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I have a 3-dimensional sample $(X_k,Y_k,Z_k), k=1, \ldots, N$ which I suspect to be uniform on some parallelepiped in $R^3$ (i.e. a set of the form [a;b]X[c;d]X[e;f], where numbers a,b,c,d,e,f are unknown).

  1. How should I estimate numbers a, b, c, d, e, f? Obviously I can try MLE, but then my estimates are biased. Does unbiased estimates exist?
  2. How can I check that my sample is indeed uniform?
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  1. For the 1D continuous uniform distribution U(a,b) the uniformly minimum variance unbiased (UMVU) estimates of a and b can be obtained in closed form by a straightforward example of maximum spacing estimation. Can't see any reason that applying this separately for each dimension wouldn't give you UMVU estimates of all parameters of your multivariate uniform distribution
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+1 Note that the MSE comes with a built-in goodness of fit estimate, so it solves problems #1 and #2 simultaneously. – whuber♦ Nov 17 '10 at 13:28
@whuber (+1) Thanks, I hadn't noticed that. I admit I've never actually used maximum spacing estimation; I'd heard of it in connection with estimation for a uniform distribution and just checked Wikipedia. – onestop Nov 17 '10 at 14:26
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Whuber and onestop thanks for the answers. Now I understand how to check that my marginal distributions X, Y and Z are uniform. However to prove that the joint distribution of (X,Y,Z) is uniform in R^3 we have to prove, that (X,Y,Z) are independent. So the question is: by given sample (X_k, Y_k, Z_k) how can I check whether random variables (X,Y,Z) are independent? Thanks! – Oleg Nov 18 '10 at 10:43
Good question (i.e. i can't immediately think of an answer). – onestop Nov 18 '10 at 12:48
And is it possible to check whether say (X_k) and (Y_k) are independent? – Oleg Nov 19 '10 at 20:53

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