Semiparametric Tail Index Estimation: A Density Quantile
Approach
Scott Holan and Tucker McElroy
KEY WORDS: Density quantile, Extreme-value theory, Quantile density, Semiparametric, Tail exponent
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.
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