levy {VGAM} | R Documentation

## 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.

### 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(b / (2 pi)) exp( -b / (2(y - a))) / (y - a)^{3/2}

where a<y<Inf and b>0. 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.

### Value

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

### Author(s)

T. W. Yee

### 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))

[Package VGAM version 1.0-3 Index]