Goodness of fit test for isotonic regression

Durot, Cécile; Tocquet, Anne-Sophie

Durot, Cécile ; Tocquet, Anne-Sophie

ESAIM: Probability and Statistics, Tome 5 (2001), pp. 119-140.

Résumé

We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis $H_{0}$ : “ $f = f_{0}$ ” against the composite alternative $H_{a}$ : “ $f \neq f_{0}$ ” 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 $f_{0}$ , since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under $H_{0}$ . We study the asymptotic power of the test under alternatives that converge to the null hypothesis.

MR Zbl

Classification : 62G08, 62G10, 62G20
Mots-clés : nonparametric regression, isotonic estimator, goodness of fit test, asymptotic power

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     author = {Durot, C\'ecile and Tocquet, Anne-Sophie},
     title = {Goodness of fit test for isotonic regression},
     journal = {ESAIM: Probability and Statistics},
     pages = {119--140},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     mrnumber = {1875667},
     zbl = {0990.62041},
     language = {en},
     url = {https://www.numdam.org/item/PS_2001__5__119_0/}
}

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Durot, Cécile; Tocquet, Anne-Sophie. Goodness of fit test for isotonic regression. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 119-140. https://www.numdam.org/item/PS_2001__5__119_0/

Bibliographie
Cité par

[1] R.E. Barlow, D.J. Bartholomew, J.M. Bremmer and H.D. Brunk, Statistical Inference under Order Restrictions. Wiley (1972). | Zbl

[2] D. Barry and J.A. Hartigan, An omnibus test for departures from constant mean. Ann. Statist. 18 (1990) 1340-1357. | MR | Zbl

[3] H.D. Brunk, On the estimation of parameters restricted by inequalities. Ann. Math. Statist. (1958) 437-454. | MR | Zbl

[4] C. Durot, Sharp asymptotics for isotonic regression. Probab. Theory Relat. Fields (to appear). | MR | Zbl

[5] R.L. Eubank and J.D. Hart, Testing goodness of fit in regression via order selection criteria. Ann. Statist. 20 (1992) 1412-1425. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[8] P. Groeneboom, Brownian motion with parabolic drift and airy functions. Probab. Theory Relat. Fields (1989) 79-109. | MR

[9] P. Hall, J.W. Kay and D.M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 (1990) 521-528. | MR

[10] W. Härdle and E. Mammen, Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 (1993) 1926-1947. | MR | Zbl

[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. | MR | Zbl

[12] L. Reboul, Estimation of a function under shape restrictions. Applications to reliability, Preprint. Université Paris XI, Orsay (1997). | MR | Zbl

[13] D. Revuz and M. Yor, Continuous martingales and Brownian Motion. Springer-Verlag (1991). | MR | Zbl

[14] J. Rice, Bandwidth choice for nonparametric regression. Ann. Statist. 4 (1984) 1215-1230. | MR | Zbl

[15] H.P. Rosenthal, On the subspace of $l_{p}$ , $p > 2$ , spanned by sequences of independent random variables. Israel J. Math. 8 (1970) 273-303. | MR | Zbl

[16] A.I. Sakhanenko, Estimates in the variance principle. Trudy. Inst. Mat. Sibirsk. Otdel (1972) 27-44. | MR | Zbl

[17] J.G. Staniswalis and T.A. Severini, Diagnostics for assessing regression models. J. Amer. Statist. Assoc. 86 (1991) 684-692. | MR | Zbl

[18] W. Stute, Nonparametric model checks for regression. Ann. Statist. 15 (1997) 613-641. | MR | Zbl

[19] A.S. Tocquet, Construction et étude de tests en régression1998).