amlpoisson | R Documentation |

Poisson quantile regression estimated by maximizing an asymmetric likelihood function.

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

`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 |

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

`digw ` |
Passed into |

`link` |
See |

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.

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).

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

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`

.

```
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)
```

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