VGAM (version 1.1-14)

amlpoisson: Poisson Regression by Asymmetric Maximum Likelihood Estimation

Description

Poisson quantile regression estimated by maximizing an asymmetric likelihood function.

Usage

amlpoisson(w.aml = 1, parallel = FALSE, imethod = 1, digw = 4,
           link = "loglink")

Value

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

and vgam.

Details

This method was proposed by Efron (1992) and full details can be obtained there. The model is essentially a Poisson regression model (see poissonff) but the usual deviance is replaced by an asymmetric squared error loss function; it is multiplied by \(w.aml\) for positive residuals. The solution is the set of regression coefficients that minimize the sum of these deviance-type values over the data set, weighted by the weights argument (so that it can contain frequencies). Newton-Raphson estimation is used here.

References

Efron, B. (1991). Regression percentiles using asymmetric squared error loss. Statistica Sinica, 1, 93--125.

Efron, B. (1992). Poisson overdispersion estimates based on the method of asymmetric maximum likelihood. Journal of the American Statistical Association, 87, 98--107.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33--50.

Newey, W. K. and Powell, J. L. (1987). Asymmetric least squares estimation and testing. Econometrica, 55, 819--847.

See Also

amlnormal, amlbinomial, extlogF1, alaplace1.

Examples

set.seed(1234)
mydat <- data.frame(x = sort(runif(nn <- 200)))
mydat <- transform(mydat, y = rpois(nn, exp(0 - sin(8*x))))
(fit <- vgam(y ~ s(x), fam = amlpoisson(w.aml = c(0.02, 0.2, 1, 5, 50)),
             mydat, trace = TRUE))
fit@extra

if (FALSE) {
# Quantile plot
with(mydat, plot(x, jitter(y), col = "blue", las = 1, main =
     paste(paste(round(fit@extra$percentile, digits = 1), collapse = ", "),
           "percentile-expectile curves")))
with(mydat, matlines(x, fitted(fit), lwd = 2)) }