laplace {VGAM} | R Documentation

## Laplace Distribution

### Description

Maximum likelihood estimation of the 2-parameter classical Laplace distribution.

### Usage

    laplace(llocation="identity", lscale="loge",
            elocation=list(), escale=list(),
            ilocation=NULL, iscale=NULL,
            method.init=1, zero=2)

### Arguments

`llocation, lscale` | Character. Parameter link functions for location parameter a and scale parameter b. See `Links` for more choices.
`elocation, escale` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`ilocation, iscale` | Optional initial values. If given, it must be numeric and values are recycled to the appropriate length. The default is to choose the value internally.
`method.init` | Initialization method. Either the value 1 or 2.
`zero` | See `CommonVGAMffArguments` for more information.

### Details

The Laplace distribution is often known as the double-exponential distribution and, for modelling, has heavier tail than the normal distribution. The Laplace density function is

f(y) = (1/(2b)) exp( -|y-a|/b )

where -Inf<y<Inf, -Inf<a<Inf and b>0. Its mean is a and its variance is 2b^2. This parameterization is called the classical Laplace distribution by Kotz et al. (2001), and the density is symmetric about a.

For `y ~ 1` (where `y` is the response) the maximum likelihood estimate (MLE) for the location parameter is the sample median, and the MLE for b is `mean(abs(y-location))` (replace location by its MLE if unknown).

### Value

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

### Warning

This family function has not been fully tested. The MLE regularity conditions do not hold for this distribution, therefore misleading inferences may result, e.g., in the `summary` and `vcov` of the object.

### Note

This family function uses Fisher scoring. Convergence may be slow for non-intercept-only models; half-stepping is frequently required.

### Author(s)

T. W. Yee

### References

Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Boston: Birkhauser.

### See Also

`rlaplace`, `alaplace2` (which differs slightly from this parameterization), `exponential`, `median`.

### Examples

    y = rlaplace(n <- 100, loc=2, scale=exp(1))
    fit = vglm(y  ~ 1, laplace, trace=TRUE, crit="l")
    coef(fit, matrix=TRUE)
    Coef(fit)
    median(y)

    x = runif(n <- 1001)
    y = rlaplace(n, loc=2, scale=exp(-1+1*x))
    fit = vglm(y  ~ x, laplace(iloc=0.2, meth=2, zero=1), trace=TRUE)
    coef(fit, matrix=TRUE)

[Package VGAM version 0.7-9 Index]