Re: Testing for product of Gaussians

In article <4263FEAB.5060000@xxxxxxxxxxxx>,
Stephen J. Herschkorn <sjherschko@xxxxxxxxxxxx> wrote:
>newsquestions2003@xxxxxxxxx wrote:

<>I'd like to know an appropriate statistical test for finding out
<>whether data in 2 dimensions comes from the product of two uncorrelated
<>normal distributions. To be more specific for a series of (x,y) pairs
<>I have a value (let's say z) created by a computer simulation. I know
<>the actual values of z should be drawn from the product of two
<>independent normal distributions with mean zero and the same standard
<>deviation in both directions. I wonder if anyone could tell me an
<>appropriate statistical test to check that computed values have indeed
<>probably come from this underlying distribution? Given that so many
<>parametric tests seem to require normal data to work properly, I'd
<>guess this is a very standard question?


>The density for such a random variable U is K0(|u|) / (pi sigma^2),
>where K0 is the modified Bessel function of the second kind and sigma
>is the common standard deviation; see
>http://mathworld.wolfram.com/NormalProductDistribution.html . You will
>want to use a chi-squared goodness-of-fit test to test if your data fit
>this distribution. I.e., divide the range up into bins and numerically
>determine the integral of the density over each bin, then apply the
>test, which you can find on the 'net or any standard statistics text.

>If you need to estimate sigma, note that U has variance sigma^4. I
>would compute the sample variance and take the square root to use as an
>estimate for sigma^2 for the probability calculations. With a
>sufficient sample size, this should suffice, though the estimator is
>surely biased. If you do need to estimate this parameter, you lose a
>degree of freedom in your chi-squared test.

The chi-squared test has very little power, and if the
parameter is estimated from other than cell frequencies,
the distribution of the test statistic is NOT chi-squared.
--
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@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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