Abstract <p>In this paper, the recent shifted-exponential variation property which is defined as the ratio of variance to the squared of shifted expectation is investigated for both three-parameter Weibull and log-logistic models. These nonnegative semicontinuous models are widely considered in engineering, economics, hydrology, demography and many other fields. It is shown that the log-logistic distribution corresponds to over-, equi-, and under-varied if and only if its only positive shape parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12004_2025_5066_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta\)</EquationSource>  </InlineEquation> is greater, equal and less than the determined value <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12004_2025_5066_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta_{1}\in(0,1/2)\)</EquationSource>  </InlineEquation>, respectively. Similar result holds for the Weibull distribution with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12004_2025_5066_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta_{1}=1\)</EquationSource>  </InlineEquation> and extends the one of two-parameter model. The Newton–Raphson method is used to determine the approximative value <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12004_2025_5066_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta_{1}=0.37100965\)</EquationSource>  </InlineEquation> of the log-logistic model; it can thus lead to the reference shifted-exponential model, as for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12004_2025_5066_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta_{1}=1\)</EquationSource>  </InlineEquation> of the Weibull one. The relative variation between Weibull and log-logistic is also mentioned. Finally, two illustrative applications are provided.</p>

The Shifted-Exponential Variation Property for the Weibull and Log-Logistic Models

期刊论文
发表日期：

2025年3月1日

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Amadou Sawadogo,
Marcelo Bourguignon,
Célestin C. Kokonendji

摘要

Abstract

In this paper, the recent shifted-exponential variation property which is defined as the ratio of variance to the squared of shifted expectation is investigated for both three-parameter Weibull and log-logistic models. These nonnegative semicontinuous models are widely considered in engineering, economics, hydrology, demography and many other fields. It is shown that the log-logistic distribution corresponds to over-, equi-, and under-varied if and only if its only positive shape parameter \(\beta\) is greater, equal and less than the determined value \(\beta_{1}\in(0,1/2)\) , respectively. Similar result holds for the Weibull distribution with \(\beta_{1}=1\) and extends the one of two-parameter model. The Newton–Raphson method is used to determine the approximative value \(\beta_{1}=0.37100965\) of the log-logistic model; it can thus lead to the reference shifted-exponential model, as for \(\beta_{1}=1\) of the Weibull one. The relative variation between Weibull and log-logistic is also mentioned. Finally, two illustrative applications are provided.