z_i = \begin {cases} X_i & \text{w.p. }\theta, \\ -X_i & \text{w.p. }1-\theta, \end {cases} \qquad 0<\theta<1 $$ and $P$ denote the number of pozitive $Z_i$, how can find umvu estimator of parameter $\theta$

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Let $$ Y_i = \begin{cases} 1 & \text{if }Z_i>0, \\ 0 & \text{if }Z_i<0. \end{cases} $$ Now show that the conditional distribution of $Z_i$ given $Y_i$ does not depend on $\theta$. Therefore $\{Y_i\}_{i=1}^n$ is sufficient for $\theta$. And indeed, it's not hard to show the same is true of $\sum\limits_{i=1}^n Y_i$. So that is sufficient. And that sum has a binomial distribution. If you've done a similar problem for the binomial distribution, then you've got it.

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