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:
| [1] | Barlow, R. E.; Brunk, H. D., The isotonic regression problem and its dual, J. Amer. Statist. Assoc., 67, 140-147 (1972) · Zbl 0236.62050 |
| [2] | Bartels, R. H.; Conn, A. R., Linearly constrained discrete \(l_1\) problems, ACM Trans. Math. Software, 6, 594-614 (1980) · Zbl 0448.49017 |
| [3] | Brunk, H. D., Conditional expectation given a σ-lattice and applications, Ann. Math. Statist., 36, 1339-1350 (1965) · Zbl 0144.39503 |
| [4] | Brunk, H. D., Uniform inequalities for conditional p-means given σ-lattices, Ann. Probab., 3, 1025-1030 (1975) · Zbl 0323.60052 |
| [5] | Hanson, D. L.; Pledger, G., Consistency in concave regression, Ann. Statist., 4, 1038-1050 (1976) · Zbl 0341.62034 |
| [6] | Hildreth, C., Point estimates of ordinates of concave functions, J. Amer. Statist. Assoc., 49, 598-619 (1954) · Zbl 0056.38301 |
| [7] | Landers, D.; Rogge, L., Isotonic approximation in \(L_s\), J. Approx. Theory, 31, 199-223 (1981) · Zbl 0467.41014 |
| [8] | Robertson, T.; Wright, F. T., Statistical inferences for ordered parameters: A personal view of isotonic regression since the work by Barlow, Bartholomew, Bremner and Brunk, Proc. Symp. Appl. Math., 23, 55-71 (1980) · Zbl 0443.62030 |
| [9] | Robertson, T.; Wright, F. T.; Dykstra, R. L., Order Restricted Statistical Inference (1988), Wiley: Wiley New York · Zbl 0645.62028 |
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