VGAM (version 1.1-9)

Expectiles-Normal: Expectiles of the Normal Distribution

Description

Density function, distribution function, and expectile function and random generation for the distribution associated with the expectiles of a normal distribution.

Usage

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)

Value

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 \(G\).

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.

Author

T. W. Yee and Kai Huang

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.

See Also

deunif, deexp, dnorm, amlnormal, lms.bcn.

Examples

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

if (FALSE) {
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) }