hypersecant {VGAM}R Documentation

Hyperbolic Secant Distribution Family Function

Description

Estimation of the parameter of the hyperbolic secant distribution.

Usage

hypersecant(link.theta="elogit", earg=if(link.theta=="elogit")
    list(min=-pi/2, max=pi/2) else list(), init.theta=NULL)
hypersecant.1(link.theta="elogit", earg=if(link.theta=="elogit")
    list(min=-pi/2, max=pi/2) else list(), init.theta=NULL)

Arguments

link.theta Parameter link function applied to the parameter theta. See Links for more choices.
earg List. Extra argument for the link. See earg in Links for general information.
init.theta Optional initial value for theta. If failure to converge occurs, try some other value. The default means an initial value is determined internally.

Details

The probability density function of the hyperbolic secant distribution is given by

f(y) =exp(theta*y + log(cos(theta ))) / (2*cosh(pi*y/2)),

for parameter pi/2 < theta < pi/2 and all real y. The mean of Y is tan(theta) (returned as the fitted values).

Another parameterization is used for hypersecant.1(). This uses

f(y) =(cos(theta)/pi) * y^(-0.5+theta/pi) * (1-y)^(-0.5-theta/pi),

for parameter pi/2 < theta < pi/2 and 0 < y < 1. Then the mean of Y is 0.5 + theta/pi (returned as the fitted values) and the variance is (pi^2 - 4*theta^2) / (8*pi^2).

For both parameterizations Newton-Raphson is same as Fisher scoring.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Author(s)

T. W. Yee

References

Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman & Hall.

See Also

elogit.

Examples

x = rnorm(n <- 200)
y = rnorm(n)  # Not very good data!
fit = vglm(y ~ x, hypersecant, trace=TRUE, crit="c")
coef(fit, matrix=TRUE)
fit@misc$earg

# Not recommended
fit = vglm(y ~ x, hypersecant(link="identity"), trace=TRUE, crit="c")
coef(fit, matrix=TRUE)
fit@misc$earg

[Package VGAM version 0.7-7 Index]