Clearly Don is correct that it depends on the nature of the distribution of the variable in a particular population (and on population size).
But given a finite population, I don't think the expected value of the maximum of a normally distributed variable is infinity. (But maybe I am all wrong....) Given a population size N and a distribution, could one not bootstrap for the maximum? I know that in the typical case, where one wants to estimate the population parameter from a sample statiistic, the maximum is not bootstrappable (if that's the word) but here, if we are given the population size and the distribution, it seems to me intuitively that it should work e.g. If we are willing to posit that IQ is normally distributed with mean 100 and sd 15 (there is evidence that this is not the case, but let's ignore that for this example) and we wanted to estimate the highest IQ in the USA (let's say the population is 220 million), why couldn't we repeatedly find the maximum of a sample of 220 million, distributed this way? As a quick shot, I ran this in R max(rnorm(220000000, 100, 15)) three times and got values of 180.23, 181.40, and 180.56. I can't prove that this is correct, but it seems sensible, and the results also make sense Or am I doing something all wrong here? Peter >>> [EMAIL PROTECTED] 3/22/2004 11:02:02 AM >>> 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? ------------------------------------------------------------ Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . ================================================================= . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
