Robust regression designs when the design space consists of finitely many points. (English) Zbl 0546.62048

Li, Ker-Chau

Ann. Stat. 12, 269-282 (1984).

The author considers the model \(Y_ i=\alpha +\beta x_ i+g(x_ i)+\epsilon_ i\), \(i=1,2,...,n\), where \(\{\epsilon_ i\}\) are uncorrelated random variables with mean 0 and variance \(\sigma^ 2\), \(x_ i\in X=\{k/2N,-k/2N|\quad k=1,2,...,N\}\) for some fixed positive integer \(N\geq 3\) and \[ g\in G=\{h|\quad | h(x)|\leq \delta,\quad\min_{\alpha,\beta}\int (h(x)-\alpha -\beta x)^ 2d\mu (x)=\int h^ 2(x)d\mu (x)\}, \] with \(\delta \geq 0\) and \(\mu\) the uniform probability measure on X. The loss function \(\omega_ 1(\alpha - {\hat\alpha })^ 2+\omega_ 2(\beta -{\hat\beta })^ 2\) is assumed where \(\omega_ 1,\omega_ 2\) are specified nonnegative constants and \({\hat\alpha }\),\({\hat\beta }\) are the least squares estimators of \(\alpha\) and \(\beta\), respectively. Designs symmetric about zero are obtained which minimize the expected loss.

Reviewer: H.Iyer

MSC:

62K05	Optimal statistical designs
62J05	Linear regression; mixed models

Keywords:

squared loss function; maximum risk among symmetric designs; robust design; nearly linear regression; designs with finite support; least squares estimators