# 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. ```r 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 $$ p = G(y) = (P(y) - y F(y)) / (2[P(y) - y F(y)] + y - \mu),p = G(y) = (P(y) - y F(y)) / (2[P(y) - y F(y)] + y - mu), $$ and $mu$ is the mean of $X$. The derivative of $G$ is $$ g(y) = (\mu F(y) - P(y)) / (2[P(y) - y F(y)] + y - \mu)^2 .g(y) = ( mu F(y) - P(y)) / (2[P(y) - y F(y)] + y - mu)^2 . $$ 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. ## Returns `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(s) T. W. Yee and Kai Huang ## See Also `deexp`, `denorm`, `dunif`, `dsc.t2`. ## Examples ```r 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 = "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") ## End(Not run) ```

eunifUC function