1\name{sc.studentt2} 2\alias{sc.studentt2} 3%- Also NEED an '\alias' for EACH other topic documented here. 4\title{ Scaled Student t Distribution with 2 df Family Function } 5\description{ 6 Estimates the location and scale parameters of 7 a scaled Student t distribution with 2 degrees of freedom, 8 by maximum likelihood estimation. 9 10 11} 12\usage{ 13sc.studentt2(percentile = 50, llocation = "identitylink", lscale = "loglink", 14 ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale") 15} 16%- maybe also 'usage' for other objects documented here. 17\arguments{ 18 \item{percentile}{ 19 A numerical vector containing values between 0 and 100, 20 which are the quantiles and expectiles. 21 They will be returned as `fitted values'. 22 23 24 } 25 \item{llocation, lscale}{ 26 See \code{\link{Links}} for more choices, 27 and \code{\link{CommonVGAMffArguments}}. 28 29 30 } 31 \item{ilocation, iscale, imethod, zero}{ 32 See \code{\link{CommonVGAMffArguments}} for details. 33 34 35 } 36} 37\details{ 38 Koenker (1993) solved for the distribution whose quantiles are 39 equal to its expectiles. 40 Its canonical form has mean and mode at 0, 41 and has a heavy tail (in fact, its variance is infinite). 42 43 44% This is called Koenker's distribution here. 45 46 47 The standard (``canonical'') form of this 48 distribution can be endowed with a location and scale parameter. 49 The standard form has a density 50 that can be written as 51 \deqn{f(z) = 2 / (4 + z^2)^{3/2}}{% 52 f(z) = 2 / (4 + z^2)^(3/2) 53 } 54 for real \eqn{y}. 55 Then \eqn{z = (y-a)/b} for location and scale parameters 56 \eqn{a} and \eqn{b > 0}. 57 The mean of \eqn{Y} is \eqn{a}{a}. 58 By default, \eqn{\eta_1=a)}{eta1=a} and 59 \eqn{\eta_2=\log(b)}{eta2=log(b)}. 60 The expectiles/quantiles corresponding to \code{percentile} 61 are returned as the fitted values; 62 in particular, \code{percentile = 50} corresponds to the mean 63 (0.5 expectile) and median (0.5 quantile). 64 65 66 Note that if \eqn{Y} has a standard \code{\link{dsc.t2}} 67 then \eqn{Y = \sqrt{2} T_2}{Y = sqrt(2) * T_2} where \eqn{T_2} 68 has a Student-t distribution with 2 degrees of freedom. 69 The two parameters here can also be estimated using 70 \code{\link{studentt2}} by specifying \code{df = 2} and making 71 an adjustment for the scale parameter, however, this \pkg{VGAM} 72 family function is more efficient since the EIM is known 73 (Fisher scoring is implemented.) 74 75 76} 77\value{ 78 An object of class \code{"vglmff"} (see \code{\link{vglmff-class}}). 79 The object is used by modelling functions such as \code{\link{vglm}}, 80 \code{\link{rrvglm}} 81 and \code{\link{vgam}}. 82 83 84} 85\references{ 86 87Koenker, R. (1993). 88When are expectiles percentiles? (solution) 89\emph{Econometric Theory}, 90\bold{9}, 526--527. 91 92 93} 94\author{ T. W. Yee } 95%\note{ 96% 97%} 98 99\seealso{ 100 \code{\link{dsc.t2}}, 101 \code{\link{studentt2}}. 102 103 104} 105\examples{ 106set.seed(123); nn <- 1000 107kdata <- data.frame(x2 = sort(runif(nn))) 108kdata <- transform(kdata, mylocat = 1 + 3 * x2, 109 myscale = 1) 110kdata <- transform(kdata, y = rsc.t2(nn, loc = mylocat, scale = myscale)) 111fit <- vglm(y ~ x2, sc.studentt2(perc = c(1, 50, 99)), data = kdata) 112fit2 <- vglm(y ~ x2, studentt2(df = 2), data = kdata) # 'same' as fit 113 114coef(fit, matrix = TRUE) 115head(fitted(fit)) 116head(predict(fit)) 117 118# Nice plot of the results 119\dontrun{ plot(y ~ x2, data = kdata, col = "blue", las = 1, 120 sub = paste("n =", nn), 121 main = "Fitted quantiles/expectiles using the sc.studentt2() distribution") 122matplot(with(kdata, x2), fitted(fit), add = TRUE, type = "l", lwd = 3) 123legend("bottomright", lty = 1:3, lwd = 3, legend = colnames(fitted(fit)), 124 col = 1:3) } 125 126fit@extra$percentile # Sample quantiles 127} 128\keyword{models} 129\keyword{regression} 130 131