Moments of the product and ratio of two correlated chi-square variables

Abstract

The exact probability density function of a bivariate chi-square distribution with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution.

Constraint-based causal discovery with mixed data

“…Although taking the minimum p value should be avoided for independent p values, as it does not account for multiple testing, it may be a reasonable choice if the p values have a high positive correlation. There has been theoretical work for deriving the true distribution of the sum or the ratio of the two test statistics, assuming their correlation is known [19,20]. A general, permutation-based method for estimating the correlation between test statistics has been proposed by Hongying Dai and Cui [18].…”

“…The sample size varied in (20,50, 100, 200, 500, 1000), and each experiment was performed 1000 times, except for case (b), which was performed 1000 times for each direction. Figure 2 shows the correlation and decision agreements (reject or not the null hypothesis) at the 5% significance level (similar results hold true for the 0.1, 1, and 10% significance levels) between all five pairs of regression models.…”

Space-time code diversity by phase rotation in multi-carrier multi-user systems

“…The product of two χ 2 distributions involves the gamma function [26] whose square approximately exhibits an exponential probability density function. In our case…”

Comment on the paper by A. H. Joarder

“…It appears that Joarder (2007) made no attempts to simplify the infinite sums. Here, I would like to point out that each of (1), (2) and (3) can be reduced to known hypergeometric functions.…”

“…The recent paper by Joarder (2007) studied the moments of the product and ratio of two correlated chi-square variables. The main results of the paper are presented in Theorems 2.1, 3.1 and 4.1.…”