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How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$. Is it a function of the order statistics?

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    $\begingroup$ It may help to start from writing the likelihood function and see what kind of estimator you get from it, and whether it is biased or unbiased $\endgroup$
    – sega_sai
    Commented Nov 25, 2018 at 19:10
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    $\begingroup$ As asked the question makes no sense: the unbiased estimator need estimate a function of $\theta$ not the entire distribution. $\endgroup$
    – Xi'an
    Commented Nov 25, 2018 at 21:05
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    $\begingroup$ Please add the self-study tag and detail which steps you took to attempt to solve the question. Else the question risks getting closed. $\endgroup$
    – Xi'an
    Commented Nov 25, 2018 at 21:06
  • $\begingroup$ Relevant: stats.stackexchange.com/questions/354893/… $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 25, 2018 at 21:43

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$2\frac {N+1}N \mathrm{max}(|X_1|, |X_2|,...,|X_N|)$ where $N$ is sample size.

Suppose $X_i \sim U(-\theta/2, \theta/2)$.

Step 1: Let $Y_i = |X_i|$. What distribution does $Y_i$ follow?

Step 2: Find the distribution of $\mathrm{max}(Y_i)$. Refer How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?

Step 3: Find the expectation of $\mathrm{max}(Y_i)$.

Then you find the answer.

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  • $\begingroup$ Can u show me how it comes? Thank you. $\endgroup$
    – Yanrique
    Commented Nov 25, 2018 at 18:39
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    $\begingroup$ I think it is self-study question, So I added the steps, instead of answers, in the Answer. $\endgroup$
    – user158565
    Commented Nov 25, 2018 at 18:48

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