# amlpoisson: Poisson Regression by Asymmetric Maximum Likelihood... In VGAM: Vector Generalized Linear and Additive Models

## Description

Poisson quantile regression estimated by maximizing an asymmetric likelihood function.

## Usage

 ```1 2``` ```amlpoisson(w.aml = 1, parallel = FALSE, imethod = 1, digw = 4, link = "loglink") ```

## 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. `link` See `poissonff`.

## 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.

## 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).

## Note

On fitting, the `extra` slot has list components `"w.aml"` and `"percentile"`. The latter is the percent of observations below the “w-regression plane”, which is the fitted values. Also, the individual deviance values corresponding to each element of the argument `w.aml` is stored in the `extra` slot.

For `amlpoisson` objects, methods functions for the generic functions `qtplot` and `cdf` have not been written yet.

About the jargon, Newey and Powell (1987) used the name expectiles for regression surfaces obtained by asymmetric least squares. This was deliberate so as to distinguish them from the original regression quantiles of Koenker and Bassett (1978). Efron (1991) and Efron (1992) use the general name regression percentile to apply to all forms of asymmetric fitting. Although the asymmetric maximum likelihood method very nearly gives regression percentiles in the strictest sense for the normal and Poisson cases, the phrase quantile regression is used loosely in this VGAM documentation.

In this documentation the word quantile can often be interchangeably replaced by expectile (things are informal here).

Thomas W. Yee

## 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`, `extlogF1`, `alaplace1`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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)) ## End(Not run) ```