VGAM (version 1.1-9)

laplace: Laplace Regression Family Function

Description

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

Usage

laplace(llocation = "identitylink", lscale = "loglink",
  ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Arguments

llocation, lscale

Character. Parameter link functions for location parameter \(a\) and scale parameter \(b\). See Links for more choices.

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.

imethod

Initialization method. Either the value 1 or 2.

zero

See CommonVGAMffArguments for information.

Author

T. W. Yee

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. Hence this family function might be withdrawn from VGAM in the future.

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 $$f(y) = \frac{1}{2b} \exp \left( - \frac{|y-a|}{b} \right) $$ where \(-\infty<y<\infty\), \(-\infty<a<\infty\) 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).

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

ldata <- data.frame(y = rlaplace(nn <- 100, 2, scale = exp(1)))
fit <- vglm(y  ~ 1, laplace, ldata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
with(ldata, median(y))

ldata <- data.frame(x = runif(nn <- 1001))
ldata <- transform(ldata, y = rlaplace(nn, 2, scale = exp(-1 + 1*x)))
coef(vglm(y ~ x, laplace(iloc = 0.2, imethod = 2, zero = 1), ldata,
          trace = TRUE), matrix = TRUE)