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

William Wu, Qidi Peng

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..

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

Abstract

We introduce a novel method for estimating the self-similarity index of a general -self-similar process with either stationary or non-stationary increments. The estimation algorithm is developed based on a modified Lamperti transformation, which transforms -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..
Paper Structure (13 sections, 13 equations, 7 figures, 10 tables)

This paper contains 13 sections, 13 equations, 7 figures, 10 tables.

Table of Contents

  1. Literature Review on the Estimation of the Self-similarity Index
  2. A Modified Lamperti Transformation: A Novel Estimation Approach
  3. Lamperti Transformation
  4. Estimation of the Self-similarity Index When $\sigma^2$ is Given
  5. Estimation of the Self-similarity Index When $\sigma^2$ is Unknown
  6. Estimation of the Self-similarity Index with Examples
  7. Fractional Brownian motion
  8. Subfractional Brownian motion
  9. Bifractional Brownian motion
  10. Trifractional Brownian Motion
  11. Code availability
  12. Real World Application: The Nile River
  13. Conclusion

Figures (7)

  • Figure 1: The results are given by left: Algorithm \ref{['alg:WP_known_sigma']}, right: Algorithm \ref{['alg:WP_unknown_sigma']}. Each $H_0$ consists of 200 simulated paths of length $N=4096$. Squared error of the estimated parameter is computed for each path and drawn into a boxplot. Outliers over 99th percentile are cut out for the sake of clear visuals.
  • Figure 2: 200 paths of fBm are simulated for each corresponding $H_0$ and $N$ value with the RMSE plotted as a heatmap distribution. The results show empirical convergence towards 0 as the path length increases and $H_0$ value decreases.
  • Figure 3: The results are given by left: Algorithm \ref{['alg:WP_known_sigma']}, right: Algorithm \ref{['alg:WP_unknown_sigma']}. Each $H_0$ consists of 200 simulated paths of length $N=4096$. Squared error of the estimated parameter is computed for each path and drawn into a boxplot. Outliers over 99th percentile are cut out for the sake of clear visuals.
  • Figure 4: 200 paths of sfBm are simulated for each corresponding $H_0$ and $N$ value with the RMSE plotted as a heatmap distribution. The results show empirical convergence towards 0 as the path length increases.
  • Figure 5: The results are given by left: Algorithm \ref{['alg:WP_known_sigma']}, right: Algorithm \ref{['alg:WP_unknown_sigma']}. Each $H_0$ consists of 200 simulated paths of length $N=4096$. Squared error of the estimated parameter is computed for each path and drawn into a boxplot. Outliers over 99th percentile are cut out for the sake of clear visuals.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Definition 1.1
  • Definition 2.1