Q751015
type: Item
instance of: scholarly article
title: Étude de l'estimateur du maximum de vraisemblance dans le cas d'un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence. (Asymptotic behaviour of maximum likelihood estimator in an autoregressive process: consistency, asymptotic distribution and expansion, rate of convergence)
zbMATH Open document ID: 0714.60014
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77357
review text: The authors consider the group G of all invertible matrices \(g=\begin{pmatrix} A(g) & b(g)\\ 0&1 \end{pmatrix}\), where \(b(g)\in R^ d\) is a column, a norm \(\|.\|\) in \(R^ d\), a probability \(\mu\) on G for which the corresponding norm \(\| A(g)\|\) is \(\mu -a.e.<1\), a sequence \(g_ n\), \(n\geq 1\), of independent \(\mu\)-distributed random variables, a fixed \(z\in R^ d\), the distribution \(\nu\) of \(\sum_{k\geq 1}A(g_ 1)...A(g_{k-1})b(g_ k)\), \(\eta: G\to [1,\infty)\), constants \(c,\gamma,\tau >0\), \(\alpha\in (0,1]\), a function \(F: G\times R^ d\to R\) satisfying \[ | F(g,x)-F(g,y)| \leq \eta (y)\| x- y\|^{\gamma}(1+\| x\|^{\tau}+\| y\|^{\tau}) \] and \[ | F(g,x)| \leq \eta (g)(\| x\|^{\gamma +\tau}+1), \] \(\begin{pmatrix} Y_ n\\ 0 \end{pmatrix} =g_ n...g_ 1\begin{pmatrix} z\\ 0\end{pmatrix}\) for \(n\geq 0\), \(S_ n=\sum^{n}_{1}F(g_ k,Y_{k-1})\), \(e=\int \int Fd\mu d\nu\) and impose \[ (*)\quad \int \eta (g)^ 5(1- \| A(g)\|^{\alpha})^{-5(\gamma +\tau)/\alpha}\exp (c\| b(g)\|^{\alpha})d\mu (g)<\infty. \] First they prove that \(\sigma^ 2=\lim_{n} n^{-1}E((S_ n-ne)^ 2)\) exists, does not depend on z, and, under a supplementary condition on F, is positive, and that \[ d(n^{-1/2}\sigma^{-1}(S_ n-ne),N)\leq Cn^{-1/2}(1+\| x\|^{\gamma \beta}\exp (\lambda \| x\|^{\alpha}) \] with constants \(C,\lambda >0\), \(\beta >1\), where d is the supremum of the modulus of the difference of the distribution functions and N is standard normal. Replacing (*) by two other conditions, each containing \(m\geq 2\) at the exponent, and requiring \(\| A(g)\| \leq \rho <1\mu\)-a.e., they prove also that \[ \sup_{n} E(| n^{- 1/2}\sum^{n}_{1}(F(g_ k,Y_{k-1})-e)|^ m)\leq C(1+\| x\|^{m(\gamma +\tau)}). \] In the proofs they use the operators \((U_ tf)(x)=\int e^{itF(g,x)}f(g,x)d\mu (g)\) and arrange a Banach space on which \(U_ t\) are quasicompact etc. Then the authors consider the particular case, corresponding to the title, of a nonrandom \(A(g_ n)=A_{\theta}=\left( \begin{…
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4176107
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: autoregressive process
zbMATH Keywords: speed of convergence
zbMATH Keywords: limit law
zbMATH Keywords: unique invariant distribution
zbMATH Keywords: convergence in distribution
author: Xavier Milhaud
author: Albert Raugi
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
label: Étude de l'estimateur du maximum de vraisemblance dans le cas d'un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence. (Asymptotic behaviour of maximum likelihood estimator in an autoregressive process: consistency, asymptotic distribution and expansion, rate of convergence)
description: scientific article; zbMATH DE number 4176107