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 0.7-9 Index]