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Self-similarity index estimation via wavelets for locally self-similar processes. (English) Zbl 0989.62045

Summary: Many naturally occurring phenomena can be effectively modeled using self-similar processes. In such applications, accurate estimation of the scaling exponent is vital, since it is this index which characterizes the nature of the self-similarity. Although estimation of the scaling exponent has been extensively studied, previous work has generally assumed that this parameter is constant. Such an assumption may be unrealistic in settings where it is evident that the nature of the self-similarity changes as the phenomenon evolves. For such applications, the scaling exponent must be allowed to vary as a function of time, and a procedure must be available which provides a statistical characterization of this progression.
We propose and describe such a procedure. Our method uses wavelets to construct local estimates of time-varying scaling exponents for locally self-similar processes. We establish a consistency result for these estimates. We investigate the effectiveness of our procedure in a simulation study, and demonstrate it applicability in the analyses of a hydrological and a geophysical time series, each of which exhibit locally self-similar behavior.

MSC:

62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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