This paper presents a new random weighting method to estimation of the stable exponent. Assume that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1, X_2, \ldots ,X_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> is a sequence of independent and identically distributed random variables with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>-stable distribution G, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\alpha \in (0,2]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation> is the stable exponent. Denote the empirical distribution function of G by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$G_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>G</mi> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> and the random weighting estimation of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$G_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>G</mi> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$H_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>H</mi> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation>. An empirical distribution function <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\widetilde{F}_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> with U-statistic structure is defined based on the sum-preserving property of stable random variables. By minimizing the Cramer-von-Mises distance between <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$H_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>H</mi> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$${\widetilde{F}}_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation>, the random weighting estimation of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation> is constructed in the sense of the minimum distance. The strong consistency and asymptotic normality of the random weighting estimation are also rigorously proved. Experimental results demonstrate that the proposed random weighting method can effectively estimate the stable exponent, resulting in higher estimation accuracy than the Zolotarev, Press, Fan and maximum likelihood methods. Copyright Springer-Verlag Berlin Heidelberg 2014
