halfnormal: The Half-Normal Distribution

Description

Density, distribution function, quantile function and random generation for the half-normal distribution with parameter theta.

Usage

dhalfnorm(x, theta=sqrt(pi/2), log = FALSE)
phalfnorm(q, theta=sqrt(pi/2), lower.tail = TRUE, log.p = FALSE)
qhalfnorm(p, theta=sqrt(pi/2), lower.tail = TRUE, log.p = FALSE)
rhalfnorm(n, theta=sqrt(pi/2))
sd2theta(sd)
theta2sd(theta)

Value

dhalfnorm gives the density,

phalfnorm gives the distribution function,

qhalfnorm gives the quantile function, and

rhalfnorm generates random deviates.

sd2theta computes a theta parameter.

theta2sd computes a sd parameter.

Arguments

x,q: vector of quantiles.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
theta: parameter of half-normal distribution.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
sd: standard deviation of the zero-mean normal distribution that corresponds to the half-normal with parameter theta.

Details

x = abs(z) follows a half-normal distribution with if z is a normal variate with zero mean. The half-normal distribution has density $$ f(x) = \frac{2 \theta}{\pi} e^{-x^2 \theta^2/\pi}$$ It has mean $E(x) = \frac{1}{\theta}$ and variance $Var(x) = \frac{\pi-2}{2 \theta^2}$.

The parameter $\theta$ is related to the standard deviation $\sigma$ of the corresponding zero-mean normal distribution by the equation $\theta = \sqrt{\pi/2}/\sigma$.

If $\theta$ is not specified in the above functions it assumes the default values of $\sqrt{\pi/2}$, corresponding to $\sigma=1$.

Examples

Run this code

# load "fdrtool" library
library("fdrtool")


## density of half-normal compared with a corresponding normal
par(mfrow=c(1,2))

sd.norm = 0.64
x  = seq(0, 5, 0.01)
x2 = seq(-5, 5, 0.01)
plot(x, dhalfnorm(x, sd2theta(sd.norm)), type="l", xlim=c(-5, 5), lwd=2,
   main="Probability Density", ylab="pdf(x)")
lines(x2, dnorm(x2, sd=sd.norm), col=8 )


plot(x, phalfnorm(x, sd2theta(sd.norm)), type="l", xlim=c(-5, 5),  lwd=2,
   main="Distribution Function", ylab="cdf(x)")
lines(x2, pnorm(x2, sd=sd.norm), col=8 )

legend("topleft", 
c("half-normal", "normal"), lwd=c(2,1),
col=c(1, 8), bty="n", cex=1.0)

par(mfrow=c(1,1))


## distribution function

integrate(dhalfnorm, 0, 1.4, theta = 1.234)
phalfnorm(1.4, theta = 1.234)

## quantile function
qhalfnorm(-1) # NaN
qhalfnorm(0)
qhalfnorm(.5)
qhalfnorm(1)
qhalfnorm(2) # NaN

## random numbers
theta = 0.72
hz = rhalfnorm(10000, theta)
hist(hz, freq=FALSE)
lines(x, dhalfnorm(x, theta))

mean(hz) 
1/theta  # theoretical mean

var(hz)
(pi-2)/(2*theta*theta) # theoretical variance


## relationship with two-sided normal p-values
z = rnorm(1000)

# two-sided p-values
pvl = 1- phalfnorm(abs(z))
pvl2 = 2 - 2*pnorm(abs(z)) 
sum(pvl-pvl2)^2 # equivalent
hist(pvl2, freq=FALSE)  # uniform distribution

# back to half-normal scores
hz = qhalfnorm(1-pvl)
hist(hz, freq=FALSE)
lines(x, dhalfnorm(x))