Re: Some questions about the Cauchy distribution

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 12/02/04 

Date: 2 Dec 2004 11:43:07 -0500

In article <41ae43b7$1@griseus.its.uu.se>, none <""john\"@(none)"> wrote:
>The mean of n random variables picked from the Cauchy distribution has
>itself the Cauchy distribution. Does this mean that there is no use
>having a larger sample in order to estimate the expected value? (Or wait
>a minute, does the expected value exist? No, right?)

The expected value does not exist. However, a larger
sample can be used to estimate the parameters of the
distribution, and even the distribution itself without
knowing its form.

>Does the existence of such distributions and those of the form
>Constant(p) / (1 + abs(x)^p) with p > 2 indicate that it is not always
>legitimate to assume that more measurements leads to a better
>approximation of the expected value? (Unless you have a good idea of the
>distribution of the measurements.)

If p > 2, the average will be a better estimate of the
mean. If you know that the distribution is of that form,
except for translation, one can do much better.

>For instance, if several hundred people sees an UFO and it is claimed
>that it must really have been an UFO is it proper to say that this
>cannot be assumed since there exists distributions where the mean value
>of many measurements are even worse than one measurement?

Here one is just "estimating" a probability, which is
bounded. The argument does not apply. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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