zbMATH — the first resource for mathematics

Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression. (English) Zbl 1337.62090
Summary: We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional \(Q\), the null hypothesis states that the regression function \(f\) satisfies the constraint \(Q[f]=0\), while the alternative corresponds to the functions for which \(Q[f]\) is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: “regular” and “irregular”. In the “regular” case, the minimax rate is equal to \(n^{-1/4}\) while in the “irregular” case, the rate depends on the smoothness class and is slower than in the “regular” case. We apply this to the problem of testing the equality of Sobolev norms of two functions observed in noisy environments.

62G10 Nonparametric hypothesis testing
62C20 Minimax procedures in statistical decision theory
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI Euclid
[1] Baraud, Y., Huet, S., and Laurent, B. Adaptive tests of linear hypotheses by model selection., Ann. Statist. , 31(1):225-251, 2003. · Zbl 1018.62037
[2] Baraud, Y., Huet, S., and Laurent, B. Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function., Ann. Statist. , 33(1):214-257, 2005. · Zbl 1065.62109
[3] Butucea, C. Goodness-of-fit testing and quadratic functional estimation from indirect observations., Ann. Statist. , 35(5) :1907-1930, 2007. · Zbl 1126.62028
[4] Butucea, C., Matias, C., and Pouet, C. Adaptive goodness-of-fit testing from indirect observations., Ann. Inst. Henri Poincaré Probab. Stat. , 45 (2):352-372, 2009. · Zbl 1168.62040
[5] Cohen, A., Numerical analysis of wavelet methods , volume 32 of Studies in Mathematics and its Applications . North-Holland Publishing Co., Amsterdam, 2003. · Zbl 1038.65151
[6] Collier, O. Minimax hypothesis testing for curve registration., Electron. J. Statist. , 6 :1129-1154, 2012. · Zbl 1334.62077
[7] Comminges, L. and Dalalyan, A. S. Tight conditions for consistent variable selection in high dimensional nonparametric regression., Journal of Machine Learning Research - Proceedings Track , 19:187-206, 2011. · Zbl 1373.62154
[8] Comminges, L. and Dalalyan, A. S. Tight conditions for consistency of variable selection in the context of high dimensionality., Ann. Statist. , 40(6), doi:10.1214/12-AOS1046, 2012. · Zbl 1373.62154
[9] Dalalyan, A. S. and Collier, O. Wilks’ phenomenon and penalized likelihood-ratio test for nonparametric curve registration., J. Mach. Learn. Res. - Proceedings Track , 22:264-272, 2012.
[10] Dalalyan, A. S., Juditsky, A., and Spokoiny, V. A new algorithm for estimating the effective dimension-reduction subspace., J. Mach. Learn. Res. , 9 :1648-1678, 2008. · Zbl 1225.62091
[11] Donoho, D. and Nussbaum, M. Minimax quadratic estimation of a quadratic functional., J. Complexity , 6(3):290-323, 1990. · Zbl 0724.62039
[12] Efromovich, S. On the limit in the equivalence between heteroscedastic regression and filtering model., Statist. Probab. Lett. , 63(3):239-242, 2003. · Zbl 1116.62339
[13] Ermakov, M. S. Minimax detection of a signal in Gaussian white noise., Teor. Veroyatnost. i Primenen. , 35(4):704-715, 1990. · Zbl 0724.62082
[14] Ermakov, M. S. Minimax detection of a signal in weighted Gaussian white noise., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , 320(Veroyatn. i Stat. 8):54-68, 226, 2004. · Zbl 1071.60063
[15] Ermakov, M. S. Nonparametric signal detection with small type I and type II error probabilities., Stat. Inference Stoch. Process. , 14(1):1-19, 2011. · Zbl 1259.94028
[16] Fan, J. On the estimation of quadratic functionals., Ann. Statist. , 19(3) :1273-1294, 1991. · Zbl 0729.62076
[17] Gaïffas, S. and Lecué, G. Optimal rates and adaptation in the single-index model using aggregation., Electron. J. Stat. , 1:538-573, 2007. · Zbl 1320.62091
[18] Gayraud, G. and Pouet, C. Minimax testing composite null hypotheses in the discrete regression scheme., Math. Methods Statist. , 10(4):375-394 (2002), 2001. Meeting on Mathematical Statistics (Marseille, 2000). · Zbl 1005.62048
[19] Gayraud, G. and Pouet, C. Adaptive minimax testing in the discrete regression scheme., Probab. Theory Related Fields , 133(4):531-558, 2005. · Zbl 1075.62029
[20] Hall, P. Central limit theorem for integrated square error of multivariate nonparametric density estimators., J. Multivariate Anal. , 14(1):1-16, 1984. · Zbl 0528.62028
[21] Horowitz, J. L. and Spokoiny, V. G. An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative., Econometrica , 69(3):599-631, 2001. · Zbl 1017.62012
[22] Ingster, Yu. and Stepanova, N. Estimation and detection of functions from anisotropic Sobolev classes., Electron. J. Stat. , 5:484-506, 2011. · Zbl 1274.62319
[23] Ingster, Yu. I. Asymptotically minimax hypothesis testing for nonparametric alternatives. I., Math. Methods Statist. , 2(2):85-114, 1993a. · Zbl 0798.62057
[24] Ingster, Yu. I. Asymptotically minimax hypothesis testing for nonparametric alternatives. II., Math. Methods Statist. , 2(3):171-189, 1993b. · Zbl 0798.62058
[25] Ingster, Yu. I. Asymptotically minimax hypothesis testing for nonparametric alternatives. III., Math. Methods Statist. , 2(4):249-268, 1993c. · Zbl 0798.62059
[26] Ingster, Yu. I. and Sapatinas, T. Minimax goodness-of-fit testing in multivariate nonparametric regression., Math. Methods Statist. , 18(3):241-269, 2009. · Zbl 1282.62100
[27] Ingster, Yu. I., Sapatinas, T., and Suslina, I. A. Minimax Signal Detection in Ill-Posed Inverse Problems., Ann. Statist. , 40(2) :1524-1549, 2012. · Zbl 1297.62097
[28] Ingster, Yu. I. and Suslina, I. A., Nonparametric goodness-of-fit testing under Gaussian models , volume 169 of Lecture Notes in Statistics . Springer-Verlag, New York, 2003. · Zbl 1013.62049
[29] Kneser, H. Sur un théorème fondamental de la théorie des jeux., C. R. Acad. Sci. Paris , 234 :2418-2420, 1952. · Zbl 0046.12201
[30] Kolyada, V. I. On the embedding of Sobolev spaces., Mat. Zametki , 54(3):48-71, 158, 1993. · Zbl 0821.46043
[31] Laurent, B., Loubes, J. M., and Marteau, C. Testing inverse problems: a direct or an indirect problem?, Journal of Statistical Planning and Inference , 141 (5) :1849-1861, 2011. · Zbl 1394.62052
[32] Laurent, B., Loubes, J. M., and Marteau, C. Non asymptotic minimax rates of testing in signal detection with heterogeneous variances., Electronic Journal of Statistics , 6:91-122, 2012. · Zbl 1334.62085
[33] Lepski, O., Nemirovski, A., and Spokoiny, V. On estimation of the \(L_r\) norm of a regression function., Probab. Theory Related Fields , 113(2):221-253, 1999. · Zbl 0921.62103
[34] Pinsker, M. S. Optimal filtration of square-integrable signals in Gaussian noise., Problems Inform. Transmission , (16):52-68, 1980. · Zbl 0452.94003
[35] Pouet, C. An asymptotically optimal test for a parametric set of regression functions against a non-parametric alternative., J. Statist. Plann. Inference , 98(1-2):177-189, 2001. · Zbl 0977.62050
[36] Samarov, A., Spokoiny, V., and Vial, C. Component identification and estimation in nonlinear high-dimensional regression models by structural adaptation., J. Amer. Statist. Assoc. , 100(470):429-445, 2005. · Zbl 1117.62419
[37] Spokoiny, V. G. Adaptive hypothesis testing using wavelets., Ann. Statist. , 24(6) :2477-2498, 1996. · Zbl 0898.62056
[38] Tsybakov, A. B., Introduction to nonparametric estimation . Springer Verlag, 2009. · Zbl 1176.62032
[39] Vershynin, R. Introduction to the non-asymptotic analysis of random matrices., Compressed sensing , 210-268, Cambridge Univ. Press, Cambridge, 2012.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.