Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\theta$

share|improve this question
add comment

1 Answer

up vote 0 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.