Expectiles of the Uniform Distribution
Density function, distribution function, and expectile function and random generation for the distribution associated with the expectiles of a uniform distribution.
deunif(x, min = 0, max = 1, log = FALSE)
peunif(q, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)
qeunif(p, min = 0, max = 1, Maxit.nr = 10, Tol.nr = 1.0e-6,
lower.tail = TRUE, log.p = FALSE)
reunif(n, min = 0, max = 1)x, q |
Vector of expectiles. See the terminology note below. |
p |
Vector of probabilities. These should lie in (0,1). |
n, min, max, log |
See |
lower.tail, log.p |
|
Maxit.nr |
Numeric.
Maximum number of Newton-Raphson iterations allowed.
A warning is issued if convergence is not obtained for all |
Tol.nr |
Numeric. Small positive value specifying the tolerance or precision to which the expectiles are computed. |
Jones (1994) elucidated on the property that the expectiles of a random variable X with distribution function F(x) correspond to the quantiles of a distribution G(x) where G is related by an explicit formula to F. In particular, let y be the p-expectile of F. Then y is the p-quantile of G where
p = G(y) = (P(y) - y F(y)) / (2[P(y) - y F(y)] + y - mu),
and mu is the mean of X. The derivative of G is
g(y) = ( mu F(y) - P(y)) / (2[P(y) - y F(y)] + y - mu)^2 .
Here, P(y) is the partial moment int^{y} x f(x) dx and 0 < p < 1. The 0.5-expectile is the mean mu and the 0.5-quantile is the median.
A note about the terminology used here.
Recall in the S language there are the dpqr-type functions
associated with a distribution, e.g.,
dunif,
punif,
qunif,
runif,
for the uniform distribution.
Here,
unif corresponds to F and
eunif corresponds to G.
The addition of “e” (for expectile) is for the ‘other’
distribution associated with the parent distribution.
Thus
deunif is for g,
peunif is for G,
qeunif is for the inverse of G,
reunif generates random variates from g.
For qeunif 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.
deunif(x) gives the density function g(x).
peunif(q) gives the distribution function G(q).
qeunif(p) gives the expectile function:
the expectile y such that G(y) = p.
reunif(n) gives n random variates from G.
T. W. Yee and Kai Huang
Jones, M. C. (1994). Expectiles and M-quantiles are quantiles. Statistics and Probability Letters, 20, 149–153.
Yee, T. W. (2012). Vector generalized linear and additive quantile and expectile regression. In preparation.
my.p <- 0.25; y <- runif(nn <- 1000)
(myexp <- qeunif(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 uniform
mymin <- 1; mymax <- 8
yy <- runif(nn, mymin, mymax)
(myexp <- qeunif(my.p, mymin, mymax))
sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy)) # Should be my.p
peunif(mymin, mymin, mymax) # Should be 0
peunif(mymax, mymin, mymax) # Should be 1
peunif(mean(yy), mymin, mymax) # Should be 0.5
abs(qeunif(0.5, mymin, mymax) - mean(yy)) # Should be 0
abs(qeunif(0.5, mymin, mymax) - (mymin+mymax)/2) # Should be 0
abs(peunif(myexp, mymin, mymax) - my.p) # Should be 0
integrate(f = deunif, lower = mymin - 3, upper = mymax + 3,
min = mymin, max = mymax) # Should be 1
## Not run:
par(mfrow = c(2,1))
yy <- seq(0.0, 1.0, len = nn)
plot(yy, deunif(yy), type = "l", col = "blue", ylim = c(0, 2),
xlab = "y", ylab = "g(y)", main = "g(y) for Uniform(0,1)")
lines(yy, dunif(yy), col = "darkgreen", lty = "dotted", lwd = 2) # 'original'
plot(yy, peunif(yy), type = "l", col = "blue", ylim = 0:1,
xlab = "y", ylab = "G(y)", main = "G(y) for Uniform(0,1)")
abline(a = 0.0, b = 1.0, col = "darkgreen", lty = "dotted", lwd = 2)
abline(v = 0.5, h = 0.5, col = "red", lty = "dashed")
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