tikuv {VGAM} | R Documentation

## 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 choices.
`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 predictors are modelled as a linear combination of the explanatory variables. For many data sets having `zero=2` is a good idea.

### Details

The short-tailed symmetric distribution of Tiku and Vaughan (1999) has a probability density function that can be written

f(y) = (K/(sqrt(2*pi)*sigma)) * [1 + (1/(2*h)) * ((y-mu)/sigma)^2]^2 * exp( -0.5 * (y-mu)^2/ sigma^2)

where h=2-d>0, K is a function of h, -Inf < y < Inf, sigma > 0. The mean of Y is E(Y) = mu and this is returned as the fitted values.

### 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`.

### Note

The density function is the product of a univariate normal density and a polynomial in the response y. The distribution is bimodal if d>0, else is unimodal. A normal distribution arises as the limit as d approaches -Inf, i.e., as h approaches Inf. Fisher scoring is implemented. After fitting the value of `d` is stored in `@misc` with component name `d`.

### Author(s)

Thomas W. Yee

### References

Akkaya, A. D. and Tiku, M. L. (2006) Short-tailed distributions and inliers. Test, 15(2), in press.

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)
    ## Not run:
    hist(y, prob=TRUE)
    lines(sy, dtikuv(sy, d=1, m=Cfit[1], s=Cfit[2]), col="red")
    ## End(Not run)

[Package VGAM version 0.7-9 Index]