VGAM (version 1.1-12)

Gensh: Generalized Secant Hyperbolic Distribution

Description

Density, distribution function, quantile function and random generation for the generalized secant hyperbolic distribution.

Usage

dgensh(x, shape, location = 0, scale = 1, tol0 = 1e-4,
       log = FALSE)
pgensh(q, shape, location = 0, scale = 1, tol0 = 1e-4,
       lower.tail = TRUE)
qgensh(p, shape, location = 0, scale = 1, tol0 = 1e-4)
rgensh(n, shape, location = 0, scale = 1, tol0 = 1e-4)

Value

dgensh gives the density,

pgensh gives the distribution function,

qgensh gives the quantile function, and

rgensh generates random deviates.

Arguments

x, q, p, n, log, lower.tail

Similar meaning as in Normal.

shape

Numeric. Shape parameter, called \(t\) in Vaughan (2002). Valid values are \(-\pi/2 < t\).

location, scale

Numeric. The location and (positive) scale parameters.

tol0

Numeric. Used to test whether the shape parameter is close enough to be treated as 0.

Warning

Numerical problems may occur when some argument values are extreme.

Author

T. W. Yee.

Details

This is an implementation of the family of symmetric densities described by Vaughan (2002). By default, the mean and variance are 0 and 1, for all \(t\). Some special (default) cases are: \(t=0\): logistic (which is similar to stats:dt with 9 degrees of freedom); \(t=-\pi/2\): the standard secant hyperbolic (whence the name); \(t=\infty\): uniform(-sqrt(3), sqrt(3)).

See Also

gensh, logistic, hypersecant, Logistic.

Examples

x <- seq(-2, 4, by = 0.01)
loc <- 1; shape <- -pi /2
if (FALSE) plot(x, dgensh(x, shape, loc), type = "l",
     main = "Blue is density, orange is the CDF",
     ylim = 0:1, las = 1, ylab = "", 
     sub = "Purple are 5, 10, ..., 95 percentiles",
     col = "blue")
abline(h = 0, col = "blue", lty = 2)
lines(qgensh((1:19) / 20, shape, loc), type = "h",
      dgensh(qgensh((1:19) / 20, shape, loc),
             shape, loc), col = "purple", lty = 3)
lines(x, pgensh(x, shape, loc), col = "orange")
abline(h = 0, lty = 2) 

pp <- (1:19) / 20  # Test two functions
max(abs(pgensh(qgensh(pp, shape, loc),
               shape,loc) - pp))  # Should be 0