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   \end{cases}, \quad x \in [l, u]\]</p><p>where <span>$f_{d_0}(x)$</span> is the probability density (mass) function of <span>$d_0$</span>.</p><p>If <span>$Z \sim d_0$</span>, and <code>X = clamp(Z, l, u)</code>, then <span>$X \sim d$</span>. Note that this implies that even if <span>$d_0$</span> is continuous, its censored form assigns positive probability to the bounds <span>$l$</span> and <span>$u$</span>. Therefore, a censored continuous distribution has atoms and is a mixture of discrete and continuous components.</p><p>The function falls back to constructing a <a href="#Distributions.Censored"><code>Distributions.Censored</code></a> wrapper.</p><p><strong>Usage</strong></p><pre><code class="language-julia hljs">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]</code></pre><p><strong>Implementation</strong></p><p>To implement a specialized censored form for distributions of type <code>D</code>, instead of overloading a method with one of the above signatures, one or more of the following methods should be implemented:</p><ul><li><code>censored(d0::D, l::T, u::T) where {T &lt;: Real}</code></li><li><code>censored(d0::D, ::Nothing, u::Real)</code></li><li><code>censored(d0::D, l::Real, ::Nothing)</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/Distributions.jl/blob/d60af7b14842064a40ce51c5886482ff5ac45b0a/src/censored.jl#L1-L41">source</a></section></article><p>In the general case, this will create a <code>Distributions.Censored{typeof(d0)}</code> structure, defined as follows:</p><article class="docstring"><header><a class="docstring-binding" id="Distributions.Censored" href="#Distributions.Censored"><code>Distributions.Censored</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Censored</code></pre><p>Generic wrapper for a <a href="#Distributions.censored"><code>censored</code></a> distribution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/Distributions.jl/blob/d60af7b14842064a40ce51c5886482ff5ac45b0a/src/censored.jl#L62-L66">source</a></section></article><p>In general, <code>censored</code> should be called instead of the constructor of <code>Censored</code>, which is not exported.</p><p>Many functions, including those for the evaluation of pdf and sampling, are defined for all censored univariate distributions:</p><ul><li><a href="../univariate/#Base.maximum-Tuple{UnivariateDistribution}"><code>maximum(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Base.minimum-Tuple{UnivariateDistribution}"><code>minimum(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Distributions.insupport-Tuple{UnivariateDistribution, Any}"><code>insupport(::UnivariateDistribution, x::Any)</code></a></li><li><a href="../univariate/#Distributions.pdf-Tuple{UnivariateDistribution, Real}"><code>pdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logpdf-Tuple{UnivariateDistribution, Real}"><code>logpdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.cdf-Tuple{UnivariateDistribution, Real}"><code>cdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logcdf-Tuple{UnivariateDistribution, Real}"><code>logcdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logdiffcdf-Tuple{UnivariateDistribution, Real, Real}"><code>logdiffcdf(::UnivariateDistribution, ::T, ::T) where {T &lt;: Real}</code></a></li><li><a href="../univariate/#Distributions.ccdf-Tuple{UnivariateDistribution, Real}"><code>ccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logccdf-Tuple{UnivariateDistribution, Real}"><code>logccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Statistics.quantile-Tuple{UnivariateDistribution, Real}"><code>quantile(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.cquantile-Tuple{UnivariateDistribution, Real}"><code>cquantile(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.invlogcdf-Tuple{UnivariateDistribution, Real}"><code>invlogcdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.invlogccdf-Tuple{UnivariateDistribution, Real}"><code>invlogccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Statistics.median-Tuple{UnivariateDistribution}"><code>median(::UnivariateDistribution)</code></a></li><li><a href="../mixture/#Base.rand-Tuple{AbstractMixtureModel}"><code>rand(::UnivariateDistribution)</code></a></li><li><a href="../mixture/#Random.rand!-Tuple{AbstractMixtureModel, AbstractArray}"><code>rand!(::UnivariateDistribution, ::AbstractArray)</code></a></li></ul><p>Some functions to compute statistics are available for the censored distribution if they are also available for its truncation:</p><ul><li><a href="../univariate/#Statistics.mean-Tuple{UnivariateDistribution}"><code>mean(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Statistics.var-Tuple{UnivariateDistribution}"><code>var(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Statistics.std-Tuple{UnivariateDistribution}"><code>std(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#StatsBase.entropy-Tuple{UnivariateDistribution}"><code>entropy(::UnivariateDistribution)</code></a></li></ul><p>For example, these functions are available for the following uncensored distributions:</p><ul><li><code>DiscreteUniform</code></li><li><code>Exponential</code></li><li><code>LogUniform</code></li><li><code>Normal</code></li><li><code>Uniform</code></li></ul><p><a href="../multivariate/#StatsBase.mode-Tuple{MvLogNormal}"><code>mode</code></a> is not implemented for censored distributions.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../truncate/">« Truncated Distributions</a><a class="docs-footer-nextpage" href="../multivariate/">Multivariate Distributions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 25 September 2024 23:42">Wednesday 25 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+censored(d0, l, u)              # d0 interval-censored to the interval [l, u]</code></pre><p><strong>Implementation</strong></p><p>To implement a specialized censored form for distributions of type <code>D</code>, instead of overloading a method with one of the above signatures, one or more of the following methods should be implemented:</p><ul><li><code>censored(d0::D, l::T, u::T) where {T &lt;: Real}</code></li><li><code>censored(d0::D, ::Nothing, u::Real)</code></li><li><code>censored(d0::D, l::Real, ::Nothing)</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/Distributions.jl/blob/5e30aff3f28ed8df93213551821d837517a13484/src/censored.jl#L1-L41">source</a></section></article><p>In the general case, this will create a <code>Distributions.Censored{typeof(d0)}</code> structure, defined as follows:</p><article class="docstring"><header><a class="docstring-binding" id="Distributions.Censored" href="#Distributions.Censored"><code>Distributions.Censored</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Censored</code></pre><p>Generic wrapper for a <a href="#Distributions.censored"><code>censored</code></a> distribution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaStats/Distributions.jl/blob/5e30aff3f28ed8df93213551821d837517a13484/src/censored.jl#L62-L66">source</a></section></article><p>In general, <code>censored</code> should be called instead of the constructor of <code>Censored</code>, which is not exported.</p><p>Many functions, including those for the evaluation of pdf and sampling, are defined for all censored univariate distributions:</p><ul><li><a href="../univariate/#Base.maximum-Tuple{UnivariateDistribution}"><code>maximum(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Base.minimum-Tuple{UnivariateDistribution}"><code>minimum(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Distributions.insupport-Tuple{UnivariateDistribution, Any}"><code>insupport(::UnivariateDistribution, x::Any)</code></a></li><li><a href="../univariate/#Distributions.pdf-Tuple{UnivariateDistribution, Real}"><code>pdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logpdf-Tuple{UnivariateDistribution, Real}"><code>logpdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.cdf-Tuple{UnivariateDistribution, Real}"><code>cdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logcdf-Tuple{UnivariateDistribution, Real}"><code>logcdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logdiffcdf-Tuple{UnivariateDistribution, Real, Real}"><code>logdiffcdf(::UnivariateDistribution, ::T, ::T) where {T &lt;: Real}</code></a></li><li><a href="../univariate/#Distributions.ccdf-Tuple{UnivariateDistribution, Real}"><code>ccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.logccdf-Tuple{UnivariateDistribution, Real}"><code>logccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Statistics.quantile-Tuple{UnivariateDistribution, Real}"><code>quantile(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.cquantile-Tuple{UnivariateDistribution, Real}"><code>cquantile(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.invlogcdf-Tuple{UnivariateDistribution, Real}"><code>invlogcdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Distributions.invlogccdf-Tuple{UnivariateDistribution, Real}"><code>invlogccdf(::UnivariateDistribution, ::Real)</code></a></li><li><a href="../univariate/#Statistics.median-Tuple{UnivariateDistribution}"><code>median(::UnivariateDistribution)</code></a></li><li><a href="../mixture/#Base.rand-Tuple{AbstractMixtureModel}"><code>rand(::UnivariateDistribution)</code></a></li><li><a href="../mixture/#Random.rand!-Tuple{AbstractMixtureModel, AbstractArray}"><code>rand!(::UnivariateDistribution, ::AbstractArray)</code></a></li></ul><p>Some functions to compute statistics are available for the censored distribution if they are also available for its truncation:</p><ul><li><a href="../univariate/#Statistics.mean-Tuple{UnivariateDistribution}"><code>mean(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Statistics.var-Tuple{UnivariateDistribution}"><code>var(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#Statistics.std-Tuple{UnivariateDistribution}"><code>std(::UnivariateDistribution)</code></a></li><li><a href="../univariate/#StatsBase.entropy-Tuple{UnivariateDistribution}"><code>entropy(::UnivariateDistribution)</code></a></li></ul><p>For example, these functions are available for the following uncensored distributions:</p><ul><li><code>DiscreteUniform</code></li><li><code>Exponential</code></li><li><code>LogUniform</code></li><li><code>Normal</code></li><li><code>Uniform</code></li></ul><p><a href="../multivariate/#StatsBase.mode-Tuple{MvLogNormal}"><code>mode</code></a> is not implemented for censored distributions.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../truncate/">« Truncated Distributions</a><a class="docs-footer-nextpage" href="../multivariate/">Multivariate Distributions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 25 September 2024 23:51">Wednesday 25 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>