VGAM (version 0.8-3)

tikuv: Short-tailed Symmetric Distribution Family Function

Description

Fits the short-tailed symmetric distribution of Tiku and Vaughan (1999).

Usage

tikuv(d, lmean="identity", lsigma="loge", emean=list(), esigma=list(),
      isigma=NULL, zero=2)

Arguments

d

The $d$ parameter. It must be a single numeric value less than 2. Then $h=2-d>0$ is another parameter.

lmean, lsigma

Link functions for the mean and standard deviation parameters of the usual univariate normal distribution (see Details below). They are $\mu$ and $\sigma$ respectively. See Links for more choic

emean, esigma

List. Extra argument for each of the links. See earg in Links for general information.

isigma

Optional initial value for $\sigma$. A NULL means a value is computed internally.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2} corresponding respectively to $\mu$, $\sigma$. If zero=NULL then all linear/additive predictor

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Under- or over-flow may occur if the data is ill-conditioned, e.g., when $d$ is very close to 2 or approaches -Inf.

Details

The short-tailed symmetric distribution of Tiku and Vaughan (1999) has a probability density function that can be written $$f(y) = \frac{K}{\sqrt{2\pi} \sigma} \left[ 1 + \frac{1}{2h} \left( \frac{y-\mu}{\sigma} \right)^2 \right]^2 \exp\left( -\frac12 (y-\mu)^2 / \sigma^2 \right)$$ where $h=2-d>0$, $K$ is a function of $h$, $-\infty < y < \infty$, $\sigma > 0$. The mean of $Y$ is $E(Y) = \mu$ and this is returned as the fitted values.

References

Akkaya, A. D. and Tiku, M. L. (2008) Short-tailed distributions and inliers. Test, 17, 282--296.

Tiku, M. L. and Vaughan, D. C. (1999) A family of short-tailed symmetric distributions. Technical report, McMaster University, Canada.

See Also

dtikuv, normal1.

Examples

m = 1.0; sigma = exp(0.5)
sy = sort(y <- rtikuv(n=1000, d=1, m=m, s=sigma))
fit = vglm(y ~ 1, fam=tikuv(d=1), trace=TRUE)
coef(fit, mat=TRUE)
(Cfit = Coef(fit))
mean(y)
hist(y, prob=TRUE)
lines(sy, dtikuv(sy, d=1, m=Cfit[1], s=Cfit[2]), col="red")