Study On Properties Of Crovella Estimation For The Heavy Tail Index

The property of power-law behavior in the tail of a distribution has important implications, and plenty of examples from the financial market show that in many practical situations the distributions are heavy-tailed. So the estimation of the tail index for heavy-tailed distributions aroused our concern, many scholars proposed several methods, but all of them have some disadvantages. Crovella presented a method based on the scaling properties of sums of heavy-tailed random variables. It has the advantages of being easy to apply, and of being relatively accurate.In chap 2 of this paper we describe the estimation presented by Crovella, analyze the consistency of the estimator, and prove that the Crovella estimator has strong consistency. In chap 3 of this paper we contrast the Crovella estimator with Hill estimator by stochastic simulation, present evidence that when we estimate the tail index of the Levy-Stable distribution, the Crovella estimator appers to be more accurate than the Hill estimator as the size of the dataset grows, especially for the tail index α close to 2.