The Annals of Statistics

On the degrees of freedom in shape-restricted regression

Mary Meyer and Michael Woodroofe

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For the problem of estimating a regression function, $\mu$ 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 $\sigma^2$ is asymptotically normal with mean $\sigma^2$ and variance $2\sigma_2/n$. For monotone regression, it is shown that the maximum likelihood estimator of $\mu$ attains the optimal rate of convergence, and a bias correction to the maximum likelihood estimator of $\sigma^2$ is derived.

Article information

Ann. Statist. Volume 28, Number 4 (2000), 1083-1104.

First available in Project Euclid: 12 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression

Asymptotic distribution bias reduction divergence effective dimension simulation Stein's identity variance estimation


Meyer, Mary; Woodroofe, Michael. On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 (2000), no. 4, 1083--1104. doi:10.1214/aos/1015956708.

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