On convex least squares estimation when the truth is linear
| Autor: | Jon A. Wellner, Yining Chen |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
regression function estimation Boundary (topology) Asymptotic distribution Mathematics - Statistics Theory Context (language use) Statistics Theory (math.ST) 02 engineering and technology 01 natural sciences Least squares Article Convexity least squares 010104 statistics & probability Adaptive estimation 62G08 density estimation 62G07 FOS: Mathematics 0202 electrical engineering electronic engineering information engineering Applied mathematics QA Mathematics 0101 mathematics 62G20 Mathematics 62E20 Pointwise convexity 020206 networking & telecommunications shape constraint Density estimation 62E20 62G07 62G08 62G10 60G15 62G20 Rate of convergence 60G15 Statistics Probability and Uncertainty 62G10 |
| Zdroj: | Electron. J. Statist. 10, no. 1 (2016), 171-209 |
| ISSN: | 1935-7524 |
| DOI: | 10.1214/15-ejs1098 |
| Popis: | We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation. Comment: 35 pages, 5 figures |
| Databáze: | OpenAIRE |
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