A property of projection residuals with applications to concave regression. (English) Zbl 0784.62060
Summary: The residuals of the \(\ell_ 2\)-projection onto a linear subspace, or more generally onto a closed, convex cone, which contains the constant vectors sum to zero, i.e. if \(y^*\) is the projection of \(y\), then \(y_ 1-y^*_ 1+y_ 2-y_ 2^*+\cdots+y_ n-y_ n^*=0\). The analogue of this result for least absolute deviations, as well as for other objective functions, is established. The application of these results in establishing the consistency of the \(\ell_ 1\) and \(\ell_ 2\) estimates of concave regression functions is discussed.
MSC:
62J05 | Linear regression; mixed models |
Keywords:
characterizing projections; concave regression; isotonic regression; \(\ell_ p\)-estimation; residuals; closed convex cone; least absolute deviations; consistency; concave regression functionsReferences:
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