VGAM (version 0.7-1)

amlexponential: Exponential Regression by Asymmetric Maximum Likelihood Estimation

Description

Exponential expectile regression estimated by maximizing an asymmetric likelihood function.

Usage

amlexponential(w.aml = 1, parallel = FALSE, method.init = 1, digw = 4,
               link = "loge", earg = list())

Value

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

Warning

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).

Details

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.

References

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

See Also

exponential, amlbinomial, amlpoisson, amlnormal, alaplace1.

Examples

nn = 2000
mydat = data.frame(x = seq(0, 1, length=nn))
mydat = transform(mydat, mu = loge(-0+1.5*x+0.2*x^2, inverse=TRUE))
mydat = transform(mydat, mu = loge(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

# 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, dig=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, dig=1), collapse=", "),
           "percentile curves are red")))
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="red"))