dist.HalfNorm: Half-Normal Distribution In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

Description

These functions provide the density, distribution function, quantile function, and random generation for the half-normal distribution.

Usage

 ```1 2 3 4``` ```dhalfnorm(x, scale=sqrt(pi/2), log=FALSE) phalfnorm(q, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE) qhalfnorm(p, scale=sqrt(pi/2), lower.tail=TRUE, log.p=FALSE) rhalfnorm(n, scale=sqrt(pi/2)) ```

Arguments

 `x,q` These are each a vector of quantiles. `p` This is a vector of probabilities. `n` This is the number of observations, which must be a positive integer that has length 1. `scale` This is the scale parameter sigma, which must be positive. `log,log.p` Logical. If `log=TRUE`, then the logarithm of the density or result is returned. `lower.tail` Logical. If `lower.tail=TRUE` (default), probabilities are Pr[X <= x], otherwise, Pr[X > x].

Details

• Application: Continuous Univariate

• Density: p(theta) = 2*sigma/pi e^-(theta^2*sigma^2/pi), theta >= 0

• Inventor: Derived from the normal or Gaussian

• Notation 1: theta ~ HALF-N(sigma)

• Notation 2: p(theta) = HN(theta | sigma)

• Parameter 1: scale parameter sigma > 0

• Mean: E(theta) = 1 / sigma

• Variance: var(theta) = (pi-2)/(2*sigma^2)

• Mode: mode(theta) = 0

The half-normal distribution is recommended as a weakly informative prior distribution for a scale parameter that may be useful as an alternative to the half-Cauchy, half-t, or vague gamma.

Value

`dhalfnorm` gives the density, `phalfnorm` gives the distribution function, `qhalfnorm` gives the quantile function, and `rhalfnorm` generates random deviates.

`dnorm`, `dnormp`, and `dnormv`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```library(LaplacesDemonCpp) x <- dhalfnorm(1) x <- phalfnorm(1) x <- qhalfnorm(0.5) x <- rhalfnorm(10) #Plot Probability Functions x <- seq(from=0.1, to=20, by=0.1) plot(x, dhalfnorm(x,0.1), ylim=c(0,1), type="l", main="Probability Function", ylab="density", col="red") lines(x, dhalfnorm(x,0.5), type="l", col="green") lines(x, dhalfnorm(x,1), type="l", col="blue") legend(2, 0.9, expression(sigma==0.1, sigma==0.5, sigma==1), lty=c(1,1,1), col=c("red","green","blue")) ```