censoredlogistic

Create a Censored Logistic Distribution

Description

Class and methods for left-, right-, and interval-censored logistic distributions using the workflow from the distributions3 package.

Usage

CensoredLogistic(location = 0, scale = 1, left = -Inf, right = Inf)

Arguments

location numeric. The location parameter of the underlying uncensored logistic distribution, typically written \(\mu\) in textbooks. Can be any real number, defaults to 0.

scale numeric. The scale parameter (standard deviation) of the underlying uncensored logistic distribution, typically written \(\sigma\) in textbooks. Can be any positive number, defaults to 1.

left numeric. The left censoring point. Can be any real number, defaults to -Inf (uncensored). If set to a finite value, the distribution has a point mass at left whose probability corresponds to the cumulative probability function of the uncensored logistic distribution at this point.

right numeric. The right censoring point. Can be any real number, defaults to Inf (uncensored). If set to a finite value, the distribution has a point mass at right whose probability corresponds to 1 minus the cumulative probability function of the uncensored logistic distribution at this point.

Details

The constructor function CensoredLogistic sets up a distribution object, representing the censored logistic probability distribution by the corresponding parameters: the latent mean location = \(\mu\) and latent standard deviation scale = \(\sigma\) (i.e., the parameters of the underlying uncensored logistic variable), the left censoring point (with -Inf corresponding to uncensored), and the right censoring point (with Inf corresponding to uncensored).

The censored logistic distribution has probability density function (PDF) \(f(x)\):

\(\Lambda((left - \mu)/\sigma)\)	if \(x \le left\)
\(1 - \Lambda((right - \mu)/\sigma)\)	if \(x \ge right\)
\(\lambda((x - \mu)/\sigma)/\sigma\)	otherwise

where \(\Lambda\) and \(\lambda\) are the cumulative distribution function and probability density function of the standard logistic distribution, respectively.

All parameters can also be vectors, so that it is possible to define a vector of censored logistic 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 CensoredLogistic 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 logistic distributions in the crch package, see dclogis, and the crps_clogis 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 CensoredLogistic distribution object.

Examples

library("crch")


## package and random seed
library("distributions3")
set.seed(6020)

## three censored logistic distributions:
## - uncensored standard logistic
## - left-censored at zero with latent location = 1 and scale = 1
## - interval-censored in [0, 5] with latent location = 2 and scale = 1
X <- CensoredLogistic(
  location = c(   0,   1, 2),
  scale    = c(   1,   1, 1),
  left     = c(-Inf,   0, 0),
  right    = c( Inf, Inf, 5)
)

X

[1] "CensoredLogistic(location = 0, scale = 1, left = -Inf, right = Inf)"
[2] "CensoredLogistic(location = 1, scale = 1, left =    0, right = Inf)"
[3] "CensoredLogistic(location = 2, scale = 1, left =    0, right =   5)"

## compute mean of the censored distribution
mean(X)

[1] 0.000000 1.313262 2.078341

## higher moments (variance, skewness, kurtosis) are not implemented yet

## support interval (minimum and maximum)
support(X)

      min max
[1,] -Inf Inf
[2,]    0 Inf
[3,]    0   5

## simulate random variables
random(X, 5)

            r_1        r_2        r_3         r_4        r_5
[1,] -0.5489266 0.71149394 -0.9934677 -0.09368233 -3.1677767
[2,]  0.3829342 2.00150190  2.0657180  0.31347300  0.0000000
[3,]  2.7294189 0.05283236  0.0000000  1.19632319  0.5573964

## 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)

[1] 0.2500000 0.2689414 0.1966119

pdf(X, x, log = TRUE)

[1] -1.386294 -1.313262 -1.626523

log_pdf(X, x)

[1] -1.386294 -1.313262 -1.626523

## cumulative distribution function (CDF)
cdf(X, x)

[1] 0.5000000 0.2689414 0.2689414

## quantiles
quantile(X, 0.5)

[1] 0 1 2

## cdf() and quantile() are inverses (except at censoring points)
cdf(X, quantile(X, 0.5))

[1] 0.5 0.5 0.5

quantile(X, cdf(X, 1))

[1] 1 1 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)

        q_0.05 q_0.5   q_0.95
[1,] -2.944439     0 2.944439
[2,]  0.000000     1 3.944439
[3,]  0.000000     2 4.944439

quantile(X, p, elementwise = TRUE)

[1] -2.944439  1.000000  4.944439

quantile(X, p, elementwise = TRUE, drop = FALSE)

      quantile
[1,] -2.944439
[2,]  1.000000
[3,]  4.944439

## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
  "theoretical" = mean(X),
  "empirical" = rowMeans(random(X, 1000))
)

     theoretical  empirical
[1,]    0.000000 -0.0299458
[2,]    1.313262  1.2761385
[3,]    2.078341  2.0830220

## evaluate continuous ranked probability score (CRPS) using scoringRules
library("scoringRules")
crps(X, x)

[1] 0.3862944 0.5822031 0.6176368