## The Annals of Statistics

### On the degrees of freedom in shape-restricted regression

#### Abstract

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

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

Dates
First available in Project Euclid: 12 March 2002

https://projecteuclid.org/euclid.aos/1015956708

Digital Object Identifier
doi:10.1214/aos/1015956708

Mathematical Reviews number (MathSciNet)
MR1810920

Zentralblatt MATH identifier
1105.62340

Subjects
Primary: 62G08: Nonparametric regression

#### Citation

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. https://projecteuclid.org/euclid.aos/1015956708

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