VGAMextra (version 0.0-6)

truncnormal: Truncated normal Distribution Family Function

Description

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

Usage

truncnormal(lmean = "identitylink", lsd = "loglink",
             min.support = -Inf, max.support = Inf, zero = "sd")

Value

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

Arguments

lmean, lsd

Link functions applied to mean and standard deviation/variance.

min.support, max.support

Vector of lower and upper truncation limits (recycled). min.support enables LHS truncation and max.support enables RHS truncation. The default imply no truncation (mimicks uninormal).

zero

See CommonVGAMffArguments for more information.

Author

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

Details

MLE of the two--parameter (univariate) normal distribution subject to lower/upper truncation. All response values are greater then min.support and/or lower than max.support.

The truncated--normal density for a response \(Y\) is

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

where \(f\) is the probability density function of standard normal distribution and \(\Phi\) is the standard normal CDF.

The mean of Y, given by

$$ \mu + [\varphi(\texttt{min.support}) + \varphi(\texttt{max.support})/\Delta \Phi(\mu,\sigma)]\cdot \sigma, $$

with \(\Delta \Phi(\mu, \sigma) = \Phi((\texttt{max.support} - \mu)/\sigma ) - \Phi( (\texttt{min.support} - \mu)/\sigma ),\) 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

uninormal, CommonVGAMffArguments.

Examples

nn <- 2000
set.seed(14290909)

## Parameters
mysd <- exp(1.0)   # sd
LL   <- -0.5       # Lower bound
UL   <-  8.0       # Upper bound

## Truncated data
ldata2 <- data.frame(x2 = runif(nn))
ldata2 <- transform(ldata2, y1 = rtruncnorm(nn, 1 + 1.5 * x2, mysd, 
                                min.support = LL, max.support = UL))
# head(ldata2)
# hist(ldata2$y1, breaks = 22, col = "blue", xlim = c(-5, 10))

##############################################################
# Fitting a truncated normal distribution - sd is intercept only
fit1 <- vglm(y1 ~ x2, truncnormal(zero = "sd", 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, truncnormal(zero = NULL, min.support = LL, max.support = UL),
             data = ldata2, trace = TRUE)
coef(fit2, matrix = TRUE)
vcov(fit2)

##############################################################
# Mimicking uninormal()
fit3 <- vglm(y1 ~ x2, truncnormal(zero = "sd"),
             data = ldata2, trace = TRUE)
coef(fit3, mat = TRUE)

# Same as
fit3bis <- vglm(y1 ~ x2, uninormal(zero = "sd"),
                 data = ldata2, trace = TRUE)
coef(fit3bis, mat = TRUE)