levy {VGAM} | R Documentation

## Levy Distribution Family Function

### Description

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

### Usage

    levy(delta = NULL, link.gamma = "loge", earg=list(),
         idelta = NULL, igamma = NULL)

### Arguments

`delta` | Location parameter. May be assigned a known value, otherwise it is estimated (the default).
`link.gamma` | Parameter link function for the (positive) gamma parameter. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`idelta` | Initial value for the delta parameter (if it is to be estimated). By default, an initial value is chosen internally.
`igamma` | Initial value for the gamma parameter. By default, an initial value is chosen internally.

### Details

The Levy distribution is one of three stable distributions whose density function has a tractable form. The formula for the density is

f(y;gamma,delta) = sqrt(gamma / (2 pi)) exp( -gamma / (2(y - delta))) / (y - delta)^{3/2}

where delta<y<Inf and gamma>0. The mean does not exist.

### Value

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

### Note

If delta is given, then only one parameter is estimated and the default is eta1=log(gamma). If delta is not given, then eta2=delta.

### Author(s)

T. W. Yee

### References

Nolan, J. P. (2005) Stable Distributions: Models for Heavy Tailed Data.

### See Also

The Nolan article is at http://academic2.american.edu/~jpnolan/stable/chap1.pdf.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### Examples

    n = 1000
    mygamma = 1         # log link ==> 0 is the answer
    delta = 0
    y = delta + mygamma / rnorm(n)^2 # This is Levy(mygamma, delta)

    # Cf. Table 1.1 of Nolan for Levy(1,0)
    sum(y > 1) / length(y)  # Should be 0.6827
    sum(y > 2) / length(y)  # Should be 0.5205

    fit = vglm(y ~ 1, levy(delta=delta), trace=TRUE) # 1 parameter
    fit = vglm(y ~ 1, levy(idelta=delta, igamma=mygamma),
               trace=TRUE)    # 2 parameters
    coef(fit, matrix=TRUE)
    Coef(fit)
    summary(fit)
    head(weights(fit, type="w"))

[Package VGAM version 0.7-9 Index]