VGAM (version 0.9-1)

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.

p

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.

See Also

dt, koenker.

Examples

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)