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 0.7-7 Index]