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)

Arguments

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 runif.

lower.tail, log.p

Same meaning as in punif or qunif.

Maxit.nr

Numeric. Maximum number of Newton-Raphson iterations allowed. A warning is issued if convergence is not obtained for all p values.

Tol.nr

Numeric. Small positive value specifying the tolerance or precision to which the expectiles are computed.

Details

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 $\int_{-\infty}^{y} x f(x) \, dx$ 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.

Value

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 $G$.

References

Jones, M. C. (1994). Expectiles and M-quantiles are quantiles. Statistics and Probability Letters, 20, 149–153.

Author

T. W. Yee and Kai Huang

Examples

my.p <- 0.25; y <- runif(nn <- 1000)
(myexp <- qeunif(my.p))
#> [1] 0.3660254
sum(myexp - y[y <= myexp]) / sum(abs(myexp - y))  # Should be my.p
#> [1] 0.264408
# 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
#> [1] 0.264408
# 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))
#> [1] 3.562178
sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy))  # Should be my.p
#> [1] 0.2429901
peunif(mymin, mymin, mymax)     #  Should be 0
#> [1] 0
peunif(mymax, mymin, mymax)     #  Should be 1
#> [1] 1
peunif(mean(yy), mymin, mymax)  #  Should be 0.5
#> [1] 0.5116885
abs(qeunif(0.5, mymin, mymax) - mean(yy))  #  Should be 0
#> [1] 0.04091519
abs(qeunif(0.5, mymin, mymax) - (mymin+mymax)/2)  #  Should be 0
#> [1] 0
abs(peunif(myexp, mymin, mymax) - my.p)  #  Should be 0
#> [1] 2.220446e-16
integrate(f = deunif, lower = mymin - 3, upper = mymax + 3,
          min = mymin, max = mymax)  # Should be 1
#> 1.000003 with absolute error < 1e-04

if (FALSE) { # \dontrun{
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 = "green", 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 = "green", lty = "dotted", lwd = 2)
abline(v = 0.5, h = 0.5, col = "red", lty = "dashed") } # }