Holonomic properties and recurrence formula for the distribution of sample correlation coefficient

Abstract

This paper presents the holonomic properties and recurrence formula for the distribution of the sample correlation coefficient. The probability density function (pdf) is holonomic. Therefore, it is computed exactly based on the holonomic gradient method (HGM). The initial values for computation are expressed in terms of Gaussian hypergeometric functions with specific parameters that can be transformed to a rational equation of gamma functions. Using the integral algorithm in the $D$-module theory, the cumulative distribution function (cdf) is also holonomic. It can be computed using HGM. Next, we derive the recurrence formula for the Gaussian hypergeometric function related to the degrees of freedom and apply it to exact computation of the pdf under a fixed population correlation coefficient and increasing degrees of freedom. We conclude with discussion of the quantile function of the sample correlation coefficient which satisfies a nonlinear differential equation of second order.