VGAM (version 1.0-4)

# 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,

## Arguments

w.aml

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.

parallel

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.

imethod

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

digw

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

See `poissonff`.

## Value

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

## Warning

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).

## 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.

`amlnormal`, `amlbinomial`, `alaplace1`.

## Examples

Run this code
``````# NOT RUN {
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

# }
# NOT RUN {
# 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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