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RE: st: Re: normal distributions

From   "Gene Fisher" <>
To   <>
Subject   RE: st: Re: normal distributions
Date   Sun, 21 Jul 2002 17:14:15 -0400

Regression coefficients are functions of sums of random variables (sum(X),
sum(X^2), sum(XY), so that the CLT applies to them.  The variance of the
regression coefficient is also a sum of random variables (the residuals, and
of X and X^2 if the X's are not fixed), so the distribution of the ratio of
the regression coefficient to its standard error tends toward a standard
normal distribution as N approaches infinity.  But we don't know so readily
how close to normal b/se(b) is for a given distribution of Y (and possibly
X) and N.  Convergence is a lot faster if the residuals are symmetrical than
if they are skewed.
Also, looking at Serfling's Approximation Theorems of Mathematical
Statistics, pp 125-126, it may be that se(b)^2 is not the asymptotic
variance of b (He talks about the sampling distribution of the correlation
coefficient).  I think it is, but I haven't looked at this in a long time.

Gene Fisher
Department of Sociology
University of Massachusetts
Thompson Hall, 200 Hicks Way
Amherst, MA  01003-9277
(413) 545-4056;

-----Original Message-----
[]On Behalf Of Michael Cha
Sent: Sunday, July 21, 2002 4:37 PM
Subject: RE: st: Re: normal distributions

Dear listers,

Could anyone tell me the relationship between regression assumption test
(Normality test in particular) and CLT (Central Limit Theorem)?

Does CLT imply that we do not have to worry about normality test for
residuals as far as sample size is large?

Or CLT does not imply anything about the regression assumption test?

Thanks in advance,


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