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

^{*}

^{†}

## Abstract

**:**

## 1. Introduction. Methods of Construction Asymptotically Efficient Estimates for Parameters of Stationary Time Series

## 2. Construction of M-Estimates for Parameters of Stationary Time Series with Suitable Asymptotical Properties

**u**. It is proved in Theorem 1 that under certain restrictions, such an estimate ${\widehat{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ can be found using the algorithm

**u**.

**Theorem**

**1.**

**A**. There exists a$\sqrt{n}$-consistent estimate ${u}_{n}^{\ast}\left({\overline{z}}_{n}\right)$ of the parameter

**u**.

**B**. Let the family of statistics${\delta}_{n}\left({\overline{z}}_{n},u\right)\in {R}^{m}$,$u\in U$, and the sequence of positive definite symmetric$q\times q$-matrix functions${\Phi}_{n}\left(u\right)$satisfy the following constraints:

**B1**. For each value of the parameter$u\in U$, the sequence of statistics${\delta}_{n}\left({\overline{z}}_{n},u\right)$is asymptotically normal with zero mean and the covariance matrix$\Psi \left(u\right)$:

**B2**. For each value of the parameter$u\in U$, the following asymptotic expansion of the statistic${\delta}_{n}\left({\overline{z}}_{n},u\right)$holds:

**Corollary**

**1.**

**a**) Let, for any $n\in {\mathbb{Z}}^{+}$, a statistic ${\tilde{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ be the root of the equation ${\delta}_{n}\left({\overline{z}}_{n};u\right)=0$ with respect to the parameter $u\in U$ with probability equal to 1.

**b**) Let the statistic ${\tilde{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ also is a $\sqrt{n}$-consistent estimate of the parameter $u\in U$. Then the statistic ${\tilde{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ is asymptotically normal with the moments $\left(0,D\left(u\right)\right)$.

**Remark**

**1.**

**a**) The statement similar to Statement (T1) of Theorem 1 was proved in [3,4] in the case when the objective function ${Q}_{n}\left({\overline{z}}_{n};u\right)$ is the likelihood function of ${\overline{z}}_{n}$ having the LAN property (2). In this case ${\delta}_{n}\left({\overline{z}}_{n};u\right)$ $\equiv $ ${\Delta}_{n}\left({\overline{z}}_{n};u\right)$, the matrix function ${\Phi}_{n}^{}\left(u\right)$ $\equiv $ ${\Gamma}_{n}\left(u\right)$ and

**u**.

**b**) It follows from the corollary of Theorem 1 that a statistic ${\tilde{u}}_{n}^{\Delta}\left({\overline{z}}_{n}\right)$, which has the property: ${\Delta}_{n}\left({\overline{z}}_{n};{\tilde{u}}_{n}^{\Delta}\left({\overline{z}}_{n}\right)\right)=0$ with probability equal to one, and at the same time is a $\sqrt{n}$-consistent estimate of the parameter $u\in U$, is asymptotically normal with the moments $\left(0,\Gamma \left(u\right)\right)$. Consequently, the statistic ${\tilde{u}}_{n}^{\Delta}\left({\overline{z}}_{n}\right)$ is the asymptotically efficient estimate of the parameter $u\in U$.

## 3. Proof of Theorem 1

**u**by a subscript.

**Lemma**

**1.**

**a**) $\underset{n\to \infty}{\mathrm{lim}}\underset{u\in U}{\mathrm{sup}}{P}_{n,u}\left\{\left|{\rho}_{n,u}\left({u}_{n}^{\ast}\right)\right|>\epsilon \right\}=0$, (

**b**) $\underset{n\to \infty}{\mathrm{lim}}\underset{u\in U}{\mathrm{sup}}{P}_{n,u}\left\{\left|{\xi}_{n,u}\left({u}_{n}^{\ast}\right)\right|>\epsilon \right\}=0$.

**Lemma**

**2.**

**a**) $\underset{n\to \infty}{\mathrm{lim}}{\mathfrak{L}}_{n}\left\{{\phi}_{n}\right\}=\underset{n\to \infty}{\mathrm{lim}}{P}_{n}\left\{{\phi}_{n}<x\right\}=F\left(x\right)$; (

**b**) for any $\epsilon >0$ $\underset{n\to \infty}{\mathrm{lim}}{P}_{n}\left\{\left|{\eta}_{n}\right|>\epsilon \right\}=0$.

## 4. Proof of Corollary

**b**) of the corollary, the statistic

**a**) of the corollary, we have that ${\widehat{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)={\tilde{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ with probability equal to one. Hence, the statistic ${\tilde{u}}_{n}^{\delta}\left({\overline{z}}_{n}\right)$ is asymptotically normal with the moments $\left(0,D\left(u\right)\right)$. □

## 5. Proof of Lemma 1

**a**) For any $\epsilon >0$, q > 0 and $u\in U$, we can write the following equation:

**b**) Since $\left|{\xi}_{n,u}\left({u}_{n}^{\ast}\right)\right|\le \Vert {\Phi}_{n,u}^{-1}\left({u}_{n}^{\ast}\right)\Vert \left|{\rho}_{n,u}\left({u}_{n}^{\ast}\right)\right|$, to prove statement (

**b**) of Lemma 1, it suffices to check that $\Vert {\Phi}_{n}^{-1}\left({u}_{n}^{\ast}\right)\Vert $ is bounded in probability. Since ${\Phi}_{n}^{-1}\left(u\right)$ satisfies conditions B2 of Theorem 1, for any $\epsilon >0$ there exists ${C}_{\epsilon}>0$ that for all n the following inequality holds: ${P}_{n,u}\left\{\Vert {\Phi}_{n,u}^{-1}\left({u}_{n}^{\ast}\right)\Vert \ge {C}_{\epsilon}\right\}<\epsilon $. So, we can write:

**a**) of Lemma 1, one can find a number ${N}_{\epsilon}$ such that $\underset{u\in U,\text{\hspace{0.17em}}n>{N}_{\epsilon}}{\mathrm{sup}}{P}_{u,n}\left\{\left|{\xi}_{u,n}\left({u}_{n}^{\ast}\right)\right|>\epsilon \right\}$< 2$\epsilon $. □

## 6. Conclusions

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**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