The Half-Normal Distribution

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

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)

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.

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.

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 \(\sigma=1\).

See also

Normal.

Examples

# 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)
#> 0.831928 with absolute error < 9.2e-15
phalfnorm(1.4, theta = 1.234)
#> [1] 0.831928

## quantile function
qhalfnorm(-1) # NaN
#> [1] NaN
qhalfnorm(0)
#> [1] 0
qhalfnorm(.5)
#> [1] 0.6744898
qhalfnorm(1)
#> [1] Inf
qhalfnorm(2) # NaN
#> Warning: NaNs produced
#> [1] NaN

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


mean(hz) 
#> [1] 1.380705
1/theta  # theoretical mean
#> [1] 1.388889

var(hz)
#> [1] 1.10687
(pi-2)/(2*theta*theta) # theoretical variance
#> [1] 1.101073


## 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
#> [1] 0
hist(pvl2, freq=FALSE)  # uniform distribution


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