A Note on Wavelet-Based Estimator of the Hurst Parameter
Abstract
:1. Introduction
- The changes of bias and variance with all different Hs, different data lengths, different s and different wavelets;
- The relations of selected with data length and H;
- The initialization for initial approximation wavelet coefficients which introduces errors in used detailed wavelet coefficients.
- The inaccurate bias correction caused by correlations of wavelet coefficients.
- The method of simulation for FBM is not enough exact that the empirical bias is caused.
2. Wavelet-Based Estimator
2.1. Definitions and Properties
- For fixed j, the are independent and identically distributed;
- The processes and , , are independent.
2.2. Two Wavelet-Based Estimators
The First Estimator
The Second Estimator
Explicit Formula of theTwo Estimators
Variance Comparison
2.3. Calculation of Wavelet Coefficients
2.4. The Initialization Method
3. Simulation of FBM
The Cholesky Method
The Circulant Embedding Method
4. Simulation Results and Discussions
4.1. Selection of Parameters
4.2. Results and Discussions on Empirical Bias
- The initialization for given in (27) introduces errors in , and the initialization errors are significant on small octaves but decrease with increasing j.
- The inaccurate bias correction for (under independent assumptions) caused by correlations of wavelet coefficients.
- The method of simulation for FBM is not enough exact that the empirical bias is caused.
- The increase of N and change of wavelet made no improvements to the empirical bias. The chosen of biorthogonal wavelet makes the empirical bias worse.
- The increase of leads to the decrease of empirical bias.
- The empirical bias increases with the decrease of H. when choosing and , the empirical bias of estimator can be ignored for . So the estimator is suitable to detect the long-range dependence (can be described by ).
Comparison of Estimators
Comparison of Simulation Methods
4.3. Analysis of the Initialization Method
4.4. Analysis of Noise Effects
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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H | Bias | Std | RMSE | Bias | Std | RMSE | ||
---|---|---|---|---|---|---|---|---|
0.05 | 7 | −0.0122 | 0.0133 | 0.0180 | 3 | −0.1450 | 0.0030 | 0.1451 |
0.10 | 6 | −0.0087 | 0.0091 | 0.0126 | 3 | −0.0801 | 0.0031 | 0.0801 |
0.15 | 6 | −0.0046 | 0.0090 | 0.0101 | 3 | −0.0499 | 0.0030 | 0.0500 |
0.20 | 5 | −0.0056 | 0.0064 | 0.0085 | 3 | −0.0330 | 0.0031 | 0.0331 |
0.25 | 5 | −0.0032 | 0.0063 | 0.0071 | 3 | −0.0226 | 0.0032 | 0.0228 |
0.30 | 5 | −0.0019 | 0.0063 | 0.0066 | 3 | −0.0160 | 0.0032 | 0.0163 |
0.35 | 4 | −0.0038 | 0.0045 | 0.0059 | 3 | −0.0115 | 0.0032 | 0.0119 |
0.40 | 4 | −0.0023 | 0.0047 | 0.0052 | 3 | −0.0081 | 0.0033 | 0.0087 |
0.45 | 4 | −0.0019 | 0.0048 | 0.0052 | 3 | −0.0060 | 0.0033 | 0.0068 |
0.50 | 4 | −0.0012 | 0.0048 | 0.0049 | 3 | −0.0044 | 0.0033 | 0.0056 |
0.55 | 3 | −0.0030 | 0.0034 | 0.0046 | 3 | −0.0030 | 0.0034 | 0.0046 |
0.60 | 3 | −0.0025 | 0.0036 | 0.0044 | 3 | −0.0025 | 0.0036 | 0.0044 |
0.65 | 3 | −0.0018 | 0.0034 | 0.0038 | 3 | −0.0018 | 0.0034 | 0.0038 |
0.70 | 3 | −0.0014 | 0.0035 | 0.0038 | 3 | −0.0014 | 0.0035 | 0.0038 |
0.75 | 3 | −0.0013 | 0.0037 | 0.0039 | 3 | −0.0013 | 0.0037 | 0.0039 |
0.80 | 3 | −0.0008 | 0.0036 | 0.0037 | 3 | −0.0008 | 0.0036 | 0.0037 |
0.85 | 3 | −0.0006 | 0.0037 | 0.0038 | 3 | −0.0006 | 0.0037 | 0.0038 |
0.90 | 3 | −0.0006 | 0.0037 | 0.0038 | 3 | −0.0006 | 0.0037 | 0.0038 |
0.95 | 3 | −0.0006 | 0.0038 | 0.0038 | 3 | −0.0006 | 0.0038 | 0.0038 |
H | n | Bias | Std | RMSE | Bias | Std | RMSE | ||
---|---|---|---|---|---|---|---|---|---|
2 | −0.0632 | 0.0473 | 0.0789 | 3 | −0.0239 | 0.0776 | 0.0811 | ||
3 | −0.0220 | 0.0305 | 0.0376 | 3 | −0.0220 | 0.0305 | 0.0376 | ||
0.3 | 4 | −0.0080 | 0.0202 | 0.0217 | 3 | −0.0186 | 0.0136 | 0.0231 | |
4 | −0.0063 | 0.0096 | 0.0115 | 3 | −0.0167 | 0.0065 | 0.0179 | ||
5 | −0.0019 | 0.0063 | 0.0066 | 3 | −0.0160 | 0.0032 | 0.0163 | ||
2 | −0.0276 | 0.0479 | 0.0553 | 3 | −0.0078 | 0.0779 | 0.0783 | ||
2 | −0.0231 | 0.0202 | 0.0307 | 3 | −0.0073 | 0.0312 | 0.0320 | ||
0.5 | 3 | −0.0048 | 0.0142 | 0.0149 | 3 | −0.0048 | 0.0142 | 0.0149 | |
3 | −0.0050 | 0.0068 | 0.0085 | 3 | −0.0050 | 0.0068 | 0.0085 | ||
4 | −0.0012 | 0.0048 | 0.0049 | 3 | −0.0044 | 0.0033 | 0.0056 | ||
2 | −0.0109 | 0.0526 | 0.0537 | 3 | −0.0056 | 0.0878 | 0.0879 | ||
2 | −0.0061 | 0.0233 | 0.0241 | 3 | −0.0010 | 0.0359 | 0.0359 | ||
0.8 | 2 | −0.0062 | 0.0106 | 0.0123 | 3 | −0.0015 | 0.0159 | 0.0160 | |
3 | −0.0010 | 0.0074 | 0.0075 | 3 | −0.0010 | 0.0074 | 0.0075 | ||
3 | −0.0008 | 0.0036 | 0.0037 | 3 | −0.0008 | 0.0036 | 0.0037 |
H | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 |
---|---|---|---|---|---|---|---|---|---|---|
2.41 | 2.18 | 2.32 | 2.27 | 2.16 | 2.19 | 2.27 | 2.22 | 2.18 | 2.13 | |
H | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 | |
1.92 | 2.08 | 2.00 | 1.94 | 1.76 | 1.95 | 1.71 | 1.89 | 1.88 |
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Wu, L. A Note on Wavelet-Based Estimator of the Hurst Parameter. Entropy 2020, 22, 349. https://doi.org/10.3390/e22030349
Wu L. A Note on Wavelet-Based Estimator of the Hurst Parameter. Entropy. 2020; 22(3):349. https://doi.org/10.3390/e22030349
Chicago/Turabian StyleWu, Liang. 2020. "A Note on Wavelet-Based Estimator of the Hurst Parameter" Entropy 22, no. 3: 349. https://doi.org/10.3390/e22030349