| CensoredNormal {crch} | R Documentation |
Create a Censored Normal Distribution
Description
Class and methods for left-, right-, and interval-censored normal distributions using the workflow from the distributions3 package.
Usage
CensoredNormal(mu = 0, sigma = 1, left = -Inf, right = Inf)
Arguments
mu |
numeric. The location parameter of the underlying uncensored
normal distribution, typically written |
sigma |
numeric. The scale parameter (standard deviation) of
the underlying uncensored normal distribution,
typically written |
left |
numeric. The left censoring point. Can be any real number,
defaults to |
right |
numeric. The right censoring point. Can be any real number,
defaults to |
Details
The constructor function CensoredNormal sets up a distribution
object, representing the censored normal probability distribution by the
corresponding parameters: the latent mean mu = \mu and
latent standard deviation sigma = \sigma (i.e., the parameters
of the underlying uncensored normal variable), the left censoring
point (with -Inf corresponding to uncensored), and the
right censoring point (with Inf corresponding to uncensored).
The censored normal distribution has probability density function (PDF) f(x):
\Phi((left - \mu)/\sigma) | if x \le left |
1 - \Phi((right - \mu)/\sigma) | if x \ge right |
\phi((x - \mu)/\sigma)/\sigma | if left < x < right
|
where \Phi and \phi are the cumulative distribution function
and probability density function of the standard normal distribution
respectively.
All parameters can also be vectors, so that it is possible to define a vector of censored normal distributions with potentially different parameters. All parameters need to have the same length or must be scalars (i.e., of length 1) which are then recycled to the length of the other parameters.
For the CensoredNormal distribution objects there is a wide range
of standard methods available to the generics provided in the distributions3
package: pdf and log_pdf
for the (log-)density (PDF), cdf for the probability
from the cumulative distribution function (CDF), quantile for quantiles,
random for simulating random variables,
crps for the continuous ranked probability score
(CRPS), and support for the support interval
(minimum and maximum). Internally, these methods rely on the usual d/p/q/r
functions provided for the censored normal distributions in the crch
package, see dcnorm, and the crps_cnorm
function from the scoringRules package.
The methods is_discrete and is_continuous
can be used to query whether the distributions are discrete on the entire support
(always FALSE) or continuous on the entire support (only TRUE if
there is no censoring, i.e., if both left and right are infinite).
See the examples below for an illustration of the workflow for the class and methods.
Value
A CensoredNormal distribution object.
See Also
dcnorm, Normal, TruncatedNormal,
CensoredLogistic, CensoredStudentsT
Examples
## package and random seed
library("distributions3")
set.seed(6020)
## three censored normal distributions:
## - uncensored standard normal
## - left-censored at zero (Tobit) with latent mu = 1 and sigma = 1
## - interval-censored in [0, 5] with latent mu = 1 and sigma = 2
X <- CensoredNormal(
mu = c( 0, 1, 1),
sigma = c( 1, 1, 2),
left = c(-Inf, 0, 0),
right = c( Inf, Inf, 5)
)
X
## compute mean of the censored distribution
mean(X)
## higher moments (variance, skewness, kurtosis) are not implemented yet
## support interval (minimum and maximum)
support(X)
## simulate random variables
random(X, 5)
## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ], main = "uncensored")
hist(x[2, ], main = "left-censored at zero")
hist(x[3, ], main = "interval-censored in [0, 5]")
## probability density function (PDF) and log-density (or log-likelihood)
x <- c(0, 0, 1)
pdf(X, x)
pdf(X, x, log = TRUE)
log_pdf(X, x)
## cumulative distribution function (CDF)
cdf(X, x)
## quantiles
quantile(X, 0.5)
## cdf() and quantile() are inverses (except at censoring points)
cdf(X, quantile(X, 0.5))
quantile(X, cdf(X, 1))
## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)
quantile(X, p, elementwise = TRUE)
quantile(X, p, elementwise = TRUE, drop = FALSE)
## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
"theoretical" = mean(X),
"empirical" = rowMeans(random(X, 1000))
)
## evaluate continuous ranked probability score (CRPS) using scoringRules
library("scoringRules")
crps(X, x)