Testing Self-Similarity Through Lamperti Transformations

Myoungji Lee (), Marc G. Genton () and Mikyoung Jun ()
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Myoungji Lee: Texas A&M University
Marc G. Genton: King Abdullah University of Science and Technology
Mikyoung Jun: Texas A&M University

Journal of Agricultural, Biological and Environmental Statistics, 2016, vol. 21, issue 3, No 3, 426-447

Abstract: Abstract Self-similar processes have been widely used in modeling real-world phenomena occurring in environmetrics, network traffic, image processing, and stock pricing, to name but a few. The estimation of the degree of self-similarity has been studied extensively, while statistical tests for self-similarity are scarce and limited to processes indexed in one dimension. This paper proposes a statistical hypothesis test procedure for self-similarity of a stochastic process indexed in one dimension and multi-self-similarity for a random field indexed in higher dimensions. If self-similarity is not rejected, our test provides a set of estimated self-similarity indexes. The key is to test stationarity of the inverse Lamperti transformations of the process. The inverse Lamperti transformation of a self-similar process is a strongly stationary process, revealing a theoretical connection between the two processes. To demonstrate the capability of our test, we test self-similarity of fractional Brownian motions and sheets, their time deformations and mixtures with Gaussian white noise, and the generalized Cauchy family. We also apply the self-similarity test to real data: annual minimum water levels of the Nile River, network traffic records, and surface heights of food wrappings.

Keywords: Fractional Brownian sheet; Hurst coefficient; Hypothesis test; Multi-self-similarity; Random fields; Stationarity (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s13253-016-0258-1

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