\(L_ p\)-linear regression, consistency and significative regression in median (Q1073511)

In this paper we are concerned with \(L_ p\)-linear regression \((L_ p\)- L.R.). We prove that the \(L_ p\)-L.R. is a continuous function of p which, as \(p\downarrow 1\), converges to a regression in median. Several characterizations of this ''significative'' regression in median are presented. We also show the a.s. consistency of the empirical \(L_ p\)- L.R. to the theoretical \(L_ p\)-L.R., uniformly for p varying in compact sets. Some comments on the statistical role of \(L_ p\)-L.R. are provided.