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

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

Zbl 0603.62090
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