Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

Exponential expectile regression estimated by maximizing an asymmetric likelihood function.

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

`w.aml` |
Numeric, a vector of positive constants controlling the expectiles. The larger the value the larger the fitted expectile 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 |

The general methodology behind this VGAM family function
is given in Efron (1992) and full details can be obtained there.
This model is essentially an exponential regression model
(see `exponential`

) 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`

.

Note that the `link`

argument of `exponential`

and
`amlexponential`

are currently different: one is the
rate parameter and the other is the mean (expectile) parameter.

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 `amlexponential`

objects, methods functions for the generic
functions `qtplot`

and `cdf`

have not been written yet.

See `amlpoisson`

about comments on the jargon, e.g.,
*expectiles* etc.

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

Thomas W. Yee

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

`exponential`

,
`amlbinomial`

,
`amlpoisson`

,
`amlnormal`

,
`extlogF1`

,
`alaplace1`

,
`lms.bcg`

,
`deexp`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | ```
nn <- 2000
mydat <- data.frame(x = seq(0, 1, length = nn))
mydat <- transform(mydat,
mu = loglink(-0 + 1.5*x + 0.2*x^2, inverse = TRUE))
mydat <- transform(mydat, mu = loglink(0 - sin(8*x), inverse = TRUE))
mydat <- transform(mydat, y = rexp(nn, rate = 1/mu))
(fit <- vgam(y ~ s(x, df=5), amlexponential(w=c(0.001, 0.1, 0.5, 5, 60)),
mydat, trace = TRUE))
fit@extra
## Not run: # These plots are against the sqrt scale (to increase clarity)
par(mfrow = c(1,2))
# Quantile plot
with(mydat, plot(x, sqrt(y), col = "blue", las = 1, main =
paste(paste(round(fit@extra$percentile, digits = 1), collapse=", "),
"percentile-expectile curves")))
with(mydat, matlines(x, sqrt(fitted(fit)), lwd = 2, col = "blue", lty=1))
# Compare the fitted expectiles with the quantiles
with(mydat, plot(x, sqrt(y), col = "blue", las = 1, main =
paste(paste(round(fit@extra$percentile, digits = 1), collapse=", "),
"percentile curves are orange")))
with(mydat, matlines(x, sqrt(fitted(fit)), lwd = 2, col = "blue", lty=1))
for (ii in fit@extra$percentile)
with(mydat, matlines(x, sqrt(qexp(p = ii/100, rate = 1/mu)),
col = "orange"))
## End(Not run)
``` |

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