Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

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

1 2 |

`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 |

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).

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

and `vgam`

.

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Hence this family function might be withdrawn from VGAM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.

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

T. W. Yee

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`

.

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)
``` |

```
Loading required package: stats4
Loading required package: splines
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] 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
```

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