Testing Self-Similarity Through Lamperti Transformations

Myoungji Lee, Marc G. Genton, Mikyoung Jun

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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. © 2016, International Biometric Society.
Original languageEnglish (US)
Pages (from-to)426-447
Number of pages22
JournalJournal of Agricultural, Biological, and Environmental Statistics
Volume21
Issue number3
DOIs
StatePublished - Jul 14 2016

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