Density, distribution function, quantile function and random generation for
the truncated Normal distribution with mean equal to mean and
standard deviation equal to sd before truncation, and truncated on
the interval [lower, upper].
Usage
dtnorm(x, mean = 0, sd = 1, lower = -Inf, upper = Inf, log = FALSE)
ptnorm(
q,
mean = 0,
sd = 1,
lower = -Inf,
upper = Inf,
lower.tail = TRUE,
log.p = FALSE
)
qtnorm(
p,
mean = 0,
sd = 1,
lower = -Inf,
upper = Inf,
lower.tail = TRUE,
log.p = FALSE
)
rtnorm(n, mean = 0, sd = 1, lower = -Inf, upper = Inf)Arguments
- x, q
vector of quantiles.
- mean
vector of means.
- sd
vector of standard deviations.
- lower
lower truncation point.
- upper
upper truncation point.
- log
logical; if TRUE, return log density or log hazard.
- lower.tail
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
- log.p
logical; if TRUE, probabilities p are given as log(p).
- p
vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.
Value
dtnorm gives the density, ptnorm gives the
distribution function, qtnorm gives the quantile function, and
rtnorm generates random deviates.
Details
The truncated normal distribution has density
$$ f(x, \mu, \sigma) = \phi(x, \mu, \sigma) / (\Phi(u, \mu, \sigma) - \Phi(l, \mu, \sigma)) $$ for \(l <= x <= u\), and 0 otherwise.
\(\mu\) is the mean of the original Normal distribution before
truncation,
\(\sigma\) is the corresponding standard deviation,
\(u\) is the upper truncation point,
\(l\) is the lower
truncation point,
\(\phi(x)\) is the density of the
corresponding normal distribution, and
\(\Phi(x)\) is the
distribution function of the corresponding normal distribution.
If mean or sd are not specified they assume the default values
of 0 and 1, respectively.
If lower or upper are not specified they assume the default
values of -Inf and Inf, respectively, corresponding to no
lower or no upper truncation.
Therefore, for example, dtnorm(x), with no other arguments, is simply
equivalent to dnorm(x).
Only rtnorm is used in the msm package, to simulate from
hidden Markov models with truncated normal distributions. This uses the
rejection sampling algorithms described by Robert (1995).
These functions are merely provided for completion, and are not optimized
for numerical stability or speed. To fit a hidden Markov model with a
truncated Normal response distribution, use a hmmTNorm
constructor. See the hmm-dists help page for further details.
References
Robert, C. P. Simulation of truncated normal variables. Statistics and Computing (1995) 5, 121–125
Author
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
