phyperPeizer: Peizer's Normal Approximation to the Cumulative Hyperbolic

In DPQ: Density, Probability, Quantile ('DPQ') Computations

Peizer's Normal Approximation to the Cumulative Hyperbolic

Description

Compute Peizer's extremely good normal approximation to the cumulative hyperbolic distribution.

This implementation corrects a typo in the reference.

Usage

phyperPeizer(q, m, n, k)

Arguments

q	vector of quantiles representing the number of white balls drawn without replacement from an urn which contains both black and white balls.
m	the number of white balls in the urn.
n	the number of black balls in the urn.
k	the number of balls drawn from the urn, hence must be in 0,1,\dots, m+n.

Value

a numeric vector, with the length the maximum of the lengths of q, m, n, k.

Author(s)

Martin Maechler

References

Johnson, Kotz & Kemp (1992): (6.93) & (6.94), p.261 CORRECTED by M.M.

See Also

phyper.

Examples

## The function is defined as

phyperPeizer <- function(q, m, n, k)
{
  ## Purpose: Peizer's extremely good Normal Approx. to cumulative Hyperbolic
  ##  Johnson, Kotz & Kemp (1992):  (6.93) & (6.94), p.261 __CORRECTED__
  ## ----------------------------------------------------------------------
  Np <- m; N <- n + m; n <- k; x <- q
  ## (6.94) -- in proper order!
  nn <- Np			;  n. <- Np     + 1/6
  mm <- N - Np                  ;  m. <- N - Np + 1/6
  r <- n                        ;  r. <- n      + 1/6
  s <- N - n                    ;  s. <- N - n  + 1/6
                                   N. <- N      - 1/6
  A <- x + 1/2                  ;  A. <- x      + 2/3
  B <- Np - x - 1/2             ;  B. <- Np - x - 1/3
  C <- n  - x - 1/2             ;  C. <- n  - x - 1/3
  D <- N - Np - n + x + 1/2     ;  D. <- N - Np - n + x + 2/3

  n <- nn
  m <- mm
  ## After (6.93):
  L <-
    A * log((A*N)/(n*r)) +
    B * log((B*N)/(n*s)) +
    C * log((C*N)/(m*r)) +
    D * log((D*N)/(m*s))
  ## (6.93) :
  pnorm((A.*D. - B.*C.) / abs(A*D - B*C) *
        sqrt(2*L* (m* n* r* s* N.)/
                  (m.*n.*r.*s.*N )))
  # The book wrongly has an extra "2*" before `m* ' (after "2*L* (" ) above
}

DPQ documentation built on Dec. 5, 2023, 3:05 a.m.