# laplace: Laplace Regression Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2``` ```laplace(llocation = "identitylink", lscale = "loglink", ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale") ```

## 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.

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 Hence this family function might be withdrawn from VGAM in the future.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

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.

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```ldata <- data.frame(y = rlaplace(nn <- 100, loc = 2, scale = exp(1))) fit <- vglm(y ~ 1, laplace, data = ldata, trace = TRUE, crit = "l") coef(fit, matrix = TRUE) Coef(fit) with(ldata, median(y)) ldata <- data.frame(x = runif(nn <- 1001)) ldata <- transform(ldata, y = rlaplace(nn, loc = 2, scale = exp(-1 + 1*x))) coef(vglm(y ~ x, laplace(iloc = 0.2, imethod = 2, zero = 1), data = ldata, trace = TRUE), matrix = TRUE) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -270.6351
VGLM    linear loop  2 :  loglikelihood = -270.63491
VGLM    linear loop  3 :  loglikelihood = -270.63491
location loge(scale)
(Intercept) 1.808314    1.013202
location    scale
1.808314 2.754406
 1.808314
VGLM    linear loop  1 :  loglikelihood = -2438.45022
VGLM    linear loop  2 :  loglikelihood = -2580.31389
Taking a modified step.
VGLM    linear loop  2 :  loglikelihood = -2065.34264
VGLM    linear loop  3 :  loglikelihood = -2001.714
VGLM    linear loop  4 :  loglikelihood = -1459.61704
VGLM    linear loop  5 :  loglikelihood = -1232.47531
VGLM    linear loop  6 :  loglikelihood = -1194.61584
VGLM    linear loop  7 :  loglikelihood = -1193.36466
VGLM    linear loop  8 :  loglikelihood = -1193.36082
VGLM    linear loop  9 :  loglikelihood = -1193.3604
VGLM    linear loop  10 :  loglikelihood = -1193.35998
VGLM    linear loop  11 :  loglikelihood = -1193.36091
Taking a modified step...
VGLM    linear loop  11 :  loglikelihood = -1193.35994
VGLM    linear loop  12 :  loglikelihood = -1193.36103
Taking a modified step.......
VGLM    linear loop  12 :  loglikelihood = -1193.35994
VGLM    linear loop  13 :  loglikelihood = -1193.36102
Taking a modified step.......
VGLM    linear loop  13 :  loglikelihood = -1193.35993
location loge(scale)
(Intercept) 2.016141   -1.017678
x           0.000000    1.018386
Warning message:
In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.