# Re: [R] product of two chi-squared

From: Spencer Graves <spencer.graves_at_pdf.com>
Date: Fri 02 Jul 2004 - 06:48:09 EST

Assuming independence, the expectation of a product is the product of the expectations. From this you could easily get moments of all orders and thence the moment generating function or characteristic function. By direct computation (or consulting Johnson and Kotz, Distributions in Statistics-1, ch. 17), we see that the chi-square is just a gamma with alpha = df/2 and scale = 2, and for the gamma distribution, the r-th moment is

E(X^r) = gamma(alpha+r)/gamma(alpha) = alpha*(alpha+1)*...*(alpha+r-1).

Therefore, the r-th moment of the product of two chi-squares with the same number of degrees of freedom is just the square of this expression. This same approach can be used to obtain moments of all orders for products of an arbitrary number of chi-squares with different numbers of degrees of freedom.

Somewhat more generally, if the two chi-squares arose in the same linear model context, if they are NOT independent, then I might expect them to be something like (X1+X2) and (X2+X3), where X1, X2, and X3 are independent. In that case, the above rule could still be used to easily get the expected value and variance, plus (with more effort) moments of higher order.

Beyond that, I know of no general result about products of chi-squares (even when they are independent). I just did a search on "querry.statindex.org" for "product of chi-square" and got nothing. When I searched for "product of gamma", I got the following:

O'Brien, Robert and Sinha, Bimal K. (1993) On shortest confidence intervals for product of gamma means Calcutta Statistical Association Bulletin, 43, 181-190

Rukhin, Andrew L. and Sinha, Bimal K. (1991) Decision-theoretic estimation of the product of gamma scales and generalized variance
Calcutta Statistical Association Bulletin, 40, 257-265 Keywords: Admissibility

I don't know if these articles would help you, but they might.

spencer graves

Giovanni Petris wrote:

>There is not enough information here: you need to know the joint
>distribution of the two. If they are independent, the expectation of
>the product is just the product of expectations - as any elementary
>textbook will tell you.
>
>Giovanni
>
>
>
>>Date: Thu, 01 Jul 2004 14:51:31 -0400
>>From: "Eugene Salinas (R)" <r-eugenesalinas@comcast.net>
>>Sender: r-help-bounces@stat.math.ethz.ch
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>>Hi,
>>
>>Does anyone know what the expectation of the product of two chi-squares
>>distributions is? Is the product of two chi-squared distributions
>>anything useful (as in a nice distribution)?
>>
>>thanks, eugene.
>>
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