Goodness of fit test for isotonic regression

Cécile Durot; Anne-Sophie Tocquet

ESAIM: Probability and Statistics (2010)

  • Volume: 5, page 119-140
  • ISSN: 1292-8100

Abstract

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We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H0: “ƒ = ƒ0” against the composite alternative Ha: “ƒ ≠ ƒ0” under the assumption that the true regression function f is decreasing. The test statistic is based on the 𝕃 1 -distance between the isotonic estimator of f and the function f0, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H0. We study the asymptotic power of the test under alternatives that converge to the null hypothesis.

How to cite

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Durot, Cécile, and Tocquet, Anne-Sophie. "Goodness of fit test for isotonic regression." ESAIM: Probability and Statistics 5 (2010): 119-140. <http://eudml.org/doc/197737>.

@article{Durot2010,
abstract = { We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H0: “ƒ = ƒ0” against the composite alternative Ha: “ƒ ≠ ƒ0” under the assumption that the true regression function f is decreasing. The test statistic is based on the $\{\mathbb L\}_\{1\}$-distance between the isotonic estimator of f and the function f0, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H0. We study the asymptotic power of the test under alternatives that converge to the null hypothesis. },
author = {Durot, Cécile, Tocquet, Anne-Sophie},
journal = {ESAIM: Probability and Statistics},
keywords = {Nonparametric regression; isotonic estimator; goodness of fit test; asymptotic power.; asymptotic power},
language = {eng},
month = {3},
pages = {119-140},
publisher = {EDP Sciences},
title = {Goodness of fit test for isotonic regression},
url = {http://eudml.org/doc/197737},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Durot, Cécile
AU - Tocquet, Anne-Sophie
TI - Goodness of fit test for isotonic regression
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 119
EP - 140
AB - We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H0: “ƒ = ƒ0” against the composite alternative Ha: “ƒ ≠ ƒ0” under the assumption that the true regression function f is decreasing. The test statistic is based on the ${\mathbb L}_{1}$-distance between the isotonic estimator of f and the function f0, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H0. We study the asymptotic power of the test under alternatives that converge to the null hypothesis.
LA - eng
KW - Nonparametric regression; isotonic estimator; goodness of fit test; asymptotic power.; asymptotic power
UR - http://eudml.org/doc/197737
ER -

References

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  1. R.E. Barlow, D.J. Bartholomew, J.M. Bremmer and H.D. Brunk, Statistical Inference under Order Restrictions. Wiley (1972).  
  2. D. Barry and J.A. Hartigan, An omnibus test for departures from constant mean. Ann. Statist.18 (1990) 1340-1357.  
  3. H.D. Brunk, On the estimation of parameters restricted by inequalities. Ann. Math. Statist. (1958) 437-454.  
  4. C. Durot, Sharp asymptotics for isotonic regression. Probab. Theory Relat. Fields (to appear).  
  5. R.L. Eubank and J.D. Hart, Testing goodness of fit in regression via order selection criteria. Ann. Statist.20 (1992) 1412-1425.  
  6. R.L. Eubank and C.H. Spiegelman, Testing the goodness of fit of a linear model via nonparametric regression techniques. J. Amer. Statist. Assoc.85 (1990) 387-392.  
  7. P. Groeneboom, Estimating a monotone density, edited by R.A. Olsen and L. Le Cam, in Proc. of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. 2. Wadsworth (1985) 539-554.  
  8. P. Groeneboom, Brownian motion with parabolic drift and airy functions. Probab. Theory Relat. Fields (1989) 79-109.  
  9. P. Hall, J.W. Kay and D.M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika77 (1990) 521-528.  
  10. W. Härdle and E. Mammen, Comparing nonparametric versus parametric regression fits. Ann. Statist.21 (1993) 1926-1947.  
  11. J.D. Hart and T.E. Wehrly, Kernel regression when the boundary region is large, with an application to testing the adequacy of polynomial models. J. Amer. Statist. Assoc.87 (1992) 1018-1024.  
  12. L. Reboul, Estimation of a function under shape restrictions. Applications to reliability, Preprint. Université Paris XI, Orsay (1997).  
  13. D. Revuz and M. Yor, Continuous martingales and Brownian Motion. Springer-Verlag (1991).  
  14. J. Rice, Bandwidth choice for nonparametric regression. Ann. Statist.4 (1984) 1215-1230.  
  15. H.P. Rosenthal, On the subspace of lp, p&gt;2, spanned by sequences of independent random variables. Israel J. Math.8 (1970) 273-303.  
  16. A.I. Sakhanenko, Estimates in the variance principle. Trudy. Inst. Mat. Sibirsk. Otdel (1972) 27-44.  
  17. J.G. Staniswalis and T.A. Severini, Diagnostics for assessing regression models. J. Amer. Statist. Assoc.86 (1991) 684-692.  
  18. W. Stute, Nonparametric model checks for regression. Ann. Statist.15 (1997) 613-641.  
  19. A.S. Tocquet, Construction et étude de tests en régression. 1. Correction du rapport de vraisemblance par approximation de Laplace en régression non-linéaire. 2. Test d'adéquation en régression isotonique à partir d'une asymptotique des fluctuations de la distance l1, Ph.D. Thesis. Université Paris Sud, Orsay (1998).  

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