# Expectiles of the Normal Distribution Density function, distribution function, and expectile function and random generation for the distribution associated with the expectiles of a normal distribution. ```r denorm(x, mean = 0, sd = 1, log = FALSE) penorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE) qenorm(p, mean = 0, sd = 1, Maxit.nr = 10, Tol.nr = 1.0e-6, lower.tail = TRUE, log.p = FALSE) renorm(n, mean = 0, sd = 1) ``` ## Arguments - `x, p, q`: See `deunif`. - `n, mean, sd, log`: See `rnorm`. - `lower.tail, log.p`: Same meaning as in `pnorm` or `qnorm`. - `Maxit.nr, Tol.nr`: See `deunif`. ## Details General details are given in `deunif` including a note regarding the terminology used. Here, `norm` corresponds to the distribution of interest, $F$, and `enorm` corresponds to $G$. The addition of ```e`'' is for the `other' distribution associated with the parent distribution. Thus `denorm` is for $g$, `penorm` is for $G$, `qenorm` is for the inverse of $G$, `renorm` generates random variates from $g$. For `qenorm` 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 `denorm(x)` gives the density function $g(x)$. `penorm(q)` gives the distribution function $G(q)$. `qenorm(p)` gives the expectile function: the value $y$ such that $G(y)=p$. `renorm(n)` gives $n$ random variates from $G$. ## Author(s) T. W. Yee and Kai Huang ## See Also `deunif`, `deexp`, `dnorm`, `amlnormal`, `lms.bcn`. ## Examples ```r my.p <- 0.25; y <- rnorm(nn <- 1000) (myexp <- qenorm(my.p)) sum(myexp - y[y <= myexp]) / sum(abs(myexp - y)) # Should be my.p # Non-standard normal mymean <- 1; mysd <- 2 yy <- rnorm(nn, mymean, mysd) (myexp <- qenorm(my.p, mymean, mysd)) sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy)) # Should be my.p penorm(-Inf, mymean, mysd) # Should be 0 penorm( Inf, mymean, mysd) # Should be 1 penorm(mean(yy), mymean, mysd) # Should be 0.5 abs(qenorm(0.5, mymean, mysd) - mean(yy)) # Should be 0 abs(penorm(myexp, mymean, mysd) - my.p) # Should be 0 integrate(f = denorm, lower = -Inf, upper = Inf, mymean, mysd) # Should be 1 ## Not run: par(mfrow = c(2, 1)) yy <- seq(-3, 3, len = nn) plot(yy, denorm(yy), type = "l", col="blue", xlab = "y", ylab = "g(y)", main = "g(y) for N(0,1); dotted green is f(y) = dnorm(y)") lines(yy, dnorm(yy), col = "green", lty = "dotted", lwd = 2) # 'original' plot(yy, penorm(yy), type = "l", col = "blue", ylim = 0:1, xlab = "y", ylab = "G(y)", main = "G(y) for N(0,1)") abline(v = 0, h = 0.5, col = "red", lty = "dashed") lines(yy, pnorm(yy), col = "green", lty = "dotted", lwd = 2) ## End(Not run) ```

enormUC function