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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Distortion function</span></span>
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<p>A <b>distortion function</b> in <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Statistics" title="Statistics">statistics</a>, for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:[0,1]\to [0,1]}">
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</math></span><img src="./0329691aecddec5158ad537f5df2e83578a302e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.972ex; height:2.843ex;" alt="{\displaystyle g:[0,1]\to [0,1]}" loading="lazy"></span>, is a <a href="Non-decreasing_function" class="mw-redirect" title="Non-decreasing function">non-decreasing function</a> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(0)=0}">
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</math></span><img src="./679b33e306b0721551440784cbc6b73cc13c77d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.349ex; height:2.843ex;" alt="{\displaystyle g(0)=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(1)=1}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle g(1)=1}</annotation>
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</math></span><img src="./e8f44832211fe2708fcaa0897d33654ea8f7abb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.349ex; height:2.843ex;" alt="{\displaystyle g(1)=1}" loading="lazy"></span>. The <b>dual distortion function</b> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {g}}(x)=1-g(1-x)}">
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<annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}(x)=1-g(1-x)}</annotation>
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</math></span><img src="./0d0a7ff0b296ceb899b58a415eea5e3e663e0cb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.73ex; height:2.843ex;" alt="{\displaystyle {\tilde {g}}(x)=1-g(1-x)}" loading="lazy"></span>.<sup id="cite_ref-PropertiesDRM_1-0" class="reference"><a href="#cite_note-PropertiesDRM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wirch_2-0" class="reference"><a href="#cite_note-Wirch-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Distortion functions are used to define <a href="Distortion_risk_measure" title="Distortion risk measure">distortion risk measures</a>.<sup id="cite_ref-Wirch_2-1" class="reference"><a href="#cite_note-Wirch-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Given a <a href="Probability_space" title="Probability space">probability space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}">
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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> we can define a new <a href="Probability_measure" title="Probability measure">probability measure</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">
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</math></span><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span> such that for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">
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</math></span><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy"></span> it follows that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).}">
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).}</annotation>
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</math></span><img src="./0f871977a012b3bac528c1c3b92513937a8bc313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.824ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).}" loading="lazy"></span> <sup id="cite_ref-PropertiesDRM_1-1" class="reference"><a href="#cite_note-PropertiesDRM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-PropertiesDRM-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-PropertiesDRM_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PropertiesDRM_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFBalbásGarridoMayoral2008" class="citation journal cs1">Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". <i>Methodology and Computing in Applied Probability</i>. <b>11</b> (3): 385. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11009-008-9089-z">10.1007/s11009-008-9089-z</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10016%2F14071">10016/14071</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53327887">53327887</a>.</cite></span>
</li>
<li id="cite_note-Wirch-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wirch_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wirch_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFJulia_L._WirchMary_R._Hardy" class="citation web cs1">Julia L. Wirch; Mary R. Hardy. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf">"Distortion Risk Measures: Coherence and Stochastic Dominance"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf">the original</a> <span class="cs1-format">(PDF)</span> on July 5, 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">March 10,</span> 2012</span>.</cite></span>
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