 # Nonparametric measures of variance explained

Sheehy, A; Gasser, T; Rousson, V (2005). Nonparametric measures of variance explained. Journal of Nonparametric Statistics, 17(7):765-776.

## Abstract

The problem of estimating variance due to regression and due to error in the context of nonparametric regression is considered. An estimator is proposed on the basis of the difference of the mean total sum of squares of the data and a nonparametric estimate for the residual variance. Asymptotic expressions are derived for the expectation and variance of the estimator, and a number of simulations have been performed to assess its finite-sample behaviour. An adaptation of the estimator for residual variance to reduce bias in situations with a high signal-to-noise ratio is also proposed and evaluated. An application of the method to a data analysis problem concerning 'tracking' in the growth of children, the motivation behind this work, is then demonstrated. Possible extensions of the estimator to more complicated situations are also considered: nonparametric regression estimation in higher dimensions and estimating that part of variance in a regression function which is orthogonal to a linear parametric model.

## Abstract

The problem of estimating variance due to regression and due to error in the context of nonparametric regression is considered. An estimator is proposed on the basis of the difference of the mean total sum of squares of the data and a nonparametric estimate for the residual variance. Asymptotic expressions are derived for the expectation and variance of the estimator, and a number of simulations have been performed to assess its finite-sample behaviour. An adaptation of the estimator for residual variance to reduce bias in situations with a high signal-to-noise ratio is also proposed and evaluated. An application of the method to a data analysis problem concerning 'tracking' in the growth of children, the motivation behind this work, is then demonstrated. Possible extensions of the estimator to more complicated situations are also considered: nonparametric regression estimation in higher dimensions and estimating that part of variance in a regression function which is orthogonal to a linear parametric model.

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