VGAM (version 0.8-4.1)

koenker: Koenker's Distribution Family Function

Description

Estimates the location and scale parameters of Koenker's distribution by maximum likelihood estimation.

Usage

koenker(percentile = 50, llocation = "identity", lscale = "loge",
        elocation = list(), escale = list(), ilocation = NULL,
        iscale = NULL, imethod = 1, zero = 2)

Arguments

percentile

A numerical vector containing values between 0 and 100, which are the quantiles and expectiles. They will be returned as `fitted values'.

llocation, lscale, elocation, escale

See Links for more choices, and CommonVGAMffArguments.

ilocation, iscale, imethod, zero

See CommonVGAMffArguments for details.

Value

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

Details

Koenker (1993) solved for the distribution whose quantiles are equal to its expectiles. This is called Koenker's distribution here. Its canonical form has mean and mode at 0 and has a heavy tail (in fact, its variance is infinite).

The standard (``canonical'') form of Koenker's distribution can be endowed with a location and scale parameter. The standard form has a density that can be written as $$f(z) = 2 / (4 + z^2)^{3/2}$$ for real $y$. Then $z = (y-a)/b$ for location and scale parameters $a$ and $b > 0$. The mean of $Y$ is $a$. By default, $\eta_1=a)$ and $\eta_2=\log(b)$. The expectiles/quantiles corresponding to percentile are returned as the fitted values; in particular, percentile = 50 corresponds to the mean (0.5 expectile) and median (0.5 quantile).

Note that if $Y$ has a standard Koenker distribution then $Y = \sqrt{2} T_2$ where $T_2$ has a Student-t distribution with 2 degrees of freedom. The two parameters here can also be estimated using studentt2 by specifying df = 2 and making an adjustment for the scale parameter, however, this VGAM family function is more efficient since the EIM is known (Fisher scoring is implemented.)

References

Koenker, R. (1993) When are expectiles percentiles? (solution) Econometric Theory, 9, 526--527.

See Also

dkoenker, studentt2.

Examples

set.seed(123); nn <- 1000
kdat <- data.frame(x2 = sort(runif(nn)))
kdat <- transform(kdat, mylocat = 1 + 3 * x2,
                        myscale = 1)
kdat <- transform(kdat, y = rkoenker(nn, loc = mylocat, scale = myscale))
fit  <- vglm(y ~ x2, koenker(perc = c(1, 50, 99)), kdat, trace = TRUE)
fit2 <- vglm(y ~ x2, studentt2(df = 2), kdat, trace = TRUE) # 'same' as fit

coef(fit, matrix = TRUE)
head(fitted(fit))
head(predict(fit))

# Nice plot of the results
plot(y ~ x2, kdat, col = "blue", las = 1,
     sub  = paste("n =", nn),
     main = "Fitted quantiles/expectiles using Koenker's distribution")
matplot(with(kdat, x2), fitted(fit), add = TRUE, type = "l", lwd = 3)
legend("bottomright", lty = 1:3, lwd = 3, legend = colnames(fitted(fit)),
       col = 1:3)

fit@extra$percentile # Sample quantiles