# Hyperbolic Secant Regression Family Function Estimation of the parameter of the hyperbolic secant distribution. ```r 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) ``` ## 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. ## Details The probability density function of the hyperbolic secant distribution is given by $$ f(y;\theta) =\exp(\theta y + \log(\cos(\theta ))) / (2 \cosh(\pi y/2)),%f(y; theta) = exp(theta*y + log(cos(theta))) / (2*cosh(pi*y/2)), $$ 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 $$ f(u;\theta)=(\cos(\theta)/\pi) \timesu^{-0.5+\theta/\pi} \times(1-u)^{-0.5-\theta/\pi},%f(u;theta)=(cos(theta)/pi)*u^(-0.5+theta/pi)*(1-u)^(-0.5-theta/pi), $$ 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 c("$\n$", "$ (pi^2 - 4*theta^2) / (8*pi^2)$"). For both parameterizations Newton-Raphson is same as Fisher scoring. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Jorgensen, B. (1997). **The Theory of Dispersion Models**. London: Chapman & Hall. Morris, C. N. (1982). Natural exponential families with quadratic variance functions. **The Annals of Statistics**, 10 (1), 65--80. ## Author(s) T. W. Yee ## See Also `gensh`, `extlogitlink`. ## Examples ```r hdata <- data.frame(x2 = rnorm(nn <- 200)) hdata <- transform(hdata, y = rnorm(nn)) # Not very good data! fit1 <- vglm(y ~ x2, hypersecant, hdata, trace = TRUE, crit = "c") coef(fit1, matrix = TRUE) fit1@misc$earg # Not recommended: fit2 <- vglm(y ~ x2, hypersecant(link = "identitylink"), hdata) coef(fit2, matrix = TRUE) fit2@misc$earg ```

hypersecant function