```Date: Jan 9, 2013 10:35 PM
Author: Paul
Subject: Test constantness of normality of residuals from linear regression

After much browsing of Wikipedia and the web, I used both normalprobability plot and Anderson-Darling to test the normality ofresiduals from a simple linear regression (SLR) of 6 data points.Results were very good.  However, SLR doesn't just assume that theresiduals are normal.  It assumes that the standard deviation of thePDF that gives rise to the residuals is constant along the horizontalaxis.  Is there a way to test for this if none of the data points havethe same value for the independent variable?  I want to be able toshow that there is no gross curves or spreading/focusing of thescatter.In electrical engineering signal theory, the horizontal axis is time.Using Fourier Transform (FT), time-frequency domains can show trends.Intuitively, I would set up the data as a scatter graph of residualsplotted against the independent variable (which would be treated astime).  Gross curves show up as low-frequency content.  There shouldbe none if residuals are truly iid.  The spectrum should look likewhite noise.  The usual way to get the power spectrum is the FT of theautocorrelation function, which itself should resemble an impulse atzero.  This just shows indepedence of samples, not constant iid normalalong the horizontal axis.As for spreading or narrowing of the scatter, I guess that can bemodelled in time as a multiplication of a truly random signal by alinear (or exponential) attenuation function.  The latter acts like amodulation envelope.  Their power spectrums will then convolve in someweird way.  I'm not sure if this is a fruitful direction foridentifying trends in the residuals.  It starts to get convolutedpretty quickly.Surely there must be a less klugy way from the world of statistics?  Irealize that my sample size will probably be too small for manyconceptual approaches.  For example, if I had a wealth of data points,I could segment the horizontal axis, then do a normality test on eachsegment.  This would generate mu's and sigma's as well, which couldthen be compared across segments.  So for the sake of conceptualgratification, I'm hoping for a more elegant test for the ideal caseof many data points.  If there is also a test for small sample sizes,so much the better (though I don't hold my breath).
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