Phi2rho (version 1.0.1)

OwenT: Owen's T Function

Description

Computes Owen's T function using the modified Euler's arctangent series, tetrachoric series or Vasicek's series.

Usage

OwenT(h, a, opt = TRUE, fun = c("mOwenT", "tOwenT", "vOwenT"))

Value

The value of computed function is returned, scalar or vector. The attribute ‘nIter’ of returned value means the number of iterations.

Arguments

h

Numeric scalar or vector.

a

Numeric scalar or vector.

opt

If TRUE, an optimized calculation is performed.

fun

The name of the internal function being used.

Author

Janez Komelj

Details

If h and a are both vectors, they must be of the same length. If one of h and a is a vector and the other is a scalar, the latter is replicated to the length of the former. The calculation is performed component-wise.

The parameter fun specifies which series is used:

“mOwenT”:

modified Euler's arctangent series (default).

“tOwenT”:

tetrachoric series.

“vOwenT”:

Vasicek's series.

The opt parameter enables checking the results in the submitted article and may be dropped later.

If fun = "mOwenT" and opt = TRUE, the external arctangent function is used, otherwise all necessary values are calculated on the fly, but usually more iterations are needed.

If fun = "tOwenT" or fun = "vOwenT", and opt = TRUE, then the parameters transformation is performed when it makes sense, which significantly reduces the number of iterations.

References

Komelj, J. (2023): The Bivariate Normal Integral via Owen's T Function as a Modified Euler's Arctangent Series, American Journal of Computational Mathematics, 13, 4, 476--504, tools:::Rd_expr_doi("10.4236/ajcm.2023.134026") (or reprint https://arxiv.org/pdf/2312.00011.pdf with better typography).

Owen, D. B. (1956): Tables for Computing Bivariate Normal Probabilities, The Annals of Mathematical Statistics, 27, 4, 1075--1090, tools:::Rd_expr_doi("10.1214/aoms/1177728074").

Owen, D. B. (1980): A table of normal integrals, Communications in Statistics -- Simulation and Computation, 9, 4, 389--419, tools:::Rd_expr_doi("10.1080/03610918008812164").

See Also

Phi2xy

Examples

OwenT(2, 0.5)
OwenT(2, 0.5, fun = "mOwenT") # modified arctangent series (default)
OwenT(2, 0.5, fun = "tOwenT") # tetrachoric series
OwenT(2, 0.5, fun = "vOwenT") # Vasicek's series

rho <- 0.6
a <- rho/sqrt(1 - rho^2)
OwenT(0.3, a)
OwenT(0.3, a, fun = "tOwenT") 
OwenT(c(-1, 0.5, 4), a, fun = "vOwenT")

OwenT(2, c(-1, -0.5, 0, 0.5, 1), fun = "vOwenT")

function (h, a, opt = TRUE, fun = c("mOwenT", "tOwenT", "vOwenT")) 
{
    chkArgs1(h = h, a = a, opt = opt)
    if (length(h) < length(a)) 
        h <- rep(h, length(a))
    if (length(a) < length(h)) 
        a <- rep(a, length(h))
    z <- h
    n <- rep(0, length(z))
    z[a == 0] <- 0
    z[is.na(a)] <- NA
    i <- a != 0 & !is.na(a)
    if (any(i)) {
        h <- h[i]
        a <- a[i]
        ph <- pnorm(h)
        fun <- match.arg(fun)
        if (fun == "mOwenT") 
            j <- abs(a) > 1
        if (fun == "tOwenT") 
            j <- opt & abs(a) > 1
        if (fun == "vOwenT") 
            j <- opt & abs(a) < 1
        pah <- h
        if (any(j)) 
            pah[j] <- pnorm(a[j] * h[j])
        pah[is.nan(pah)] <- 0
        w <- eval(call(fun, h, a, ph, pah, opt))
        z[i] <- w
        n[i] <- attr(w, "nIter")
    }
    attr(z, "nIter") <- n
    return(z)
  }