VGAM (version 1.0-1)

levy: Levy Distribution Family Function

Description

Estimates the scale parameter of the Levy distribution by maximum likelihood estimation.

Usage

levy(location = 0, lscale = "loge", iscale = NULL)

Arguments

location

Location parameter. Must have a known value. Called $a$ below.

lscale

Parameter link function for the (positive) scale parameter $b$. See Links for more choices.

iscale

Initial value for the $b$ parameter. By default, an initial value is chosen internally.

Value

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

Details

The Levy distribution is one of three stable distributions whose density function has a tractable form. The formula for the density is $$f(y;b) = \sqrt{\frac{b}{2\pi}} \exp \left( \frac{-b}{2(y - a)} \right) / (y - a)^{3/2}$$ where $a0$. Note that if $a$ is very close to min(y) (where y is the response), then numerical problem will occur. The mean does not exist. The median is returned as the fitted values.

References

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

See Also

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

Examples

nn <- 1000; loc1 <- 0; loc2 <- 10
myscale <- 1  # log link ==> 0 is the answer
ldata <- data.frame(y1 = loc1 + myscale/rnorm(nn)^2,  # Levy(myscale, a)
                    y2 = rlevy(nn, loc = loc2, scale = exp(+2)))

# Cf. Table 1.1 of Nolan for Levy(1,0)
with(ldata, sum(y1 > 1) / length(y1))  # Should be 0.6827
with(ldata, sum(y1 > 2) / length(y1))  # Should be 0.5205

fit1 <- vglm(y1 ~ 1, levy(location = loc1), data = ldata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
summary(fit1)
head(weights(fit1, type = "work"))

fit2 <- vglm(y2 ~ 1, levy(location = loc2), data = ldata, trace = TRUE)
coef(fit2, matrix = TRUE)
Coef(fit2)
c(median = with(ldata, median(y2)), fitted.median = head(fitted(fit2), 1))