The Annals of Mathematical Statistics

On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations

T. W. Anderson

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Abstract

Let $x_t(t = 1, 2, \cdots)$ be defined recursively by \begin{equation*}\tag{1.1}x_t = ax_{t-1} + u_t,\quad t = 1, 2, \cdots,\end{equation*} where $x_0$ is a constant, $\varepsilon u_t = 0, \varepsilon u^2_t = \sigma^2$ and $\varepsilon u_tu_s = 0, t \neq s$. ($\varepsilon$ denotes mathematical expectation.) An estimate of $\alpha$ based on $x_1, \cdots, x_T$ (which is the maximum likelihood estimate of $\alpha$ if the $u$'s are normally distributed) is \begin{equation*}\tag{1.2}\hat \alpha = \bigg(\sum^T_{t=1} x_tx_{t-1}\bigg)\bigg/\big(\sum^T_{t=1} x^2_{t-1}\bigg).\end{equation*} If $|\alpha| < 1, \sqrt T (\hat \alpha - \alpha)$ has a limiting normal distribution with mean 0 under fairly general conditions such as independence of the $u$'s and uniformly bounded moments of the $u$'s of order $4 + \epsilon$, for some $\epsilon > 0$. (See [2], Chapter II, for example.) If $|\alpha| > 1$, White [3] has shown $(\hat \alpha - \alpha)|\alpha|^T/(\alpha^2 - 1)$ has a limiting Cauchy distribution under the assumption that $x_0 = 0$ and the $u$'s are normally distributed; he has also found the distribution when $x_0 \neq 0$. His results can be easily modified and restated in the following form $(\Sigma^T_{t=1} x^2_{t-1})^{\frac{1}{2}}(\hat \alpha - \alpha)$ has a limiting normal distribution if the $u$'s are normally distributed and if $|\alpha| \neq 1$. Peculiarly, for $|\alpha| = 1$ this statistic has a limiting distribution which is not normal (and is not even symmetric for $x_0 = 0)$. One purpose of this paper is to characterize the limiting distributions for $|\alpha| > 1$ when the $u$'s are not necessarily normally distributed; it will be shown that for $|\alpha| > 1$ the results depend on the distribution of the $u$'s. Central limit theorems are not applicable. Secondly, the limiting distribution for $|\alpha| < 1$ will be shown to hold under the assumption that the $u$'s are independently, identically distributed with finite variance. This was conjectured by White.

Article information

Source
Ann. Math. Statist., Volume 30, Number 3 (1959), 676-687.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706198

Digital Object Identifier
doi:10.1214/aoms/1177706198

Mathematical Reviews number (MathSciNet)
MR107347

Zentralblatt MATH identifier
0092.36502

JSTOR
links.jstor.org

Citation

Anderson, T. W. On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations. Ann. Math. Statist. 30 (1959), no. 3, 676--687. doi:10.1214/aoms/1177706198. https://projecteuclid.org/euclid.aoms/1177706198


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