# CensoredDistribution

CensoredDistribution[{xmin,xmax},dist]

represents the distribution of values that come from dist and are censored to be between xmin and xmax.

CensoredDistribution[{{xmin,xmax},{ymin,ymax},},dist]

represents the distribution of values that come from the multivariate distribution dist and are censored to be between xmin and xmax, ymin and ymax, etc.

# Details

• CensoredDistribution[{xmin,xmax},dist] is equivalent to TransformedDistribution[f,xdist], where f is given by Piecewise[{{xmin,x<=xmin},{x,xmin<x<xmax},{xmax,x>=xmax}}].
• Common cases for {xmin,xmax} include:
•  {-∞,xmax} censoring from above, right-censoring {xmin,∞} censoring from below, left-censoring {xmin,xmax} doubly censored, interval-censoring {-∞,∞},None no censoring, uncensored
• CensoredDistribution can be used with such functions as Mean, CDF, RandomVariate, etc.

# Background & Context

• CensoredDistribution[{xmin,xmax},dist] represents a statistical distribution modeling data that is known to be taken from the univariate distribution dist for all in the interval and that is assumed to be constantly equal to (respectively, constantly equal to ) for (respectively for ). The terms uncensored, left-censored, right-censored, and doubly censored are used to describe univariate censorings for which {xmin,xmax} has the form {-,}, {xmin,}, {-,xmax}, and {xmin,xmax}, respectively, while univariate dist may be either continuous (e.g. NormalDistribution, GammaDistribution, or BetaDistribution) or discrete (e.g. PoissonDistribution, BinomialDistribution, or BernoulliDistribution) and may be defined in terms of transformations, censoring, or truncations (by way of TransformedDistribution, CensoredDistribution, and TruncatedDistribution, respectively) of known distributions.
• The multivariate CensoredDistribution[{{,},{,}, ,{,}},dist] is defined analogously and thus represents the distribution of vectors taken from the multivariate distribution dist and whose component is censored to be in the interval . As in the univariate case, multivariate dist may again be either continuous (e.g. MultinormalDistribution) or discrete (e.g. MultivariateHypergeometricDistribution), and may also be defined as a copula or product (using CopulaDistribution and ProductDistribution, respectively) of known distributions.
• Censored distributions arise when modeling data for which the values are only partially known (i.e. those datasets containing only partially observed or accuracy-constrained data), and the analysis of datasets containing censored values dates back to the eighteenth-century smallpox investigations of Daniel Bernoulli. The existence of such data is relatively common in fields such as medicine and physiology, as well as in reliability and manufacturing, where failure predictions must sometimes be made without having observed actual failure. Censored distributions are also commonly utilized tools in survival analysis, and a variety of specialized statistical tools (e.g. censored regression) exists to analyze such datasets.
• By definition, CensoredDistribution[{xmin,xmax},dist] is equivalent to TransformedDistribution[f,xdist], where f is given by Piecewise[{{xmin,x<=xmin},{x,xmin<x<xmax},{xmax,x>=xmax}}]. CensoredDistribution is often confused with TruncatedDistribution, though the two are fundamentally different in the sense that censoring puts the probability at the end of the censoring interval, while the probability is distributed over the truncation interval via truncation.

# Examples

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## Basic Examples(2)

Define a left-censored discrete distribution:

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Probability density function:

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Define a right-censored continuous distribution:

 In[1]:=

Cumulative distribution function:

 In[2]:=
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