VGAMextra (version 0.0-6)

trunclognormal: Truncated Log-normal Distribution Family Function

Description

Maximum likelihood estimate of the two--parameter lognormal distribution with lower/upper truncation.

Usage

trunclognormal(lmeanlog = "identitylink", lsdlog = "loglink",
                min.support = 1e-6, max.support = Inf, zero = "sdlog")

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

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).

Author

Victor Miranda, Siqi (Vicky) Liu and Thomas W. Yee.

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

$$f(y; \mu, \sigma) = f_N(y; \mu, \sigma) / [\Phi(\texttt{max.support}, \mu, \sigma) - \Phi(\texttt{min.support},\mu, \sigma) ], $$

where \(f_N\) is the ordinary lognormal density (see lognormal) and \(\Phi\) is the standard normal CDF.

The mean of Y, given by $$ \exp{(\mu + \sigma^2/2)} \cdot [\Phi(((\log(\texttt{max.support}) - \mu)/\sigma) - \sigma) - \Phi(((\log(\texttt{min.support}) - \mu)/\sigma) - \sigma) ] / \Delta \Phi(\mu,\sigma),$$

with \(\Delta \Phi(\mu, \sigma) = \Phi( (\log(\texttt{max.support}) - \mu)/\sigma ) - \Phi( (\log(\texttt{min.support}) - \mu)/\sigma ),\) are returned as the fitted values.

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.

See Also

lognormal, uninormal, CommonVGAMffArguments, Lognormal.

Examples

##########
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)