# Truncated Log-normal Distribution Family Function Maximum likelihood estimate of the two--parameter lognormal distribution with lower/upper truncation. ```r trunclognormal(lmeanlog = "identitylink", lsdlog = "loglink", min.support = 1e-6, max.support = Inf, zero = "sdlog") ``` ## Arguments - `lmeanlog, lsdlog, zero`: Same as `lognormal`. - `min.support, max.support`: Positive lower and upper truncation limits (recycled). `min.support` enables LHS truncation; `max.support` enables RHS truncation (default is none). ## Details MLE of the two--parameter (univariate) lognormal distribution subject to lower/upper truncation. All response values are greater than `min.support` and lower than `max.support`. Default values of `min.support`, `max.suppport` should effectively reproduce `lognormal`. The truncated--lognormal density for a response $Y$ is $$ f(y; \mu, \sigma) = f_N(y; \mu, \sigma) /[\Phi(\texttt{max.support}, \mu, \sigma) -\Phi(\texttt{min.support},\mu, \sigma) ],f(y; \mu, \sigma) = f_N(y; \mu, \sigma) /[\Phi(\texttt{max.support}, \mu, \sigma) -\Phi(\texttt{min.support},\mu, \sigma) ], $$ where $f_N$ is the ordinary lognormal density (see `lognormal`) and $\Phi$ is the standard normal CDF. The mean of Y, given by $$ \exp{(\mu + \sigma^2/2)} \cdot[\Phi(((\log(\texttt{max.support}) - \mu)/\sigma) - \sigma) -\Phi(((\log(\texttt{min.support}) - \mu)/\sigma) - \sigma) ] /\Delta \Phi(\mu,\sigma),\exp{(\mu + \sigma^2/2)} \cdot[\Phi(((\log(\code{max.support}) - \mu)/\sigma) - \sigma) -\Phi(((\log(\code{min.support}) - \mu)/\sigma) - \sigma) ] /\Delta \Phi(\mu,\sigma), $$ with c("$\n$", "$ \\Delta \\Phi(\\mu, \\sigma) = \n$", "$ \\Phi( (\\log(\\texttt{max.support}) - \\mu)/\\sigma ) -\n$", "$ \\Phi( (\\log(\\texttt{min.support}) - \\mu)/\\sigma ),\n$") are returned as the fitted values. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Nadarajah, S. and Kotz, S. (2003). `R` Programs for Computing Truncated Distributions. **Journal of Statistical Software, Code Snippets**, 16 (2), 1--8. Cohen, A.C. (1991) **Truncated and Censored Samples: Theory and Applications**, New York, USA. Marcel Dekker. ## Author(s) Victor Miranda, Siqi (Vicky) Liu and Thomas W. Yee. ## See Also `lognormal`, `uninormal`, `CommonVGAMffArguments`, `Lognormal`. ## Examples ```r ########## set.seed(10470923) nn <- 3000 ## Parameters mysdlog <- exp(-1.5) # sdlog LL <- 3.5 # Lower bound UL <- 8.0 # Upper bound ## Truncated data ldata2 <- data.frame(x2 = runif(nn)) ldata2 <- transform(ldata2, y1 = rtrunclnorm(nn, 1 + 1.5 * x2, mysdlog, min.support = LL, max.support = UL)) # head(ldata2) # hist(ldata2$y1, breaks = 22, col = "blue", xlim = c(0, 10)) ############################################################## # Fitting a truncated lognormal distribution - sd is intercept only fit1 <- vglm(y1 ~ x2, trunclognormal(zero = "sdlog", min.support = LL, max.support = UL), data = ldata2, trace = TRUE) coef(fit1, matrix = TRUE) vcov(fit1) ############################################################## # Fitting a truncated lognormal distribution - zero = NULL fit2 <- vglm(y1 ~ x2, trunclognormal(zero = NULL, min.support = LL, max.support = UL), data = ldata2, trace = TRUE) coef(fit2, matrix = TRUE) vcov(fit2) ############################################################## # Mimicking lognormal() fit3 <- vglm(y1 ~ x2, trunclognormal(zero = "sdlog"), data = ldata2, trace = TRUE) coef(fit3, mat = TRUE) # Same as fit3bis <- vglm(y1 ~ x2, lognormal(zero = "sdlog"), data = ldata2, trace = TRUE) coef(fit3bis, mat = TRUE) ```

trunclognormalff function