Second-order analysis of regret for sequential estimation of the autoregressive parameter in a first-order autoregressive model

Abstract

This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance ${\sigma }^2\in (0,\infty )$. This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models.

Keywords:

• Best fixed sample size
• least squares estimator
• martingale
• point estimation
• regret
• stopping rule

SUBJECT CLASSIFICATIONS:

• 62L10
• 62G20
• 62G35

Acknowledgments

We thank Professor Nitis Mukhopadhyay for suggesting that we consider boostrap methods to compute the regret for real data. We also thank him for making many valuable suggestions that led to a substantial improvement in the presentation of this article.