Exponential expectile regression estimated by maximizing an asymmetric likelihood function.
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.
Integer, either 1 or 2 or 3. Initialization method. Choose another value if convergence fails.
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
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
weights argument (so that it can contain frequencies).
Newton-Raphson estimation is used here.
An object of class
The object is used by modelling functions such as
Note that the
link argument of
amlexponential are currently different: one is the
rate parameter and the other is the mean (expectile) parameter.
w.aml has more than one value then the value returned by
deviance is the sum of all the (weighted) deviances taken over
See Equation (1.6) of Efron (1992).
On fitting, the
extra slot has list components
"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
w.aml is stored in the
amlexponential objects, methods functions for the generic functions
cdf have not been written yet.
amlpoisson about comments on the jargon, e.g.,
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.
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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 ## 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)