extraDistr (version 1.10.0)

TruncNormal: Truncated normal distribution

Description

Density, distribution function, quantile function and random generation for the truncated normal distribution.

Usage

dtnorm(x, mean = 0, sd = 1, a = -Inf, b = Inf, log = FALSE)
ptnorm(
  q,
  mean = 0,
  sd = 1,
  a = -Inf,
  b = Inf,
  lower.tail = TRUE,
  log.p = FALSE
)
qtnorm(
  p,
  mean = 0,
  sd = 1,
  a = -Inf,
  b = Inf,
  lower.tail = TRUE,
  log.p = FALSE
)
rtnorm(n, mean = 0, sd = 1, a = -Inf, b = Inf)

Arguments

x, q

vector of quantiles.

mean, sd

location and scale parameters. Scale must be positive.

a, b

lower and upper truncation points (a < x <= b, with a = -Inf and b = Inf by default).

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, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function $$ f(x) = \frac{\phi(\frac{x-\mu}{\sigma})} {\Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})} $$

Cumulative distribution function $$ F(x) = \frac{\Phi(\frac{x-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})} {\Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})} $$

Quantile function $$ F^{-1}(p) = \Phi^{-1}\left(\Phi\left(\frac{a-\mu}{\sigma}\right) + p \times \left[\Phi\left(\frac{b-\mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)\right]\right) $$

For random generation algorithm described by Robert (1995) is used.

References

Robert, C.P. (1995). Simulation of truncated normal variables. Statistics and Computing 5(2): 121-125. https://arxiv.org/abs/0907.4010

Burkardt, J. (17 October 2014). The Truncated Normal Distribution. Florida State University. https://people.sc.fsu.edu/~jburkardt/presentations/truncated_normal.pdf

Examples

x <- rtnorm(1e5, 5, 3, b = 7)
hist(x, 100, freq = FALSE)
curve(dtnorm(x, 5, 3, b = 7), -8, 8, col = "red", add = TRUE)
hist(ptnorm(x, 5, 3, b = 7))
plot(ecdf(x))
curve(ptnorm(x, 5, 3, b = 7), -8, 8, col = "red", lwd = 2, add = TRUE)

R <- 1e5
partmp <- par(mfrow = c(2,4), mar = c(2,2,2,2))

hist(rtnorm(R), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x), -5, 5, col = "red", add = TRUE)

hist(rtnorm(R, a = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 0), -1, 5, col = "red", add = TRUE)

hist(rtnorm(R, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, b = 0), -5, 5, col = "red", add = TRUE)

hist(rtnorm(R, a = 0, b = 1), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 0, b = 1), -1, 2, col = "red", add = TRUE)

hist(rtnorm(R, a = -1, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = -1, b = 0), -2, 2, col = "red", add = TRUE)

hist(rtnorm(R, mean = -6, a = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, mean = -6, a = 0), -2, 1, col = "red", add = TRUE)

hist(rtnorm(R, mean = 8, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, mean = 8, b = 0), -2, 1, col = "red", add = TRUE)

hist(rtnorm(R, a = 3, b = 5), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 3, b = 5), 2, 5, col = "red", add = TRUE)

par(partmp)