Page 26 - Contributed Paper Session (CPS) - Volume 4

CPS2107 Junfan Tao et al.

Joint asymptotic normality of stopping times and
sequential estimators in monitoring
autoregressive processes
2
3
1
1
K. Nagai , K. Hitomi , Y. Nishiyama , J. Tao
1 Yokohama National University, Yokohama, Japan
2 Kyoto Institute of Technology, Kyoto, Japan
3 Kyoto University, Kyoto, Japan

Abstract
We consider the joint asymptotic properties of stopping times and sequential
estimators for a stationary rst-order autoregressive process (AR(1)) with independent
and identically distributed (i.i.d.) errors with mean 0 and nite variance. Lai and
Siegmund (1983) de ned two stopping times based on the observed Fisher
information. The rst stopping time is de ned to be the rst time at which the observed
Fisher information with known variance of errors exceeds a prescribed level. The
second one is de ned by replacing the variance of errors with its estimator. They
derived the almost sure convergence of the stopping times to some constant for a
stationary AR(1). Using a functional central limit theorem for nonlinear ergodic
stationary processes and Skorohod’s representation theorem, we show that the
stopping times, the sequential least square estimators, and the estimator of the
variance of errors have the joint asymptotic normality. We also nd that the asymptotic
variance of the rst stopping time is strictly greater than that of the second one.

Keywords
Statistical process monitoring; Observed Fisher information; Fixed accuracy
estimation; Functional central limit theorem; Skorohod’s representation
theorem

1. Introduction
Consider a AR(1) process {xn} on a probability space (Ω,F,P),
(1)
We assume that are independent, identically distributed random
variables with and that an initial value x0
∈ L is independent of . We consider two cases; the stationary case: |β| < 1
2
and the unit root case: β = ±1.
The least square estimate is

̂
2
= ∑ / ∑ −1 (2)
−1

=1 =1

It’s well known that when the process is a stationary AR(1), the least square
ˆ
estimate β N has asymptotic normality; as N → ∞,

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