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A graphical test for local self-similarity in univariate data

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The Pareto distribution, or power-law distribution, has long been used to model phenomena in many fields, including wildfire sizes, earthquake seismic moments and stock price changes. Recent observations have brought the fit of the Pareto into question, however, particularly in the upper tail where it often overestimates the frequency of the largest events. This paper proposes a graphical self-similarity test specifically designed to assess whether a Pareto distribution fits better than a tapered Pareto or another alternative. Unlike some model selection methods, this graphical test provides the advantage of highlighting where the model fits well and where it breaks down. Specifically, for data that seem to be better modeled by the tapered Pareto or other alternatives, the test assesses the degree of local self-similarity at each value where the test is computed. The basic properties of the graphical test and its implementation are discussed, and applications of the test to seismological, wildfire, and financial data are considered.
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Keywords: Pareto distribution; goodness-of-fit testing; power law; self-similarity; tapered Pareto distribution

Document Type: Research Article

Affiliations: Department of Statistics,University of California, 8125 Math Sciences Bldg., Box 951554Los AngelesCA90095-1554, USA

Publication date: November 1, 2011

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