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# Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property †

Institute of Earthquake Prediction Theory and Mathematical Geophysics of Rassian Academy of Sciences, 113556 Moscow, Russia
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Presented at the 7th International conference on Time Series and Forecasting, Gran Canaria, Spain, 19–21 July 2021.
Academic Editors: Ignacio Rojas, Fernando Rojas, Luis Javier Herrera and Hector Pomare
Eng. Proc. 2021, 5(1), 19; https://doi.org/10.3390/engproc2021005019
Published: 28 June 2021
(This article belongs to the Proceedings of The 7th International conference on Time Series and Forecasting)
The publication is devoted to studying asymptotic properties of statistical estimates of the distribution parameters $u\in {R}^{q}$ of a multidimensional random stationary time series satisfying the strong mixing conditions. We consider estimates ${\stackrel{^}{u}}_{n}^{\delta }\left({\overline{z}}_{n}\right)$, ${\overline{z}}_{n}={\left({z}_{1}^{\mathrm{T}},\dots ,{z}_{n}^{\mathrm{T}}\right)}^{\mathrm{T}}\in {R}^{mn}$ that provide in asymptotic $n\to \infty$ the maximum values for some objective functions ${Q}_{n}\left({\overline{z}}_{n};u\right)$, which have properties similar to the well-known property of local asymptotic normality. These estimates are constructed by solving the equations ${\delta }_{n}\left({\overline{z}}_{n};u\right)=0$, where ${\delta }_{n}\left({\overline{z}}_{n};u\right)$ are arbitrary functions for which ${\delta }_{n}\left({\overline{z}}_{n};u\right)-\underset{h}{\mathrm{grad}}{Q}_{n}\left({\overline{z}}_{n};u+{n}^{-1/2}h\right)\to 0$$\left(n\to \infty \right)$ in ${P}_{n,u}\left({\overline{z}}_{n}\right)$-probability uniformly on $u\in U$, were $U$ is compact in ${R}^{q}$. In many cases, the estimates ${\stackrel{^}{u}}_{n}^{\delta }\left({\overline{z}}_{n}\right)$ have the same asymptotic properties as well-known M-estimates defined by equations ${\stackrel{^}{u}}_{n}^{Q}\left({\overline{z}}_{n}\right)$$=\underset{u\in U}{arg max}{Q}_{n}\left({\overline{z}}_{n};u\right)$ but often can be much simpler computationally. We consider an algorithmic method for constructing estimates ${\stackrel{^}{u}}_{n}^{\delta }\left({\overline{z}}_{n}\right)$, which is similar to the accumulation method first proposed by R. Fischer and rigorously developed by L. Le Cam. The main theoretical result of the article is the proof of the theorem, in which conditions of the asymptotic normality of estimates ${\stackrel{^}{u}}_{n}^{\delta }\left({\overline{z}}_{n}\right)$ are formulated, and the expression is proposed for their matrix of asymptotic mean-square deviations $\underset{n\to \infty }{\mathrm{lim}}n{\mathrm{E}}_{n,u}\left\{\left({\stackrel{^}{u}}^{\delta }\left({\overline{z}}_{n}\right)-u\right){\left({\stackrel{^}{u}}^{\delta }\left({\overline{z}}_{n}\right)-u\right)}^{\mathrm{T}}\right\}$. View Full-Text
MDPI and ACS Style

Kushnir, A.; Varypaev, A. Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property. Eng. Proc. 2021, 5, 19. https://doi.org/10.3390/engproc2021005019

AMA Style

Kushnir A, Varypaev A. Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property. Engineering Proceedings. 2021; 5(1):19. https://doi.org/10.3390/engproc2021005019

Chicago/Turabian Style

Kushnir, Alexander, and Alexander Varypaev. 2021. "Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property" Engineering Proceedings 5, no. 1: 19. https://doi.org/10.3390/engproc2021005019

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