# The Truncated Log-Normal Distribution Density, distribution function, quantile function and random generation for the truncated log-normal distribution ```r 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) ``` ## 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`. ## Details Consider $Y \sim$ `Lognormal`$(\mu_Y, \sigma_Y )$ restricted to $(A, B )$, that is, c("$0 < A = \\code{min.support} < X < B \n$", "$ = \\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 $$ f(y; \mu, \sigma, A, B) = (y^{-1} / \sigma) \cdot\phi(y^*) / [ \Phi(B^*) - \Phi(A^*) ],f(y; \mu, \sigma, A, B) = (1 / \sigma) \cdot\phi(y^*) / [ \Phi(B^*) - \Phi(A^*) ], $$ where c("$y^* = \n$", "$ [log(y) - \\mu_Y]/ \\sigma_Y$"), c("$\n$", "$ A^* = [log(A) - \\mu_Y] / \\sigma_Y$"), and c("$B^* = [\\log(B) - \n$", "$ \\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. ## Returns `dtrunclnorm()` returns the density, `ptrunclnorm()` gives the distribution function, `qtrunclnorm()` gives the quantiles, and `rtrunclnorm()` generates random deviates. ## References Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) **Continuous Univariate Distributions**, Second Edition, (Chapter 13) Wiley, New York. ## Author(s) Victor Miranda and Thomas W. Yee. ## See Also `Lognormal`, `truncnormal`. ## Examples ```r ############### ## 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) ## Not run: ## 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") ## End(Not run) ## 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)) ```

trunclognormalUC function