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 (θ y + \log (\cos (θ))) / (2 \cosh (π 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 $f (y) = (\cos (θ) / π) \times y^{- 0.5 + θ / π} \times (1 - y)^{- 0.5 - θ / π},$ 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.

Examples

Run this code

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

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

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