VGAMextra (version 0.0-6)

truncLognormal: The Truncated Log-Normal Distribution

Description

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

Usage

dtrunclnorm(x, meanlog = 0, sdlog = 1, min.support = 0, max.support = Inf, log = FALSE)
ptrunclnorm(q, meanlog = 0, sdlog = 1, min.support = 0, max.support = Inf) 
qtrunclnorm(p, meanlog = 0, sdlog = 1, min.support = 0, max.support = Inf, log.p = FALSE)
rtrunclnorm(n, meanlog = 0, sdlog = 1, min.support = 0, max.support = Inf)

Value

dtrunclnorm() returns the density,

ptrunclnorm() gives the distribution function,

qtrunclnorm() gives the quantiles, and

rtrunclnorm() generates random deviates.

Arguments

x, q, p, n, meanlog, sdlog

Same as Lognormal.

min.support, max.support

Lower and upper truncation limits.

log, log.p

Same as Lognormal.

Author

Victor Miranda and Thomas W. Yee.

Details

Consider \(Y \sim\) Lognormal\((\mu_Y, \sigma_Y )\) restricted to \((A, B )\), that is, \(0 < A = \code{min.support} < X < B = \code{max.support}\). The (conditional) random variable \(Y = X \cdot I_{(A , B)} \) has a log--truncated normal distribution. Its p.d.f. is given by

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

where \(y^* = [\log(y) - \mu_Y]/ \sigma_Y\), \(A^* = [\log(A) - \mu_Y] / \sigma_Y\), and \(B^* = [\log(B) - \mu_Y] / \sigma_Y\).

Its mean is: $$\exp(\mu + \sigma^2/2) \cdot \{\Phi[(\log(B) - \mu) / \sigma - \sigma] - \Phi[(\log(A) - \mu) / \sigma - \sigma] \} / \{ \Phi[(\log(B) - \mu) / \sigma] - \Phi[(\log(A) - \mu) / \sigma] \}. $$

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

Lognormal, truncnormal.

Examples

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

mymeanlog <- exp(0.5)    # meanlog
mysdlog   <- exp(-1.5)   # sdlog
LL   <- 3.5              # Lower bound
UL   <- 8.0              # Upper bound

## Quantiles:
pp <- 1:10 / 10
(quants <- qtrunclnorm(p = pp , min.support = LL, max.support = UL, 
                        mymeanlog, mysdlog))
sum(pp - ptrunclnorm(quants, min.support = LL, max.support = UL,
                      mymeanlog, mysdlog))     # Should be zero

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

set.seed(230723)
nn <- 3000

## Truncated log-normal data
trunc_data <- rtrunclnorm(nn, mymeanlog, mysdlog, LL, UL)

## non-truncated data - reference
nontrunc_data <- rtrunclnorm(nn, mymeanlog, mysdlog, 0, Inf)

if (FALSE) {
## Densities
plot.new()
par(mfrow = c(1, 2))
plot(density(nontrunc_data), main = "Non-truncated Log--normal", 
     col = "green", xlim = c(0, 15), ylim = c(0, 0.40))
abline(v = c(LL, UL), col = "black", lwd = 2, lty = 2)
plot(density(trunc_data), main = "Truncated Log--normal", 
     col = "red", xlim = c(0, 15), ylim = c(0, 0.40))


## Histograms
plot.new()
par(mfrow = c(1, 2))
hist(nontrunc_data, main = "Non-truncated Log--normal", col = "green", 
       xlim = c(0, 15), ylim = c(0, 0.40), freq = FALSE, breaks = 22,
       xlab = "mu = exp(0.5), sd = exp(-1.5), LL = 3.5, UL = 8")
abline(v = c(LL, UL), col = "black", lwd = 4, lty = 2)

hist(trunc_data, main = "Truncated Log--normal", col = "red",
     xlim = c(0, 15), ylim = c(0, 0.40), freq = FALSE, 
     xlab = "mu = exp(0.5), sd = exp(-1.5), LL = 3.5, UL = 8")
}


## Area under the estimated densities
# (a) truncated data
integrate(approxfun(density(trunc_data)), 
          lower = min(trunc_data) - 0.1, 
          upper = max(trunc_data) + 0.1)

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