VGAM (version 0.7-5)

hypersecant: 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.

Value

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

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 $$f(y)=(\cos(\theta)/\pi) \times y^{-0.5+\theta/\pi} \times (1-y)^{-0.5-\theta/\pi},$$ 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.

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