Comptes Rendus
A nonparametric lack-of-fit test for heteroscedastic regression models
Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 215-217.

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residual plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing one to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.

Dans le cadre de la régression univariée, nous proposons un outil nonparamétrique général permettant de tester si une fonction connue m est un bon candidat pour la fonction de régression au vu des données. Ce test est basé sur la longueur maximale des suites ordonnées (par rapport à la covariable) des résidus de même signe. Aucune hypothèse n'est faite sur l'homoscédasticité des erreurs. De plus, ce test ne nécessite pas la présence de données répétées. Nous donnons ici la loi de la statistique test sous l'hypothèse nulle que la fonction considérée m est la vraie fonction de régression ainsi que sous une certaine classe d'hypothèses alternatives.

Published online:
DOI: 10.1016/j.crma.2010.12.009

Jean-Baptiste Aubin 1; Samuela Leoni-Aubin 1

1 INSA Lyon, ICJ, 20, rue Albert-Einstein, 69621 Villeurbanne cedex, France
     author = {Jean-Baptiste Aubin and Samuela Leoni-Aubin},
     title = {A nonparametric lack-of-fit test for heteroscedastic regression models},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {215--217},
     publisher = {Elsevier},
     volume = {349},
     number = {3-4},
     year = {2011},
     doi = {10.1016/j.crma.2010.12.009},
     language = {en},
AU  - Jean-Baptiste Aubin
AU  - Samuela Leoni-Aubin
TI  - A nonparametric lack-of-fit test for heteroscedastic regression models
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 215
EP  - 217
VL  - 349
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2010.12.009
LA  - en
ID  - CRMATH_2011__349_3-4_215_0
ER  - 
%0 Journal Article
%A Jean-Baptiste Aubin
%A Samuela Leoni-Aubin
%T A nonparametric lack-of-fit test for heteroscedastic regression models
%J Comptes Rendus. Mathématique
%D 2011
%P 215-217
%V 349
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2010.12.009
%G en
%F CRMATH_2011__349_3-4_215_0
Jean-Baptiste Aubin; Samuela Leoni-Aubin. A nonparametric lack-of-fit test for heteroscedastic regression models. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 215-217. doi : 10.1016/j.crma.2010.12.009.

[1] J.V. Bradley Distribution-Free Statistical Tests, Prentice–Hall, 1968

[2] P. Deheuvels On the Erdos–Renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings, Z. Wahrsch. Verw. Gebiete, Volume 70 (1985), pp. 91-115

[3] L. Gordon; M.F. Schilling; M.S. Waterman An extreme value theory for long head runs, Probab. Theory Related Fields, Volume 72 (1986), pp. 279-287

[4] J. Hart Nonparametric Smoothing and Lack-of-Fit Tests, Springer-Verlag, New York, 1997

[5] F. Kianifard; W.H. Swallow A review of the development and application of recursive residuals in linear models, J. Amer. Statist. Assoc., Volume 91 (1996) no. 433, pp. 391-400

[6] M. Muselli Useful inequalities for the longest run distribution, Statist. Probab. Lett., Volume 46 (2000), pp. 239-249

[7] J.W. Neill; D.E. Johnson Testing for lack of fit in regression—A review, Comm. Statist. Theory Methods, Volume 13 (1984) no. 4, pp. 485-511

[8] J. Riordan An Introduction to Combinatorial Analysis, John Wiley and Sons, 1958

[9] M.F. Schilling The longest run of heads, College Math. J., Volume 21 (1990), pp. 196-207

Cited by Sources:

Comments - Policy