Selection of variables in a multivariate inverse regression problem (Q1819509)

We assume that a set of p response variables \({\mathfrak y}=(y_ 1,...,y_ p)\) is linearly determined by a set of q explanatory variables \({\mathfrak x}=(x_ 1,...,x_ q)\), and that a calibration sample of N observations of \({\mathfrak x}\) and \({\mathfrak y}\) is available. A new observation \({\mathfrak y}\) is available at a single unknown \({\mathfrak x}\), and it is desired to estimate \({\mathfrak x}.\) Under this set-up we propose two methods for selection of the ''best'' subset of \({\mathfrak y}\). One is based on the asymptotic mean squared error of the classical estimate. The other uses Akaike's information criterion for selection of models. The two methods are applied to the wheat quality data analysed by \textit{P. J. Brown} [J. R. Stat. Soc., Ser. B 44, 287-321 (1982; Zbl 0511.62083)].