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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].

62L05 Sequential statistical design
62F25 Parametric tolerance and confidence regions