Page 27 - Contributed Paper Session (CPS) - Volume 4

CPS2107 Junfan Tao et al.

. (3)
For the case that are normally distributed, the observed Fisher
information about β is given b

. (4)
Lai and Siegmund (1983) considered a sequentially observed AR(1) process

and proposed to evaluate the least square estimator at the stopping time τ1c
de ned by

, (5)
for some predetermined c > 0. Later, we de ne a feasible stopping time τ2c in
(10) by replacing σ with its estimator in (5).

2
For the stopping time de ned in (5), we de ne the sequential least square
estimate by setting N = τ1c in (7). The asymptotic normality of β τ1c have been
ˆ
shown by Lai and Siegmund (1983). One of their main results is as c → ∞,
(6)
uniformly in β ∈ [−1,1], which allows us to obtain the confidence intervals
about β with fixed accuracy.
1
Lai and Siegmund (1983) also showed
2
2
τ1c/c → σ /γ(0) = 1 − β , (7)
where γ(·) is the covariance function of {xn}.
Suppose we sequentially observe {xn} from the stationary AR(1) model in
(1). When the initial value x0 possesses the stationary distribution, then {xn} has
the covariance function
. (8)
We assume that the initial value x0 is a L random variable and independent of

2
. Let s N be the estimator of σ ;
2
2
(9)
As well as τ1c in (5), we set a feasible stopping time ;

. (10)

1 To obtain the uniform asymptotic normality, Lai and Siegmund (1983) assumed that the
initial value x 0 is not dependent on β and for each
xed n ≥ 0. Since these assumptions do not hold when {x n} is a strongly stationary process, we
discard them and use the fruitful theory of the ergodic stationary processes. Then we obtain
the asymptotic normality of the stopping times instead of the uniformity in the asymptotic
normality of β τ1c.
ˆ
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