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)
weights(fit, type="w")[1:4,]

[Package VGAM version 0.7-7 Index]