[Paper Review] Estimation of ordinal pattern probabilities in fractional Brownian motion

Mathieu Sinn, Karsten Keller|ArXiv.org|Jan 10, 2008

Financial Risk and Volatility Modeling19 references3 citations

TL;DR

This paper provides a rigorous statistical analysis of the Zero Crossing (ZC) estimator for the Hurst parameter in fractional Brownian motion (fBm), showing its strong consistency and asymptotic normality for H < 3/4. It establishes confidence intervals for H, demonstrates that the ZC estimator has lower bias than the HEAF estimator despite higher variance, and proves invariance of ordinal pattern probabilities under monotonic transformations.

ABSTRACT

For equidistant discretizations of fractional Brownian motion (fBm), the probabilities of ordinal patterns of order d=2 are monotonically related to the Hurst parameter H. By plugging the sample relative frequency of those patterns indicating changes between up and down into the monotonic relation to H, one obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found considerable attention in mathematical and applied research. In this paper, we generally discuss the estimation of ordinal pattern probabilities in fBm. As it turns out, according to the sufficiency principle, for ordinal patterns of order d=2 any reasonable estimator is an affine functional of the sample relative frequency of changes. We establish strong consistency of the estimators and show them to be asymptotically normal for H&lt;3/4. Further, we derive confidence intervals for the Hurst parameter. Simulation studies show that the ZC estimator has larger variance but less bias than the HEAF estimator of the Hurst parameter.

Motivation & Objective

• To establish the statistical properties of ordinal pattern probability estimators in fractional Brownian motion (fBm), particularly focusing on the Zero Crossing (ZC) estimator of the Hurst parameter H.
• To prove strong consistency and asymptotic normality of the ZC estimator for H < 3/4, extending theoretical foundations for its use in time series analysis.
• To derive exact and asymptotic expressions for the variance of the sample relative frequency of changes, enabling precise confidence interval construction for H.
• To compare the ZC estimator with the HEAF estimator in terms of bias, variance, and finite-sample performance across different H values and sample sizes.
• To demonstrate that ordinal pattern probability estimators are invariant under monotonic transformations of fBm, supporting robustness in real-world applications.

Proposed method

• Uses the sufficiency principle to show that any reasonable estimator for d=2 ordinal patterns in fBm must be an affine functional of the sample relative frequency of changes (i.e., up-down or down-up transitions).
• Applies the delta method and central limit theorem to derive asymptotic normality of the ZC estimator under the condition H < 3/4.
• Derives exact and asymptotically equivalent formulas for the variance of the sample relative frequency of changes, which are used to construct confidence intervals for H.
• Employs simulation studies with 50,000 realizations of fBm across varying H values (0.55 to 0.95) and sample sizes (n = 128 to 8192) to evaluate bias, variance, and coverage probability of confidence intervals.
• Compares the ZC estimator to the HEAF estimator by plugging sample autocovariance into a monotonic functional relation to H, analyzing both bias and variance trade-offs.
• Proves invariance of ordinal pattern probabilities under monotonic transformations of fBm, showing that the statistical properties of the estimators are preserved.

Experimental results

Research questions

• RQ1 What are the theoretical properties—specifically consistency and asymptotic normality—of the ZC estimator for the Hurst parameter in fBm?
• RQ2 How does the variance of the sample relative frequency of changes behave, and can it be accurately approximated for finite samples?
• RQ3 What is the coverage probability of confidence intervals for H constructed from the ZC estimator, especially for H ≥ 3/4 where asymptotic normality is not guaranteed?
• RQ4 How does the ZC estimator compare to the HEAF estimator in terms of bias and variance across different sample sizes and H values?
• RQ5 Are ordinal pattern probability estimators invariant under monotonic transformations of fBm, and what are the implications for robust time series analysis?

Key findings

• The ZC estimator is strongly consistent and asymptotically normal for H < 3/4, with asymptotic variance derived from the sample relative frequency of changes.
• Confidence intervals for H constructed from the ZC estimator achieve approximately 95% coverage even for small sample sizes and H > 3/4, despite theoretical limitations.
• The ZC estimator exhibits significantly lower bias than the HEAF estimator, particularly for H > 0.85, where the HEAF estimator's bias increases substantially.
• For n = 8192, the ZC estimator's variance is 2 to 8 times larger than that of the HEAF estimator, indicating lower efficiency but superior bias performance.
• The sample relative frequency of changes has an exact variance formula, and asymptotically equivalent expressions are derived, enabling precise inference.
• Ordinal pattern probability estimators are invariant under unknown monotonic transformations of fBm, making them robust for real-world time series with non-linear dynamics.