X-13ARIMA-SEATS Seasonal Adjustment Program

Tail Exponent Estimation via Broadband Log Density-Quantile Regression

Scott Holan⁽¹⁾ and Tucker McElroy⁽²⁾

ABSTRACT

Heavy tail probability distributions are important in many scientific disciplines such as hydrology, geology, and physics and therefore feature heavily in statistical practice. Rather than specifying a family of heavy-tailed distributions for a given application, it is more common to use a nonparametric approach, where the distributions are classified according to the tail behavior. Through the use of the logarithm of Parzen’s density-quantile function, this work proposes a consistent, flexible estimator of the tail exponent. The approach we develop is based on a Fourier series estimator and allows for separate estimates of the left and right tail exponents. The theoretical properties for the tail exponent estimator are determined, and we also provide some results of independent interest that may be used to establish weak convergence of stochastic processes. We assess the practical performance of the method by exploring its finite sample properties in simulation studies. The overall performance is competitive with classical tail index estimators, and, in contrast with these, our method obtains somewhat better results in the case of lighter heavy-tailed distributions.

KEY WORDS: Density-quantile; Extreme-value theory; Fourier series estimator; Quantile density; Tail index.

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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