Donald Burrill wrote: > Surely this depends heavily on "the population"; which I think must > be finite, or at any rate bounded, for the question even to make > sense. (What's the population maximum of a normal distribution? > Infinity, I should think -- which is unlikely to be well estimated by > any sample value.) Perhaps there were some (implied?) conditions, or > constraints, in the context of the original question? > -- Don. > > On Fri, 19 Mar 2004, Kaplon, Howard wrote: > >> A student asked a colleague of mine if the sample maximum was the >> best point estimator of the population maximum. While I was tempted >> to say yes, the idea that this must surely be a biased >> under-estimator came to mind. Can someone answer the problem and >> point me to a source? >
It is tempting to refer to the well-known results for a uniform distribution with unknown bounds (two separate parameters). Then the sample maximum is maximum-likelihood and super-efficient, but not unbiased. Various adjustments to the sample maximum have been suggested. I would suggest looking in the book by Jonhnson, Kotz and Balakrishnan (Continuous Univariate Distibutions, Vol 2). Of course there are other instances where other bounded distributions arise. Some of these can give rise to instances where maximum-likelihood estimation fails (in its raw form). David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
