# [R] product of two chi-squared

Spencer Graves spencer.graves at pdf.com
Thu Jul 1 22:48:09 CEST 2004

```      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

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 at comcast.net>
>>Sender: r-help-bounces at 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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