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<p>In this paper, we explore the mean inactivity time (MIT) with respect to an item at a random time. We demonstrate that the MIT at random times closely aligns with established measures of variability. Our findings include a decomposition result, which shows that the MIT, like other variability measures, can be represented using covariance. Additionally, under the proportional reversed hazard rates (PRH) model, we show that the MIT, depending on the proportionality parameter, includes Gini’s mean difference and cumulative Tsallis entropy as specific cases. And the empirical cumulative Tsallis entropy is also proposed to estimate the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1719_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(E(X_{(T)})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On variability of the mean inactivity time at random time

  • 期刊论文

  • 发表日期：

    2025年5月8日

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  • Bin Lu

摘要

In this paper, we explore the mean inactivity time (MIT) with respect to an item at a random time. We demonstrate that the MIT at random times closely aligns with established measures of variability. Our findings include a decomposition result, which shows that the MIT, like other variability measures, can be represented using covariance. Additionally, under the proportional reversed hazard rates (PRH) model, we show that the MIT, depending on the proportionality parameter, includes Gini’s mean difference and cumulative Tsallis entropy as specific cases. And the empirical cumulative Tsallis entropy is also proposed to estimate the \(E(X_{(T)})\) E ( X ( T ) ) .