VGAM (version 1.0-4)

hypersecant: Hyperbolic Secant Distribution Family Function

Description

Estimation of the parameter of the hyperbolic secant distribution.

Usage

hypersecant(link.theta = extlogit(min = -pi/2, max = pi/2), init.theta = NULL)
hypersecant01(link.theta = extlogit(min = -pi/2, max = pi/2), init.theta = NULL)

Arguments

link.theta

Parameter link function applied to the parameter $\theta$. See Links for more choices.

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 $$f(y;\theta )=\exp (\theta y+\log (\cos (\theta )))/(2\cosh (\pi y/2)),$$ for parameter $-\pi /2<\theta <\pi /2$ and all real $y$. The mean of $Y$ is $\tan (\theta )$ (returned as the fitted values). Morris (1982) calls this model NEF-HS (Natural Exponential Family-Hyperbolic Secant). It is used to generate NEFs, giving rise to the class of NEF-GHS (G for Generalized).

Another parameterization is used for hypersecant01(): let $Y=(logitU)/\pi$. Then this uses $$f(u;\theta )=(\cos (\theta )/\pi )\times u^{-0.5+\theta /\pi }\times (1-u{)}^{-0.5-\theta /\pi },$$ for parameter $-\pi /2<\theta <\pi /2$ and $0<u<1$. Then the mean of $U$ 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.

Morris, C. N. (1982) Natural exponential families with quadratic variance functions. The Annals of Statistics, 10(1), 65--80.

See Also

extlogit.

Examples

# NOT RUN {
hdata <- data.frame(x2 = rnorm(nn <- 200))
hdata <- transform(hdata, y = rnorm(nn))  # Not very good data!
fit1 <- vglm(y ~ x2, hypersecant, data = hdata, trace = TRUE, crit = "coef")
coef(fit1, matrix = TRUE)
fit1@misc$earg

# Not recommended:
fit2 <- vglm(y ~ x2, hypersecant(link = "identitylink"), data = hdata, trace = TRUE)
coef(fit2, matrix = TRUE)
fit2@misc$earg
# }