Robust regression designs when the design space consists of finitely many points (Q797941)

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.