Hyperbolic Secant Regression Family Function
Estimation of the parameter of the hyperbolic secant distribution.
hypersecant(link.theta = extlogitlink(min = -pi/2, max = pi/2), init.theta = NULL) hypersecant01(link.theta = extlogitlink(min = -pi/2, max = pi/2), init.theta = NULL)
link.theta |
Parameter link function applied to the parameter theta.
See |
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. |
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 = (logit U) / pi.
Then this uses
f(u;theta) = (cos(theta)/pi) * u^(-0.5+theta/pi) * (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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
T. W. Yee
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.
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
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