VGAMextra (version 0.0-6)

truncNormal: The Truncated Normal Distribution

Description

Density, distribution function, quantile function and random numbers generator for the truncated normal distribution

Usage

dtruncnorm(x, mean = 0, sd = 1, min.support = -Inf, max.support = Inf, log = FALSE)
ptruncnorm(q, mean = 0, sd = 1, min.support = -Inf, max.support = Inf) 
qtruncnorm(p, mean = 0, sd = 1, min.support = -Inf, max.support = Inf, log.p = FALSE)
rtruncnorm(n, mean = 0, sd = 1, min.support = -Inf, max.support = Inf)

Value

dtruncnorm() returns the density,

ptruncnorm() gives the distribution function,

qtruncnorm() gives the quantiles, and

rtruncnorm() generates random deviates.

dtruncnorm is computed from the definition, as in 'Details'. [pqr]truncnormal are computed based on their relationship to the normal distribution.

Arguments

x, q, p, n, mean, sd

Same as Normal.

min.support, max.support

Lower and upper truncation limits.

log, log.p

Same as Normal.

Author

Victor Miranda and Thomas W. Yee.

Details

Consider \(X \sim\) N(\(\mu\), \(\sigma^2\)), with \(A < X < B\), i.e., \(X\) restricted to \((A , B)\). We denote \(A\) = min.support and \(B\) = max.support.

Then the conditional random variable \(Y = X \cdot I_{(A , B)} \) has a truncated normal distribution. Its p.d.f. is given by

$$ f(y; \mu, \sigma, A, B) = (1 / \sigma) \cdot \phi(y^*) / [ \Phi(B^*) - \Phi(A^*) ], $$

where \(y^* = (y - \mu)/ \sigma\), \(A^* = (A - \mu) / \sigma\), and \(B^* = (B - \mu) / \sigma\).

Its mean is $$\mu + \sigma \cdot [ \phi(A) - \phi(B) ] / [\Phi(B) - \Phi(A)]. $$

Here, \(\Phi\) is the standard normal c.d.f and \(\phi\) is the standard normal p.d.f.

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Second Edition (Chapter 13). Wiley, New York.

See Also

Normal, truncweibull.

Examples

###############
## Example 1 ##

mymu <- 2.1   # mu
mysd <- 1.0   # sigma
LL   <- -1.0  # Lower bound
UL   <- 3.0   # Upper bound

## Quantiles:
pp <- 1:10 / 10
(quants <- qtruncnorm(p = pp , min.support = LL, max.support = UL, 
                        mean = mymu,  sd = mysd))
sum(pp - ptruncnorm(quants, min.support = LL, max.support = UL,
                      mean = mymu, sd = mysd))     # Should be zero

###############
## Example 2 ##

## Parameters
set.seed(230723)
nn <- 3000
mymu <- 12.7    # mu
mysigma <- 3.5  # sigma
LL <- 6     # Lower bound
UL <- 17    # Upper bound

## Truncated-normal data
trunc_data <- rtruncnorm(nn, mymu, mysigma, LL, UL)

## non-truncated data - reference
nontrunc_data <- rnorm(nn, mymu, mysigma)


if (FALSE) {
## Densities
par(mfrow = c(1, 2))
plot(density(nontrunc_data), main = "Non-truncated ND", 
     col = "green", xlim = c(0, 25), ylim = c(0, 0.15))
abline(v = c(LL, UL), col = "black", lwd = 2, lty = 2)
plot(density(trunc_data), main = "Truncated ND", 
     col = "red", xlim = c(0, 25), ylim = c(0, 0.15))


## Histograms
plot.new()
par(mfrow = c(1, 2))
hist(nontrunc_data, main = "Non-truncated ND", col = "green", 
     xlim = c(0, 25), ylim = c(0, 0.15), freq = FALSE, breaks = 22,
     xlab = "mu = 12.7, sd = 3.5, LL = 6, UL = 17")
abline(v = c(LL, UL), col = "black", lwd = 4, lty = 2)
hist(trunc_data, main = "Truncated ND", col = "red",
     xlim = c(0, 25), ylim = c(0, 0.15), freq = FALSE,
     xlab = "mu = 12.7, sd = 3.5, LL = 6, UL = 17")

}


## Area under the estimated densities
# (a) truncated data
integrate(approxfun(density(trunc_data)), 
          lower = min(trunc_data) - 1, 
          upper = max(trunc_data) + 1)

# (b) non-truncated data
integrate(approxfun(density(nontrunc_data)), 
          lower = min(nontrunc_data), 
          upper = max(nontrunc_data))