Efficient on-line estimation of autoregressive parameters

Abstract

New procedures for estimating autoregressive parameters in AR(m) models are proposed. The proposed method allows for incorporation of auxiliary information into the estimation process and, under certain regularity conditions are consistent and asymptotically efficient. Also, these procedures are naturally on-line and do not require storing all the data. Theoretical results are presented in the case when m = 1. Two important particular cases are considered in details: linear procedures and likelihood procedures with the LS truncations. A specific example is also presented to briefly discuss some practical aspects of applications of the procedures of this type.

Real Time Anomaly Detection And Categorisation

“…Performing this correction requires knowledge of the AR(1) parameter φ. If it is unknown it can be estimated robustly either on a batch of the data or sequentially using SA-estimates (see (Sharia, 2010))…”

A New Recursive Estimation Method for Single Input Single Output Models

“…The analysis is illustrated in the case of an AR(1) process. The approach, based on Robbins–Monro stochastic approximation, is generalized by Sharia () to m ‐parameter truncation estimation (see also Sharia, , where other references can be found). It is partially illustrated on AR( m ) estimation.…”

“…), z t in () for γ t = 1/ t , and use Theorems 1 and 4 of Ljung () by checking a set of conditions, denoted by C , Appendix . Like Ouakasse and Mélard (), we follow Benveniste et al () to analyze a simplified algorithm, unlike Sharia, (). Note that the fully parametrized models was not considered before, even for the original RML algorithm.…”

Truncated stochastic approximation with moving bounds: convergence

“…Such a procedure is obviously consistent since θt ∈ [ θt − δ t , θt + δ t ] and θt ± δ t → θ. However, to construct an efficient estimator, care should be taken to ensure that the truncation intervals do not shrink to θt too rapidly, for otherwise θt will have the same asymptotic properties as θt (see [28] for details in the case of AR processes). Since this paper is concerned with the convergence, details of this application is not discussed here.…”

“…For example, an idea of truncations with shrinking bounds goes back to [10] and [13]. Truncations with expanding bounds were considered in [1] and also, in the context of recursive parametric estimation, in [23] (see also [28]). Truncations with adaptive truncation sets of the Robbins-Monro SA were introduced by Chen and Zhu in [5], and further explored and extended in [6], [2], [30], [31], [17].…”