Expectiles-Koenker: Expectiles/Quantiles of the Koenker Distribution

Description

Density function, distribution function, and quantile/expectile function and random generation for the Koenker distribution.

Usage

dkoenker(x, location = 0, scale = 1, log = FALSE)
pkoenker(q, location = 0, scale = 1, log = FALSE)
qkoenker(p, location = 0, scale = 1)
rkoenker(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.

Value

dkoenker(x) gives the density function. pkoenker(q) gives the distribution function. qkoenker(p) gives the expectile and quantile function. rkoenker(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 Koenker distribution. Further details about this distribution are given in koenker.

Examples

Run this code

my_p = 0.25; y = rkoenker(nn <- 5000)
(myexp = qkoenker(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 = rkoenker(nn, myloc, myscale)
(myexp = qkoenker(my_p, myloc, myscale))
sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy)) # Should be my_p
pkoenker(mean(yy), myloc, myscale)  #  Should be 0.5
abs(qkoenker(0.5, myloc, myscale) - mean(yy)) #  Should be 0
abs(pkoenker(myexp, myloc, myscale) - my_p)  #  Should be 0
integrate(f = dkoenker, lower = -Inf, upper = Inf,
          locat = myloc, scale = myscale) #  Should be 1

y <- seq(-7, 7, len = 201)
max(abs(dkoenker(y) - dt(y / sqrt(2), df = 2) / sqrt(2))) # Should be 0
plot(y, dkoenker(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)