Hyperbolic Secant Regression Family Function

Estimation of the parameter of the hyperbolic secant distribution.

Usage

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 $$f(y;\theta) = \exp(\theta y + \log(\cos(\theta ))) / (2 \cosh(\pi y/2)),$$ 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 $$f(u;\theta)=(\cos(\theta)/\pi) \times u^{-0.5+\theta/\pi} \times (1-u)^{-0.5-\theta/\pi},$$ 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.

Value

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

T. W. Yee

Examples

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")
#> Iteration 1: coefficients = -0.114019836, -0.037647484
#> Iteration 2: coefficients = -0.11373775, -0.03834265
#> Iteration 3: coefficients = -0.113740444, -0.038351822
#> Iteration 4: coefficients = -0.113740481, -0.038351962
#> Iteration 5: coefficients = -0.113740481, -0.038351964
coef(fit1, matrix = TRUE)
#>             extlogitlink(theta, min = -1.5707963267949, max = 1.5707963267949)
#> (Intercept)                                                        -0.11374048
#> x2                                                                 -0.03835196
fit1@misc$earg
#> $theta
#> $theta$theta
#> 
#> 
#> $theta$min
#> -pi/2
#> 
#> $theta$max
#> pi/2
#> 
#> $theta$bminvalue
#> NULL
#> 
#> $theta$bmaxvalue
#> NULL
#> 
#> $theta$inverse
#> [1] FALSE
#> 
#> $theta$deriv
#> [1] 0
#> 
#> $theta$short
#> [1] TRUE
#> 
#> $theta$tag
#> [1] FALSE
#> 
#> attr(,"function.name")
#> [1] "extlogitlink"
#> 

# Not recommended:
fit2 <- vglm(y ~ x2, hypersecant(link = "identitylink"), hdata)
coef(fit2, matrix = TRUE)
#>                   theta
#> (Intercept) -0.08913193
#> x2          -0.02966554
fit2@misc$earg
#> $theta
#> $theta$theta
#> 
#> 
#> $theta$inverse
#> [1] FALSE
#> 
#> $theta$deriv
#> [1] 0
#> 
#> $theta$short
#> [1] TRUE
#> 
#> $theta$tag
#> [1] FALSE
#> 
#> attr(,"function.name")
#> [1] "identitylink"
#>