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Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ Find the maximum likelihood estimators for $\theta$ and $\rho$. Are these two unbiased estimators?

Can anyone help with this problem? It looks like my one weakness with MLE is figuring them out when order statistics are introduced. Moreover, I have no idea how to determine if they're unbiased (I can't even figure out the respective pdf's-which I definitely need right?). I'd really appreciate any help someone can offer.

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    $\begingroup$ Could you do it if it was instead on $[\alpha,\beta]$? What about an even simpler problem -- what if it was on $[0,\beta]$? $\endgroup$ – Glen_b Mar 4 '15 at 1:57
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    $\begingroup$ If it's the order statistics that bother you, just write them down on a deck of cards, shuffle it, and deal them out: now you have iid (unordered) data. Moreover, you haven't lost any information because you can always re-sort the values. $\endgroup$ – whuber♦ Mar 4 '15 at 2:11
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    $\begingroup$ @John The $[\alpha,\beta]$ problem is identical to the $[\theta-\rho,\theta+\rho]$ problem: it's merely a one-to-one reparameterization (and you know that MLEs are invariant under such changes). However, you won't succeed by looking at partial derivatives--that's one of the points of this exercise. To check bias, compute the expected values of the estimates. $\endgroup$ – whuber♦ Mar 4 '15 at 2:14
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    $\begingroup$ Not a trick question at all. When you have trouble getting started, dealing first with simpler questions often helps identify where your problems lie most clearly (because you leave fewer places for your misconceptions to hide) -- and both misunderstandings I guessed you might have seem to have been revealed by asking about the simpler problem. I suggest tackling the $[0,\beta]$ problem (with some actual data), so here's a sample of size n=3: 0.064, 0.895, 0.271. Start by drawing the likelihood function. Be very careful about bounds. Where is the likelihood highest? $\endgroup$ – Glen_b Mar 4 '15 at 2:22
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    $\begingroup$ To repeat whuber's point, you do not need to write down the density of the order statistics vector, you can assume wlog that you have an iid vector from $U(\theta-\rho,\theta+\rho)$. Then write the likelihood, which is a product of indicator functions time $(2\rho)^{-n}$, simplify, and look at its variations in $\theta$ and in $\rho$. The unbiasedness issue only comes when you have solved the above and all you have to do is take an expectation. $\endgroup$ – Xi'an Mar 4 '15 at 13:02

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