VGAM (version 1.1-2)

tikuv: Short-tailed Symmetric Distribution Family Function

Description

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

Usage

tikuv(d, lmean = "identitylink", lsigma = "loglink", isigma = NULL,
      zero = "sigma")

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 choices.

isigma

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

zero

A vector specifying which linear/additive predictors are modelled as intercept-only. The values can be from the set {1,2}, corresponding respectively to \(\mu\), \(\sigma\). If zero = NULL then all linear/additive predictors are modelled as a linear combination of the explanatory variables. For many data sets having zero = 2 is a good idea. See CommonVGAMffArguments for information.

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, uninormal.

Examples

# NOT RUN {
m <- 1.0; sigma <- exp(0.5)
tdata <- data.frame(y = rtikuv(n = 1000, d = 1, m = m, s = sigma))
tdata <- transform(tdata, sy = sort(y))
fit <- vglm(y ~ 1, tikuv(d = 1), data = tdata, trace = TRUE)
coef(fit, matrix = TRUE)
(Cfit <- Coef(fit))
with(tdata, mean(y))
# }
# NOT RUN {
 with(tdata, hist(y, prob = TRUE))
lines(dtikuv(sy, d = 1, m = Cfit[1], s = Cfit[2]) ~ sy, data = tdata, col = "orange") 
# }