<p>This article proposes a new method of truncated estimation to estimate the tail index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha (0<\alpha \le 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the extremely heavy-tailed distribution with infinite mean or variance. We not only present two truncated estimators <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\hat{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\hat{\alpha }^{\prime }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>α</mi> <mo stretchy="false">^</mo> </mover> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> for estimating <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha (0<\alpha \le 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha (1<\alpha \le 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> respectively, but also prove their asymptotic statistical properties. The numerical simulation results show that the two truncated estimators have better performance in estimating error and the type error I than that of the three known estimators, Hill estimator, QQ estimator and the moment estimator.</p>

A new method for estimating the tail index using truncated sample mean

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发表日期：

2025年1月13日

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Tang Fuquan,
Han Dong

摘要

This article proposes a new method of truncated estimation to estimate the tail index $\alpha (0<\alpha \le 2)$ $α (0 < α \leq 2)$ of the extremely heavy-tailed distribution with infinite mean or variance. We not only present two truncated estimators $\hat{\alpha }$ $\hat{α}$ and $\hat{\alpha }^{\prime }$ ${\hat{α}}^{'}$ for estimating $\alpha (0<\alpha \le 1)$ $α (0 < α \leq 1)$ and $\alpha (1<\alpha \le 2)$ $α (1 < α \leq 2)$ respectively, but also prove their asymptotic statistical properties. The numerical simulation results show that the two truncated estimators have better performance in estimating error and the type error I than that of the three known estimators, Hill estimator, QQ estimator and the moment estimator.