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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?

Tim
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Yanrique
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    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 – sega_sai Nov 25 '18 at 19:10
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    As asked the question makes no sense: the unbiased estimator need estimate a function of $\theta$ not the entire distribution. – Xi'an Nov 25 '18 at 21:05
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    Please add the `self-study` tag and detail which steps you took to attempt to solve the question. Else the question risks getting closed. – Xi'an Nov 25 '18 at 21:06
  • Relevant: https://stats.stackexchange.com/questions/354893/sufficient-statistics-for-uniform-%ce%b8-%ce%b8 – kjetil b halvorsen Nov 25 '18 at 21:43

1 Answers1

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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.

user158565
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