Laplace Regression Family Function

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

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

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.

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 Hence this family function might be withdrawn from VGAM in the future.

Note

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

Examples

ldata <- data.frame(y = rlaplace(nn <- 100, 2, scale = exp(1)))
fit <- vglm(y  ~ 1, laplace, ldata, trace = TRUE)
#> Iteration 1: loglikelihood = -263.08587
#> Iteration 2: loglikelihood = -263.08587
#> Iteration 3: loglikelihood = -263.08587
coef(fit, matrix = TRUE)
#>             location loglink(scale)
#> (Intercept) 1.482043      0.9377115
Coef(fit)
#> location    scale 
#> 1.482043 2.554130 
with(ldata, median(y))
#> [1] 1.482043

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)
#> Iteration 1: loglikelihood = -2425.44872
#> Iteration 2: loglikelihood = -2509.45689
#> Taking a modified step.
#> Iteration  2 :  loglikelihood = -2036.69137
#> Iteration 3: loglikelihood = -1991.03747
#> Iteration 4: loglikelihood = -1452.19732
#> Iteration 5: loglikelihood = -1226.14944
#> Iteration 6: loglikelihood = -1180.47346
#> Iteration 7: loglikelihood = -1178.5229
#> Iteration 8: loglikelihood = -1178.51723
#> Iteration 9: loglikelihood = -1178.51699
#> Iteration 10: loglikelihood = -1178.51775
#> Taking a modified step...
#> Iteration  10 :  loglikelihood = -1178.51694
#> Iteration 11: loglikelihood = -1178.51765
#> Taking a modified step...
#> Iteration  11 :  loglikelihood = -1178.51689
#> Iteration 12: loglikelihood = -1178.51776
#> Taking a modified step.....
#> Iteration  12 :  loglikelihood = -1178.51687
#> Iteration 13: loglikelihood = -1178.51779
#> Taking a modified step.......
#> Iteration  13 :  loglikelihood = -1178.51687
#>             location loglink(scale)
#> (Intercept) 2.001053      -1.112487
#> x           0.000000       1.160548