VGAM (version 1.1-5)

sc.studentt2: Scaled Student t Distribution with 2 df Family Function

Description

Estimates the location and scale parameters of a scaled Student t distribution with 2 degrees of freedom, by maximum likelihood estimation.

Usage

sc.studentt2(percentile = 50, llocation = "identitylink", lscale = "loglink",
             ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")

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

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. 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 this 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 dsc.t2 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

dsc.t2, studentt2.

Examples

# NOT RUN {
set.seed(123); nn <- 1000
kdata <- data.frame(x2 = sort(runif(nn)))
kdata <- transform(kdata, mylocat = 1 + 3 * x2,
                          myscale = 1)
kdata <- transform(kdata, y = rsc.t2(nn, loc = mylocat, scale = myscale))
fit  <- vglm(y ~ x2, sc.studentt2(perc = c(1, 50, 99)), data = kdata)
fit2 <- vglm(y ~ x2,    studentt2(df = 2), data = kdata)  # 'same' as fit

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

# Nice plot of the results
# }
# NOT RUN {
 plot(y ~ x2, data = kdata, col = "blue", las = 1,
     sub  = paste("n =", nn),
     main = "Fitted quantiles/expectiles using the sc.studentt2() distribution")
matplot(with(kdata, x2), fitted(fit), add = TRUE, type = "l", lwd = 3)
legend("bottomright", lty = 1:3, lwd = 3, legend = colnames(fitted(fit)),
       col = 1:3) 
# }
# NOT RUN {
fit@extra$percentile  # Sample quantiles
# }