Product Moments of Sample Variances and Correlation for Variables with Bivariate Normal Distribution

Email: juan.romero@cide.edu Abstract: A general result to obtain the product moments of two sample variances and the sample correlation when the data follow a bivariate normal distribution is derived; the result is expressed in terms of the hypergeometric function. As corollaries, two general equations are stated, one to obtain the moments of the correlation sample and one to obtain the moments of the ratio of two sample variances. To evaluate the product moments in short closed forms, three theorems have been established. The results are used to obtain the expectation and variance for the ratio of two correlated sample variances. Finally, some examples of particular product moments are provided and some validations were carried out.


Introduction
We are interested in the product moments of the two sample variances ( 2 1 S , 2 2 S ) and sample correlation coefficient (R) of the bivariate normal distribution, so we want to derive ( ) 2 2 1 2 a b c E S S R for finite a, b, c. One approach to obtain the product moments of the two sample variances and the correlation coefficient was discussed by Joarder (2006), however his results involves an infinite series that does not consider an important term, without the missing term, the result of Joarder (2006) does not work to get some product moments, for instance, we are not able to get the first moment of R. Here we expand the result of Joarder (2006) to derive a more general result. There are different expressions for moments of R, Ghosh (1966) and Rady et al. (2005) obtained the first four moments of R, here we derived a general equation for any moment of R that do not involves an infinite sum and it is expressed in terms of the well-known hypergeometric function.
Let X 1 ,…,X n be iid N p (µ,Σ) where Σ is positive defined and n > p, the sums of squares and cross product matrix is given by: A, is said to have a Wishart distribution with parameters p, m = n-1 and Σ(pxp), A∼W p (m,Σ) and the Probability Density Function (PDF) is given by (Anderson, 2003):

Product Moments of Sample Variances and Sample Correlation
Throughout the paper, we will use the generalized hypergeometric function (e.g., Bailey, 1964 , ; ; (1 ) , ; ; , And the following results:

Theorem 1
For any finite a, b and c. If c is even, the product moments of the sample variances and sample correlation,

Proof
We use the joint distribution of 2 1 S , 2 2 S and R given by Joarder (2006): (1 ) The product moments, for any a, b, c are given by: The first two integrals of Equation 6 can be expressed as: To obtain the last two integrals of Equation 6 we use the beta function: The product moments are given by the expression: To express If c is odd redefine k = 2j-1 then ( ) The following expression will be used in the next two theorems, let:

Theorem 2
For m > 2 and -1 < ρ < 1, we have: If c is even, we have: If c is even redefine k = 2j: If c is odd redefine k = 2j-1: To evaluate the product moments Joarder (2006) derived a result expressed in his paper by Theorem 3.1, which only holds when c is even and the expression for b k,m is different (Equation 8). The next Theorem 3, considers the case when c is not even.

Theorem 3
For m> 2 and -1< ρ <1 and b k,m defined in (8), we have for c even: For c odd, we have: , m > 2, k > 0 and under this scenario the proof was provided by Joarder (2006).
If c is odd, let z = ρ 2 , then: Differencing identity (9) with respect to z, we got (viii): Differencing now (10) with respect to z, we obtain (ix): Differencing (11) Differencing now (12) with respect to z, we obtain (xi): Next Theorem 4, will be useful to evaluate some product moments.

Corollary 1
The moments of the sample correlation, if c is even, are given by: ( ) If c is odd, are given by: ( ) Proof Use a = 0, b = 0 in Equation 3 and 4, then replace m = n-1.
The first moment of R, is obtained using, c = 1 in Equation 17 is a well-known equation for the first moment of R (Ghosh, 1966 In the literature there are expressions for the first four moments of R (Ghosh, 1966;Soper et al., 1917). The second, third and four moment of R may be expressed as: The equation of the third moment given by Ghosh (1966) is not correct. Equation 22 comes from the equation reported by Soper et al. (1917) The expressions for the second, third and four moment of R, that we derived, are shorter than existing equations. Algebraically it is not easy to verify if Equation 18-20 are equal to Equation 21-23 respectively, as an exercise we wrote a program using the software R to get values of the above expectations for some parameters, the results obtained do not show differences between the equations derived here and the equations reported in the literature. In Table 1, we show some data reported by Soper et al. (1917)  , be the ratio of two sample variances.
The statistic W is widely used to test the homogeneity of two variances and sometimes is useful to know the mean and variance of W. With the next corollary 2, we can obtain the mean and variance of W.

Corollary 2
For m > 2a, the moments of the ratio of two sample

Proof
Use b = -a and c = 0 in Equation 3. The expectation of the ratio of two sample variances, W, is given by: ( 2) 1 2 (1 )  The second moment of the ratio W, is given by: In the case of ρ = 0, we have: ( ) If furthermore, the two variances are equal the expectation and variance of W become the expectation and variance of a central F-distribution with parameters d 1 = d 2 = m. In the future, it will be useful to study the variance of W to test homogeneity of variances for two correlated samples.

Conclusion
We derive a general result to get any product moments of the sample variances and sample correlation coefficient when the data come from a bivariate normal distribution, the final expression is given in terms of the hypergeometric function which is a well-known function and there exists computational routines to be evaluated. A general expression to get the moments of the correlation sample is obtained and a validation of the result was carried out. Finally and equation to get the moments of the ratio of two sample variances was derived and as a particular case, the expectation and variance of the ratio of two sample variances were obtained.

Ethics
This article is original and contains unpublished material. There are no ethical issues involved in any aspect of this paper.