Class and methods for left-, right-, and interval-censored normal distributions using the workflow from the distributions3 package.
CensoredNormal(mu = 0, sigma = 1, left = -Inf, right = Inf)
numeric. The location parameter of the underlying uncensored
normal distribution, typically written μ in textbooks.
Can be any real number, defaults to
numeric. The scale parameter (standard deviation) of
the underlying uncensored normal distribution,
typically written σ in textbooks.
Can be any positive number, defaults to
numeric. The left censoring point. Can be any real number,
numeric. The right censoring point. Can be any real number,
The constructor function
CensoredNormal sets up a distribution
object, representing the censored normal probability distribution by the
corresponding parameters: the latent mean
mu = μ and
latent standard deviation
sigma = σ (i.e., the parameters
of the underlying uncensored normal variable), the
-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):
|Φ((left - μ)/σ)||if x ≤ left|
|1 - Φ((right - μ)/σ)||if x ≥ right|
|φ((x - μ)/σ)/σ||if left < x < right|
where Φ and φ 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.
CensoredNormal distribution objects there is a wide range
of standard methods available to the generics provided in the distributions3
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
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
dcnorm, and the
function from the scoringRules package.
can be used to query whether the distributions are discrete on the entire support
FALSE) or continuous on the entire support (only
there is no censoring, i.e., if both
right are infinite).
See the examples below for an illustration of the workflow for the class and methods.
CensoredNormal distribution object.
## 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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.