In article <3BAF09BF.1057.32F8FF@localhost>,
J.Russell <[EMAIL PROTECTED]> wrote:
>The requirement for the CLT to hold is that there should be a mean
>and st deviation for the background distribution. This I checked in
>Introduction to the Theory of Statistics by Mood, Graybill and Boes
>For a Cauchy distribution we can not determine the standard
>deviation and thus taking random sampling from this technically
>infinite population does not obey the CLT. I have created my own
>sampling to persuade me of this and it holds.
>However once we have swapped from a theoretically infinite
>population to a finite though large population then the mean and st.
>Deviation are calculable so the CLT holds. So for sampling from a
>specific sample of a Cauchy distribution the CLT holds.
The users of statistics seem to jump to totally incorrect
conclusions based on the Central Limit Theorem.
The mean of a random sample of independent non-normal items
is NOT normally distributed. Even in well-behaved cases,
unless the third moment about the mean is 0, the convergence
is only on the order of 1/sqrt(n). The relative error is
greatest in the tails.
For sampling without replacement, the "asymptotics" never
holds, as on runs out of sample. It is nevertheless true
that similar results hold with large variance and not too
bad tails.
However, with a sample size of 100, and with a typical
lottery of the type now being used, given the typical
payoffs, the results will not be approximately normal.
The rate of convergence to approximate normality can
be very slow, even if convergence occurs.
Unnecessary attempts to use normality should be resisted,
and procedures ostensibly based on normality can work quite
well without it. Transformations to make the data normal
are almost always destructive of whatever structure was
originally present.
--
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, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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