laplace {VGAM} | R Documentation |
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")
Arguments
llocation , lscale |
Character.
Parameter link functions for location parameter |
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 |
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).
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. Hence this
family function might be withdrawn from VGAM in the future.
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
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)