Expectiles-Exponential: Expectiles of the Exponential Distribution

Description

Density function, distribution function, and expectile function and random generation for the distribution associated with the expectiles of an exponential distribution.

Usage

deexp(x, rate = 1, log = FALSE)
peexp(q, rate = 1, log = FALSE)
qeexp(p, rate = 1, Maxit_nr = 10, Tol_nr = 1.0e-6)
reexp(n, rate = 1)

Arguments

x, p, q

See deunif.

n, rate, log

See rexp.

Maxit_nr, Tol_nr

See deunif.

Value

deexp(x) gives the density function $g(x)$. peexp(q) gives the distribution function $G(q)$. qeexp(p) gives the expectile function: the value $y$ such that $G(y)=p$. reexp(n) gives $n$ random variates from $G$.

Details

General details are given in deunif including a note regarding the terminology used. Here, exp corresponds to the distribution of interest, $F$, and eexp corresponds to $G$. The addition of ``e'' is for the `other' distribution associated with the parent distribution. Thus deexp is for $g$, peexp is for $G$, qeexp is for the inverse of $G$, reexp generates random variates from $g$.

For qeexp the Newton-Raphson algorithm is used to solve for $y$ satisfying $p = G(y)$. Numerical problems may occur when values of p are very close to 0 or 1.

Examples

Run this code

my_p = 0.25; y = rexp(nn <- 1000)
(myexp = qeexp(my_p))
sum(myexp - y[y <= myexp]) / sum(abs(myexp - y))  # Should be my_p

par(mfrow=c(2,1))
yy = seq(-0, 4, len = nn)
plot(yy, deexp(yy),  col = "blue", ylim = 0:1, xlab = "y", ylab = "g(y)",
     type = "l", main = "g(y) for Exp(1); dotted green is f(y) = dexp(y)")
lines(yy, dexp(yy), col="darkgreen", lty="dotted", lwd=2) # 'original'

plot(yy, peexp(yy), type = "l", col = "blue", ylim = 0:1,
     xlab = "y", ylab = "G(y)", main = "G(y) for Exp(1)")
abline(v = 1, h = 0.5, col = "red", lty = "dashed")
lines(yy, pexp(yy), col = "darkgreen", lty = "dotted", lwd = 2)

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