Validation of linear regression models

Abstract. The assumption that a parametric class of functions fits the data structure sufficiently well is common in fitting curves and surfaces to regression data. One then derives a parameter estimate resulting from a least squares fit, say, and in a second step various kinds of χ 2 goodness of fit measures, to assess whether the deviation between data and estimated surface is due to random noise and not to systematic departures from the model. In this paper we show that commonly-used χ 2 -measures are invalid in regression models, particularly when inhomogeneous noise is present. Instead we present a bootstrap algorithm which is applicable in problems described by noisy versions of Fredholm integral equations of the first kind. We apply the suggested method to the problem of recovering the luminosity density in the Milky Way from data of the DIRBE experiment on board the COBE satellite.

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“…to converge (here it is assumed that the scheme of grid points can be described asymptotically by a distribution H (Dette & Munk 1998)) gives…”

Section: Common χ 2 Methods Of Judging the Quality Of Fitmentioning

confidence: 99%

New statistical goodness of fit techniques in noisy inhomogeneous inverse problems

Bissantz

Munk

2001

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“…For further details on such testing problems in a regression context cf. Dette and Munk [6]. Observe that the variance of T BR n under fixed alternatives is of order n −1 .…”

Section: Fixed Alternativesmentioning

confidence: 96%

Density testing in a contaminated sample

Holzmann

Bissantz

Munk

2007

Journal of Multivariate Analysis

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We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z 1 , Z 2 , . . . which are observed under additional noise with density . The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Z i . In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g = f * instead on the initial density of interest f.

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“…proposed by Hodges and Lehmann [16] for testing whether multinomial cell probabilities are approximately equal. Dette and Munk [9] have extended this approach for the purpose of validating a model in a nonparametric regression framework. For methodological aspects and a more recent discussion we refer to Goutis and Robert [14], Dette and Munk [10], and Liu and Lindsay [21].…”

Section: Introductionmentioning

confidence: 99%

“…See also Berger and Delampady [1] who employ the term "precise hypothesis'' instead of "neighborhood hypothesis'', whereas Liu and Lindsay [21] coined the phrase "tubular models''. Mostly related to the present approach is the work of Dette and Munk [9] and Munk and Dette [25] who consider L 2 -neighborhood hypotheses in nonparametric regression models.…”

Section: Introductionmentioning

confidence: 99%

The one- and multi-sample problem for functional data with application to projective shape analysis

Munk

Paige

Pang

et al. 2008

Journal of Multivariate Analysis

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In this paper tests are derived for testing neighborhood hypotheses for the one-and multi-sample problem for functional data. Our methodology is used to generalize testing in projective shape analysis, which has traditionally involving data consisting of finite number of points, to the functional case. The one-sample test is applied to the problem of scene identification, in the context of the projective shape of a planar curve.

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Validation of linear regression models

Cited by 63 publications

References 35 publications

New statistical goodness of fit techniques in noisy inhomogeneous inverse problems

New statistical goodness of fit techniques in noisy inhomogeneous inverse problems

Density testing in a contaminated sample

The one- and multi-sample problem for functional data with application to projective shape analysis

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