Expectiles of the Exponential Distribution
Density function, distribution function, and expectile function and random generation for the distribution associated with the expectiles of an exponential distribution.
deexp(x, rate = 1, log = FALSE)
peexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE)
qeexp(p, rate = 1, Maxit.nr = 10, Tol.nr = 1.0e-6,
lower.tail = TRUE, log.p = FALSE)
reexp(n, rate = 1)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.
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.
T. W. Yee and Kai Huang
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
## Not run: 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)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.