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 - δ)^{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.

Examples

nn = 1000; delta = 0
mygamma = 1         # log link ==> 0 is the answer
ldat = data.frame(y = delta + mygamma/rnorm(nn)^2) # Levy(mygamma, delta)

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

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

[Package VGAM version 0.8-4 Index]