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Corrected confidence sets for sequentially designed experiments: Examples. (English) Zbl 0954.62096
Ghosh, Subir (ed.), Multivariate analysis, design of experiments, and survey sampling. A tribute to Jagdish N. Srivastava. New York, NY: Marcel Dekker. Stat., Textb. Monogr. 159, 135-161 (1999).
This paper is a continuation of the authors’ article, Stat. Sin. 7, No. 1, 53-74 (1997; Zbl 0904.62093). They consider a model of the form $y_k=x_k' \theta+ \sigma\varepsilon_k, \quad k=1,2, \dots,$ where $$x_k= (x_{k, 1}, \dots,x_{k,p})'$$ are design variables, $$\theta= (\theta_1, \dots, \theta_p)'$$ is a vector of unknown parameters, $$\sigma >0$$ may be known, and $$\varepsilon_1, \varepsilon_2, \dots$$ are i.i.d. standard normal. The design vectors $$x_k$$, $$k=1,2, \dots$$, may be chosen adaptively; that is, each $$x_k$$ may be of the form $x_k=x_k(u_1, \dots, u_k,y_1, \dots, y_{k-1}), \quad k=1,2, \dots,$ where $$u_1,u_2, \dots$$ are independent of $$\varepsilon_1, \varepsilon_2, \dots$$ and have a known distribution. Putting $$y_n=(y, \dots, y_n)'$$, $$X_n=(x_1, \dots, x_n)$$, and $$\varepsilon_n= (\varepsilon_1, \dots, \varepsilon_n)'$$, the model equation becomes $y_n=X_n \theta+ \varepsilon_n, \quad n=1,2, \dots,$ and the usual estimators for $$\theta$$ and $$\sigma^2$$ are $\widehat\theta_n =(X_n'X_m)^{-1} X_ny_n\quad \text{and} \quad \widehat \sigma^2_n= \|y_n-X_n \widehat\theta_n \|^2/(n-p).$ It is the purpose of this paper to explain how approximate expressions for the sampling distributions of these estimators may be obtained. The case when a stopping time is applied is considered, too. The accuracy of the approximation is assessed by simulations. The presentation is largely informal; only the last short section contains outlines of some proofs.
For the entire collection see [Zbl 0927.00053].

##### MSC:
 62L05 Sequential statistical design 62F25 Parametric tolerance and confidence regions