VGAM (version 1.1-6)

amlpoisson: Poisson Regression by Asymmetric Maximum Likelihood Estimation


Poisson quantile regression estimated by maximizing an asymmetric likelihood function.


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



Numeric, a vector of positive constants controlling the percentiles. The larger the value the larger the fitted percentile value (the proportion of points below the ``w-regression plane''). The default value of unity results in the ordinary maximum likelihood (MLE) solution.


If w.aml has more than one value then this argument allows the quantile curves to differ by the same amount as a function of the covariates. Setting this to be TRUE should force the quantile curves to not cross (although they may not cross anyway). See CommonVGAMffArguments for more information.


Integer, either 1 or 2 or 3. Initialization method. Choose another value if convergence fails.


Passed into Round as the digits argument for the w.aml values; used cosmetically for labelling.


See poissonff.


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


If w.aml has more than one value then the value returned by deviance is the sum of all the (weighted) deviances taken over all the w.aml values. See Equation (1.6) of Efron (1992).


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.


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.


Run this code
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))

# }
# 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)) 
# }

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