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On the degrees of freedom in shape-restricted regression. (English) Zbl 1105.62340
Summary: For the problem of estimating a regression function, μ say, subject to shape constraints, like monotonicity or convexity, it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model. Inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares. These generalize equalities from the case of linear regression. As an application, it is shown that the maximum likelihood estimator of the error variance σ 2 is asymptotically normal with mean σ 2 and variance 2σ 2 /n. For monotone regression, it is shown that the maximum likelihood estimator of μ attains the optimal rate of convergence, and a bias correction to the maximum likelihood estimator of σ 2 is derived.
MSC:
62G08Nonparametric regression
62F12Asymptotic properties of parametric estimators