Very weak expansions for sequential confidence levels. (English) Zbl 0603.62089

Woodroofe, Michael

Ann. Stat. 14, 1049-1067 (1986).

Let \(X_ 1,X_ 2,..\). be i.i.d. random variables whose common distribution function \(F_{\omega}\) depends on an unknown parameter \(\omega\in \Omega\) and \(\{F_{\omega}\), \(\omega\in \Omega \}\) forms a one-parameter exponential family. Suppose that each \(F_{\omega}\) has a finite mean \(\theta =\theta (\omega)\) and a finite positive variance \(\sigma^ 2=\sigma^ 2(\omega)\). To consider the problem of setting approximate confidence bounds for \(\theta\) when a sequential sample is taken, we put \({\hat \theta}_ n=(X_ 1+...+X_ n)/n\) and denote a consistent sequence of positive estimators of \(\sigma^ 2\) by \({\hat \sigma}^ 2_ n\). Let \(\gamma\) be the desired confidence coefficient and c the \(\gamma\) th quantile of the standard normal distribution. Define \(c_ n=c+n^{-1/2} b_ n({\hat \theta}_ n)\) where \(b_ n({\hat \theta})\) is an appropriate function. Further, let \(\{t(a), a\geq 1\}\) be a family of stopping times. Then the confidence curves of confidence intervals are defined by \[ \gamma_ a(\omega)=P_{\omega}(\sqrt{t(a)}({\hat \theta}_{t(a)}-\theta)/{\hat \sigma}_{t(a)}\leq c_{t(a)}) \] for \(\omega\) in some subset \(\Omega^ 0\) of \(\Omega\), and the average coverage probability under density \(\xi\) is defined by \({\bar \gamma}_ a(\xi)=\int \gamma_ a(\omega)\xi (\omega)d\omega\). In this paper, the author obtained some weak asymptotic expansions of \({\bar\gamma}_ a(\xi)\) which not only show the effect of the method, but provide a method for changing the confidence limits to reduce this effect.

Reviewer: K.-I.Yoshihara

MSC:

62L12

Sequential estimation

Keywords:

posterior distributions; estimation following sequential testing; one- parameter exponential family; approximate confidence bounds; sequential sample; standard normal; stopping times; confidence curves; average coverage probability; weak asymptotic expansions

Citations:

Zbl 0603.62090