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Censored Distributions

In censoring of data, values exceeding an upper limit (right censoring) or falling below a lower limit (left censoring), or both (interval censoring) are replaced by the corresponding limit itself. The package provides the censored function, which creates the most appropriate distribution to represent a censored version of a given distribution.

A censored distribution can be constructed using the following signature:

Distributions.censored — Function
censored(d0::UnivariateDistribution; [lower::Real], [upper::Real])
censored(d0::UnivariateDistribution, lower::Real, upper::Real)

A censored distribution d of a distribution d0 to the interval $[l, u]=$[lower, upper] has the probability density (mass) function:

\[f(x; d_0, l, u) = \begin{cases} P_{Z \sim d_0}(Z \le l), & x = l \\ f_{d_0}(x), & l < x < u \\ P_{Z \sim d_0}(Z \ge u), & x = u \\ \end{cases}, \quad x \in [l, u]\]

where $f_{d_0}(x)$ is the probability density (mass) function of $d_0$.

If $Z \sim d_0$, and X = clamp(Z, l, u), then $X \sim d$. Note that this implies that even if $d_0$ is continuous, its censored form assigns positive probability to the bounds $l$ and $u$. Therefore, a censored continuous distribution has atoms and is a mixture of discrete and continuous components.

The function falls back to constructing a Distributions.Censored wrapper.

Usage

censored(d0; lower=l)           # d0 left-censored to the interval [l, Inf)
censored(d0; upper=u)           # d0 right-censored to the interval (-Inf, u]
censored(d0; lower=l, upper=u)  # d0 interval-censored to the interval [l, u]
censored(d0, l, u)              # d0 interval-censored to the interval [l, u]

Implementation

To implement a specialized censored form for distributions of type D, instead of overloading a method with one of the above signatures, one or more of the following methods should be implemented:

  • censored(d0::D, l::T, u::T) where {T <: Real}
  • censored(d0::D, ::Nothing, u::Real)
  • censored(d0::D, l::Real, ::Nothing)
source

In the general case, this will create a Distributions.Censored{typeof(d0)} structure, defined as follows:

In general, censored should be called instead of the constructor of Censored, which is not exported.

Many functions, including those for the evaluation of pdf and sampling, are defined for all censored univariate distributions:

Some functions to compute statistics are available for the censored distribution if they are also available for its truncation:

For example, these functions are available for the following uncensored distributions:

  • DiscreteUniform
  • Exponential
  • LogUniform
  • Normal
  • Uniform

mode is not implemented for censored distributions.