pprodt2 function

The distribution of the product of two jointly normal or t variables

Consider the product $W=X_1 X_2$ from a bivariate random variable $(X_1, X_2)$ having joint normal or Student's t distribution, with 0 location and unit scale parameters. Functions are provided for the distribution function of $W$ in the normal and the t case, and the quantile function for the t case.

pprodn2(x, rho)
pprodt2(x, rho, nu)
qprodt2(p, rho, nu, tol=1e-5, trace=0)

Arguments

• x: a numeric vector
• p: a numeric vector of probabilities
• rho: a scalar value representing the correlation (or the matching term in the t case when correlation does not exists)
• nu: a positive scalar representing the degrees of freedom
• tol: the desired accuracy (convergence tolerance), passed to function uniroot
• trace: integer number for controlling tracing information, passed on to uniroot

Details

Function pprodt2 implements formulae in Theorem 1 of Wallgren (1980). Corresponding quantiles are obtained by qprodt2 by solving the pertaining non-linear equations with the aid of uniroot, one such equation for each element of p.

Function pprodn2 implements results for the central case in Theorem 1 of Aroian et al. (1978).

Returns

a numeric vector

References

Aroian, L.A., Taneja, V.S, & Cornwell, L.W. (1978). Mathematical forms of the distribution of the product of two normal variables. Communications in statistics. Theory and methods, 7, 165-172

Wallgren, C. M. (1980). The distribution of the product of two correlated t variates. Journal of the American Statistical Association, 75, 996-1000

Author(s)

Adelchi Azzalini

See Also

uniroot

Examples

p <-  pprodt2(-3:3, 0.5, 8)
qprodt2(p, 0.5, 8)