On Asymptotic Normality of Sequential LS-Estimates of Unstable Autoregressive Processes

Abstract

For estimating the unknown parameters in an unstable autoregressive AR(p), the article proposes sequential least squares estimates (LSEs) with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix that holds both inside the stability region of AR(p) process and on the part of its boundary. The asymptotic distribution of the stopping time is derived. Numerical results for AR(3) processes are given.

Keywords:

• Asymptotic normality
• Autoregressive process
• Least squares estimate
• Sequential estimation

Subject Classifications:

• 62L10
• 62L12

ACKNOWLEDGMENTS

This research was carried out at the Department of Mathematics, the Strasbourg University, France. The second author is partially supported by the RFFI-DFG-Grant 02-01-04001. The authors thank the referee for valuable remarks and suggestions.

Notes

Recommended by Sangyeol Lee