1\name{Expectiles-sc.t2}
2\alias{Expectiles-sc.t2}
3\alias{dsc.t2}
4\alias{psc.t2}
5\alias{qsc.t2}
6\alias{rsc.t2}
7\title{ Expectiles/Quantiles of the Scaled Student t Distribution with 2 Df}
8\description{
9  Density function, distribution function, and
10  quantile/expectile function and random generation for the
11  scaled Student t distribution with 2 degrees of freedom.
12
13
14}
15\usage{
16dsc.t2(x, location = 0, scale = 1, log = FALSE)
17psc.t2(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
18qsc.t2(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
19rsc.t2(n, location = 0, scale = 1)
20}
21%- maybe also 'usage' for other objects documented here.
22\arguments{
23  \item{x, q}{
24  Vector of expectiles/quantiles.
25  See the terminology note below.
26
27
28  }
29  \item{p}{
30  Vector of probabilities. % (tau or \eqn{\tau}).
31  These should lie in \eqn{(0,1)}.
32
33
34  }
35  \item{n, log}{See \code{\link[stats:Uniform]{runif}}.}
36  \item{location, scale}{
37  Location and scale parameters.
38  The latter should have positive values.
39  Values of these vectors are recyled.
40
41
42  }
43  \item{lower.tail, log.p}{
44  Same meaning as in \code{\link[stats:TDist]{pt}}
45  or \code{\link[stats:TDist]{qt}}.
46
47
48  }
49}
50\details{
51  A Student-t distribution with 2 degrees of freedom and
52  a scale parameter of \code{sqrt(2)} is equivalent to
53  the standard form of this distribution
54  (called Koenker's distribution below).
55  Further details about this distribution are given in
56  \code{\link{sc.studentt2}}.
57
58
59}
60\value{
61  \code{dsc.t2(x)} gives the density function.
62  \code{psc.t2(q)} gives the distribution function.
63  \code{qsc.t2(p)} gives the expectile and quantile function.
64  \code{rsc.t2(n)} gives \eqn{n} random variates.
65
66
67}
68\author{ T. W. Yee and Kai Huang }
69
70%\note{
71%}
72
73\seealso{
74  \code{\link[stats:TDist]{dt}},
75  \code{\link{sc.studentt2}}.
76
77
78}
79
80\examples{
81my.p <- 0.25; y <- rsc.t2(nn <- 5000)
82(myexp <- qsc.t2(my.p))
83sum(myexp - y[y <= myexp]) / sum(abs(myexp - y))  # Should be my.p
84# Equivalently:
85I1 <- mean(y <= myexp) * mean( myexp - y[y <= myexp])
86I2 <- mean(y >  myexp) * mean(-myexp + y[y >  myexp])
87I1 / (I1 + I2)  # Should be my.p
88# Or:
89I1 <- sum( myexp - y[y <= myexp])
90I2 <- sum(-myexp + y[y >  myexp])
91
92# Non-standard Koenker distribution
93myloc <- 1; myscale <- 2
94yy <- rsc.t2(nn, myloc, myscale)
95(myexp <- qsc.t2(my.p, myloc, myscale))
96sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy))  # Should be my.p
97psc.t2(mean(yy), myloc, myscale)  # Should be 0.5
98abs(qsc.t2(0.5, myloc, myscale) - mean(yy))  # Should be 0
99abs(psc.t2(myexp, myloc, myscale) - my.p)  # Should be 0
100integrate(f = dsc.t2, lower = -Inf, upper = Inf,
101          locat = myloc, scale = myscale)  # Should be 1
102
103y <- seq(-7, 7, len = 201)
104max(abs(dsc.t2(y) - dt(y / sqrt(2), df = 2) / sqrt(2)))  # Should be 0
105\dontrun{ plot(y, dsc.t2(y), type = "l", col = "blue", las = 1,
106     ylim = c(0, 0.4), main = "Blue = Koenker; orange = N(0, 1)")
107lines(y, dnorm(y), type = "l", col = "orange")
108abline(h = 0, v = 0, lty = 2) }
109}
110\keyword{distribution}
111