sn (version 2.1.1)

pprodt2: The distribution of the product of two jointly normal or t variables

Description

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.

Usage

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

Value

a numeric vector

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

Author

Adelchi Azzalini

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).

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

See Also

uniroot

Examples

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