Non symmetric Rosenblatt process over a compact. (English) Zbl 07532212
Summary: In this short note, we give the representation of the non symmetric Rosenblatt process as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval. Based on this representation, we obtain a least square-type estimator for an unknown parameter of the drift coefficient of a simple model driven by the non symmetric Rosenblatt process.
MSC:
| 60G18 | Self-similar stochastic processes |
| 60G12 | General second-order stochastic processes |
| 62M86 | Inference from stochastic processes and fuzziness |
| 62-XX | Statistics |
References:
| [1] | Bai, S.; Taqqu, M. S., Generalized Hermite processes, discrete chaos and limit theorems, Stochastic Processes and Their Applications, 124, 4, 1710-39 (2014) · Zbl 1282.60043 · doi:10.1016/j.spa.2013.12.011 |
| [2] | Bai, S.; Taqqu, M. S., Behavior of the generalized Rosenblatt process at extreme critical exponent values, The Annals of Probability, 45, 2, 1278-324 (2017) · Zbl 1392.60026 · doi:10.1214/15-AOP1087 |
| [3] | Bell, D.; Nualart, D., Noncentral limit theorem for the generalized Hermite process, Electronic Communications in Probability, 22, 13 (2017) · Zbl 1386.60190 · doi:10.1214/17-ECP99 |
| [4] | Bertin, K.; Torres, S.; Tudor, C. A., Drift parameter estimation in fractional diffusions driven by perturbed random walks, Statistics & Probability Letters, 81, 2, 243-9 (2011) · Zbl 1233.62148 · doi:10.1016/j.spl.2010.10.003 |
| [5] | Embrechts, P.; Maejima, M., Selfsimilar processes (2002), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1008.60003 |
| [6] | Kleinow, T.; Hall, P.; Hardle, W.; Schmidt, P., Semiparametric bootstrap approach to hypothesis tests and confidence intervals for the Hurst coefficient, Statistical Inference for Stochastic Processes, 3, 263-76 (2000) · Zbl 0981.62035 |
| [7] | Maejima, M.; Tudor, C. A., On the distribution of the Rosenblatt process, Statistics & Probability Letters, 43, 1490-5 (2012) · Zbl 1287.60024 · doi:10.1016/j.spl.2013.02.019 |
| [8] | Maejima, M.; Tudor, C. A., Selfsimilar processes with stationary increments in the second wiener chaos, Probability and Mathematical Statistics, 32, 1, 167-86 (2012) · Zbl 1261.60041 |
| [9] | Mishura, Y., Lecture notes in Mathematics, 1929, Stochastic calculus for fractional Brownian motion and related processes (2008), Berlin: Springer-Verlag, Berlin · Zbl 1138.60006 |
| [10] | Nualart, D., The Malliavin calculus and related topics (1995), New York: Springer-Verlag, New York · Zbl 0837.60050 |
| [11] | Pipiras, V.; Taqqu, M. S., Regularization and integral representations of Hermite processes, Statistics & Probability Letters, 80, 23, 2014-23 (2010) · Zbl 1216.60029 · doi:10.1016/j.spl.2010.09.008 |
| [12] | Prakasa Rao, B. L. S., Statistical inference for diffusion type processes: Kendall’s library of statistics 8 (2010), New York, NY: Wiley, New York, NY · Zbl 1211.62143 |
| [13] | Samorodnitsky, G.; Taqqu, M., Stable non-Gaussian random processes: Stochastic models with infinite variance (1994), London: Chapman and Hall, London · Zbl 0925.60027 |
| [14] | Taqqu, M. S., Weak convergence to fractional Brownian motion and to the Rosenblatt process, Zeitschrift für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 31, 4, 287-302 (1975) · Zbl 0303.60033 · doi:10.1007/BF00532868 |
| [15] | Taqqu, M. S., Dependence in probability and statistics, A bibliographical guide to self-similar processes and long-range dependence, 137-62 (1986), Boston, MA: Birkhauser, Boston, MA · Zbl 0596.60054 |
| [16] | Tudor, C. A., Analysis of the Rosenblatt process, ESAIM: Probability and Statistics, 12, 230-57 (2008) · Zbl 1187.60028 · doi:10.1051/ps:2007037 |
| [17] | Tudor, C. A., Analysis of variations for self-similar processes. A stochastic calculus approach. Probability and its applications (2013), Cham: Springer, Cham · Zbl 1308.60004 |
| [18] | Wu, W. B., Unit root testing for functionals of linear processes, Econometric Theory, 22, 1-14 (2005) · Zbl 1083.62098 · doi:10.1017/S0266466606060014 |
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