Estimation of the Self-similarity Index of Non-stationary Increments Self-similar Processes via Lamperti Transformations

TL;DR

This paper presents a new method using Lamperti transformations to estimate the self-similarity index of various self-similar processes, including non-stationary ones, with applications demonstrated on historical water level data.

Contribution

A novel estimation algorithm based on modified Lamperti transformations for self-similarity index of both stationary and non-stationary processes.

Findings

Estimator shows consistency in simulations

Method successfully applied to historical Nile water data

Implementation available in Python

Abstract

We introduce a novel method for estimating the self-similarity index of a general $H$-self-similar process with either stationary or non-stationary increments. The estimation algorithm is developed based on a modified Lamperti transformation, which transforms $H$-self-similar processes to stationary ones. As an application, we show how to use this approach to estimate the self-similarity index of fractional Brownian motion, subfractional Brownian motion, bifractional Brownian motion, and trifractional Brownian motion. Simulation study is performed to support the consistency of our estimators. Implementation in Python is publicly shared. Application on the estimation of the self-similarity index of the Nile river water level data from the year 900 to 1200 C.E..