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Suppose I sample many times independently from each of two normal distributions. Given the means, standard deviations, and number of samples from each distribution, how can I calculate the probability that the highest sampled value came from the first distribution?

I can solve this question by simulating it numerically, but haven't figured out an analytic expression.

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  • $\begingroup$ Mark, could you please clarify an important distinction? Where you write that the "means" and "standard deviations" are "given," are you referring to the means and SDs of the data or do these refer to the means and SDs of the distributions themselves? The latter is a simple probability problem while the former is a thorny statistical problem (with a different solution). $\endgroup$ – whuber Aug 27 '12 at 13:37
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    $\begingroup$ @whuber I meant the distributions are given; I only wanted help with the probability problem. $\endgroup$ – Mark Eichenlaub Aug 29 '12 at 2:39
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The answer works in the same way as jbowman's (now deleted answer) except that X$_1$ and X$_2$ are replaced by M$_1$ and M$_2$ where M$_1$ has the distribution of the maximum of the n$_1$ samples from group 1 and M$_2$ is the maximum of the n$_2$ samples from group 2. So now we have to replace the distribution of the difference of two independent normals to the distribution of the difference between the maxima for the two normals. Unfortunately the exact distribution for the maximum is not simple but the answer is to calculate the probability that M$_1$-M$_2$>0. Actually for M$_1$ it is F$_1$$^n$(x) and F$_2$$^n$(x) for M$_2$ where F$_1$ and F$_2$ are the respective normal distributions for group 1 and group 2. This is explained below in my comment.

As a further note asymptotically M$_1$ and M$_2$ can be appropriately normalized to have their distributions of the form of a Gumbel distribution (i.e. F(x) =exp(-e$^-$$^x$)). For large n$_1$ and n$_2$ that could be useful in getting an approximate answer to the desired probability.

As requested by the OP here are some links to books on extreme value theory at the amazon.com site.

Gumbel

Leadbetter et al

Galambos

Coles

deHaan

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  • $\begingroup$ Thanks. Do you have a reference on the distribution of the maximum from a sample? $\endgroup$ – Mark Eichenlaub Aug 24 '12 at 22:26
  • $\begingroup$ @MarkEichenlaub Yes there are many books on extreme value theory that will give you the asymptotic results. Gumbel's was the first. This is based on Gnedenko's theorem which has been extended to stationary stochastic processes as described in the book by Leadbetter et al. I should mention for IID samples from cdf F(x) the cumulative distribution of the maximum of a sample of size n is exactly F$^n$(x) because P[M$_n$<=x] = P[X$_i$<=x for all i]. So in your case you do have the exact distribution for M$_1$ and M$_2$, but it is not in a nice form. I will add refs to my answer. $\endgroup$ – Michael R. Chernick Aug 24 '12 at 22:37
  • $\begingroup$ I'm a little troubled by the fact that the means and SD are estimated from the samples. Are you implying that the estimated distributions should be used for $F_1$ and $F_2$? If so, this answer cannot possibly be correct (although there's a chance it's a good approximation for large samples), because clearly the maximum of an iid normal sample, conditional upon its mean and SD, is bounded, whereas $F_i$ is not bounded. If not, then precisely how should one obtain $F_1$ and $F_2$? $\endgroup$ – whuber Aug 26 '12 at 22:33
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    $\begingroup$ @whuber The OP assumed the distributions had their means and variances known. $\endgroup$ – Michael R. Chernick Aug 26 '12 at 23:06

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