VGAM (version 0.8-7)

slash: Slash Distribution Family Function

Description

Estimates the two parameters of the slash distribution by maximum likelihood estimation.

Usage

slash (lmu="identity", lsigma="loge", emu=list(), esigma=list(),
       imu=NULL, isigma=NULL, iprobs = c(0.1, 0.9), nsimEIM=250,
       zero=NULL, smallno = .Machine$double.eps*1000)

Arguments

lmu, lsigma

Parameter link functions applied to the $\mu$ and $\sigma$ parameters, respectively. See Links for more choices.

emu, esigma

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

imu, isigma

Initial values. A NULL means an initial value is chosen internally. See CommonVGAMffArguments for more information.

iprobs

Used to compute the initial values for mu. This argument is fed into the probs argument of quantile, and then a grid between these two points is used to evaluate the lo

nsimEIM, zero

See CommonVGAMffArguments for more information.

smallno

Small positive number, used to test for the singularity.

Value

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

Details

The standard slash distribution is the distribution of the ratio of a standard normal variable to an independent standard uniform(0,1) variable. It is mainly of use in simulation studies. One of its properties is that it has heavy tails, similar to those of the Cauchy. The general slash distribution can be obtained by replacing the univariate normal variable by a general normal $N(\mu,\sigma)$ random variable. It has a density that can be written as $$f(y) = \left{ \begin{array}{cl} 1/(2 \sigma \sqrt(2 \pi)) & if y=\mu, \ 1-\exp(-(((y-\mu)/\sigma)^2)/2))/(\sqrt(2 pi) \sigma ((y-\mu)/\sigma)^2) & if y \ne \mu. \end{array} \right .$$ where $\mu$ and $\sigma$ are the mean and standard deviation of the univariate normal distribution respectively.

References

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley. Kafadar, K. (1982) A Biweight Approach to the One-Sample Problem Journal of the American Statistical Association, 77, 416--424.

See Also

rslash.

Examples

sdata = data.frame(y = rslash(n=1000, mu=4, sigma=exp(2)))
fit = vglm(y ~ 1, slash, sdata, trace=TRUE) 
coef(fit, matrix=TRUE)
Coef(fit)
summary(fit)