X-12-ARIMA

Semiparametric Tail Index Estimation: A Density Quantile Approach

Scott Holan⁽¹⁾ and Tucker S. McElroy⁽²⁾

ABSTRACT:

Heavy tail probability distributions are important in many scientific disciplines, such as hydrology, geology, and physics among others. To this end many heavy tail distributions are commonly used in practice. In order to determine an appropriate family of distributions for a specified application it is useful to classify the probability law via its tail behavior. Through the use of Parzen's density-quantile function, this work proposes a semiparametric estimator of the tail index. The method we develop is useful when little or nothing is known about the distribution a priori. Furthermore the approach we develop allows for separate estimates of the left and right tail indices. In the development of the asymptotic theory of the tail index estimator we provide results of independent interest that may be used to establish weak convergence of stochastic processes. Finally, we present theoretical properties for the tail index estimator and explore its finite sample properties through simulation.

KEYWORDS:

Density quantile, Extreme-value theory, Quantile density, Semiparametric, Tail exponent

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⁽¹⁾ Scott Holan is a professor in the Department of Statistics, University of Missouri-Columbia.

⁽²⁾ Tucker S. McElroy is Mathematical Statistican, Center for Statistical Research and Methodology U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email : Tucker.S.McElroy@census.gov