Expectiles-sc.t2: Expectiles/Quantiles of the Scaled Student t Distribution with 2 Df

Description

Density function, distribution function, and quantile/expectile function and random generation for the scaled Student t distribution with 2 degrees of freedom.

Usage

dsc.t2(x, location = 0, scale = 1, log = FALSE)
psc.t2(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qsc.t2(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rsc.t2(n, location = 0, scale = 1)

Arguments

x, q

Vector of expectiles/quantiles. See the terminology note below.

Vector of probabilities. These should lie in $(0,1)$.

n, log

See runif.

location, scale

Location and scale parameters. The latter should have positive values. Values of these vectors are recyled.

lower.tail, log.p

Same meaning as in pt or qt.

Value

dsc.t2(x) gives the density function. psc.t2(q) gives the distribution function. qsc.t2(p) gives the expectile and quantile function. rsc.t2(n) gives $n$ random variates.

Details

A Student-t distribution with 2 degrees of freedom and a scale parameter of sqrt(2) is equivalent to the standard form of this distribution (called Koenker's distribution below). Further details about this distribution are given in sc.studentt2.

Examples

Run this code

my.p <- 0.25; y <- rsc.t2(nn <- 5000)
(myexp <- qsc.t2(my.p))
sum(myexp - y[y <= myexp]) / sum(abs(myexp - y))  # Should be my.p
# Equivalently:
I1 <- mean(y <= myexp) * mean( myexp - y[y <= myexp])
I2 <- mean(y >  myexp) * mean(-myexp + y[y >  myexp])
I1 / (I1 + I2)  # Should be my.p
# Or:
I1 <- sum( myexp - y[y <= myexp])
I2 <- sum(-myexp + y[y >  myexp])

# Non-standard Koenker distribution
myloc <- 1; myscale <- 2
yy <- rsc.t2(nn, myloc, myscale)
(myexp <- qsc.t2(my.p, myloc, myscale))
sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy))  # Should be my.p
psc.t2(mean(yy), myloc, myscale)  # Should be 0.5
abs(qsc.t2(0.5, myloc, myscale) - mean(yy))  # Should be 0
abs(psc.t2(myexp, myloc, myscale) - my.p)  # Should be 0
integrate(f = dsc.t2, lower = -Inf, upper = Inf,
          locat = myloc, scale = myscale)  # Should be 1

y <- seq(-7, 7, len = 201)
max(abs(dsc.t2(y) - dt(y / sqrt(2), df = 2) / sqrt(2)))  # Should be 0
plot(y, dsc.t2(y), type = "l", col = "blue", las = 1,
     ylim = c(0, 0.4), main = "Blue = Koenker; orange = N(0, 1)")
lines(y, dnorm(y), type = "l", col = "orange")
abline(h = 0, v = 0, lty = 2)