R: Short-tailed Symmetric Distribution Family Function

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 = "identitylink", lsigma = "loge", 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.

### 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. (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

    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")
    ## End(Not run)

[Package VGAM version 1.0-3 Index]