On uniform consistency of nonparametric tests

M. S. Ermakov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 28, Tome 486 (2019), pp. 98-147

Résumé

We provide necessary and sufficient conditions of uniform consistency of nonparametric sets of alternatives for widespread nonparametric tests. Nonparametric sets of alternatives can be defined both in terms of distribution function and density (or signals in the problem of signal detection in Gaussian white noise).

Bibliographie

[1] T. Anderson, “The integral of a symmetric unimodal function”, Proc. Amer. Math. Soc., 6:1 (1955), 170–176 | DOI | MR | Zbl

[2] F. Autin, M. Clausel, J. Jean-Marc Freyermuth, C. Marteau, Maxiset point of view for signal detection in inverse problems, 2018, arXiv: 1803.05875 | MR

[3] A. Cohen, R. DeVore, G. Kerkyacharian, D. Picard, “Maximal spaces with given rate of convergence for thresholding algorithms”, Appl. Comput. Harmon. Anal., 11 (2001), 167–191 | DOI | MR | Zbl

[4] H. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996 | MR | Zbl

[5] M. S. Ermakov, “Minimaksnoe obnaruzhenie signala v gaussovskom belom shume”, Teoriya veroyatn. i ee primen., 35:4 (1990), 704–715 | MR

[6] M. S. Ermakov, “Asimptoticheskaya minimaksnost kriteriev khi-kvadrat”, Teoriya veroyatn. i ee primen., 42:4 (1997), 668–695 | DOI | MR | Zbl

[7] M. S. Ermakov, “On asymptotic minimaxity of kernel-based tests”, ESAIM Probab. Stat., 7 (2003), 279–312 | DOI | MR | Zbl

[8] M. S. Ermakov, “Minimaksnoe obnaruzhenie signala v vesovom gaussovskom belom shume”, Zap. nauchn. sem. POMI, 320, 2004, 54–68 | Zbl

[9] M. S. Ermakov, “O sostoyatelnoi proverke gipotez”, Zap. nauchn. sem. POMI, 442, 2015, 48–74

[10] M. S. Ermakov, “Ob asimptoticheski minimaksnom obnaruzhenii signala v gaussovskom belom shume”, Zap. nauchn. sem. POMI, 474, 2018, 124–138

[11] I. A. Ibragimov, R. Z. Khasminskii, “Ob otsenke beskonechnomernogo parametra v gaussovom belom shume”, Dokl. AN SSSR, 236:5 (1977), 1053–1055 | MR | Zbl

[12] Yu. I. Ingster, “O sravnenii minimaksnykh svoistv testov Kolmogorova, $\omega^2$ i $\chi^2$”, Teoriya veroyatn. i ee primen., 32:2 (1987), 374–378 | MR | Zbl

[13] Yu. I. Ingster, I. A. Suslina, Nonparametric Goodness-of-fit Testing under Gaussian Models, Lecture Notes in Statistics, 169, Springer, N.Y., 2002 | MR

[14] I. M. Johnstone, Gaussian estimation. Sequence and wavelet models, Book Draft, 2015 http://statweb.stanford.edu/ĩmj/

[15] G. Kerkyacharian, D. Picard, “Density estimation by kernel and wavelets methods: optimality of Besov spaces”, Statist. Probab. Lett., 18 (1993), 327–336 | DOI | MR | Zbl

[16] G. Kerkyacharian, D. Picard, Minimax or maxisets?, Bernoulli, 8 (2002), 219–253 | MR | Zbl

[17] L. Le Cam, L. Schwartz, “A necessary and sufficient conditions for the existence of consistent estimates”, Ann. Math. Statist., 31 (1960), 140–150 | DOI | MR | Zbl

[18] L. Le Cam, “Convergence of estimates under dimensionality restrictions”, Ann. Statist., 1 (1973), 38–53 | DOI | MR | Zbl

[19] E. L. Lehmann, J. P. Romano, Testing Statistical Hypothesis, Springer Verlag, NY, 2005 | MR

[20] V. Rivoirard, “Maxisets for linear procedures”, Statist. Probab. Lett., 67 (2004), 267–275 | DOI | MR | Zbl

[21] G. R. Shorack, J. A. Wellner, Empirical Processes with Application to Statistics, J. Wiley Sons, NY, 1986 | MR

[22] L. Schwartz, “On Bayes procedures”, Z. Wahrsch. Verw. Gebiete, 4 (1965), 10–26 | DOI | MR | Zbl

[23] A. Tsybakov, Introduction to Nonparametric Estimation, Springer, Berlin, 2009 | MR | Zbl

[24] P. L. Ulyanov, “O ryadakh Khaara”, Matem. sb., 63(105):2 (1964), 356–391 | Zbl

[25] D. M. Chibisov, “K issledovaniyu asimptoticheskoi moschnosti kriteriev soglasiya”, Teoriya veroyatn. i ee primen., 10:3 (1965), 460–478