×

On normal approximations for the two-sample problem on multidimensional tori. (English) Zbl 1393.60027

Summary: In this paper, quantitative central limit theorems for \(U\)-statistics on the \(q\)-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the \(U\)-statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The rates of convergence to Gaussianity for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by G. Peccati and M. S. Taqqu [Wiener chaos: Moments, cumulants and diagrams. A survey with computer implementation. Milano: Bocconi University Press; Milano: Springer (2011; Zbl 1231.60003)] and further developed by G. Peccati and C. Zheng [Electron. J. Probab. 15, Paper No. 48, 1487–1527 (2010; Zbl 1228.60031)] and the first author and G. Peccati [ibid. 19, Paper No. 66, 42 p. (2014; Zbl 1316.60089)]. Particular cases of the proposed framework allow to consider the two-sample problem on the circle as well as the local two-sample problem on \(\mathbb{R}^q\) through a local homeomorphism argument.

MSC:

60F05 Central limit and other weak theorems
62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing
60H07 Stochastic calculus of variations and the Malliavin calculus

Software:

circular; CircStats
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, N.; Hall, P.; Titterington, D., Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates, J. Multivariate Anal., 50, 41-54, (1994) · Zbl 0798.62055
[2] Azmoodeh, E.; Campese, S.; Poly, G., Fourth moment theorems for Markov diffusion generators, J. Funct. Anal., 266, 2341-2359, (2014) · Zbl 1292.60078
[3] Baldi, P., Adaptive density estimation for directional data using needlets, Ann. Statist., 37.6A, 3362-3395, (2009) · Zbl 1369.62061
[4] Baldi, P., Asymptotics for spherical needlets, Ann. Statist., 37, 3, 1150-1171, (2009) · Zbl 1160.62087
[5] Bourguin, S.; Peccati, G., A portmanteau inequality on the Poisson space, Electron. J. Probab., 19, 66, 1-42, (2014) · Zbl 1316.60089
[6] Bourguin, S., Gaussian approximations of nonlinear statistics on the sphere, J. Math. Anal. Appl., 436, 1121-1148, (2016) · Zbl 1383.60024
[7] Cammarota, V.; Marinucci, D., On the limiting behaviour of needlets polyspectra, Ann. Inst. H. Poincaré Probab. Statist., 51, 3, 1159-1189, (2015) · Zbl 1325.60014
[8] Cox, D. R., Simple approximate tests for Poisson variates, Biometrika, 40, 354-360, (1953) · Zbl 0051.10808
[9] DasGupta, A., Asymptotic Theory of Statistics and Probability, (2008), Springer · Zbl 1154.62001
[10] Dehling, Herold; Fried, Roland, Asymptotic distribution of two-sample empirical \(U\)-quantiles with applications to robust tests for shifts in location, J. Multivariate Anal., 105, 124-140, (2012) · Zbl 1250.62021
[11] Deshpande, J. V.; Mukhopadhyay, M.; Naik-Nimbalkar, U. V., Testing of two sample proportional intensity assumption for non-homogeneous Poisson processes, J. Statist. Plann. Inference, 81, 237-251, (1999) · Zbl 1057.62529
[12] Durastanti, C., Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields, Stat. Methods Appl., 25, 4, 651-673, (2016) · Zbl 1359.60037
[13] Durastanti, C., Adaptive density estimation on the circle by nearly tight frame, (Pesenson, I.; Le Gia, Q.; Mayeli, A.; Mhaskar, H.; Zhou, D. X., Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis, (2017), Birkhäuser, Cham) · Zbl 1408.62061
[14] Durastanti, C.; Lan, X.; Marinucci, D., Needlet-Whittle estimates on the unit sphere, Electron. J. Stat., 7, 597-646, (2013) · Zbl 1337.62287
[15] Durastanti, C.; Marinucci, D.; Peccati, G., Normal approximations for wavelet coefficients on spherical Poisson fields, J. Math. Anal. Appl., 409, 212-227, (2014) · Zbl 1306.42056
[16] Durastanti, C., A simple proposal for radial 3D needlets, Phys. Rev. D, 103532, (2014)
[17] Eplett, W. J.R., The small sample distribution of a Mann-Whitney type statistic for circular data, Ann. Statist., 7, 2, 446-453, (1979) · Zbl 0414.62036
[18] Fromont, M.; Laurent, B.; Reynaud-Bouret, P., The two-sample problem for Poisson processes: adaptive tests with a nonasymptotic wild bootstrap approach, Ann. Statist., 41, 3, 1431-1461, (2013) · Zbl 1273.62102
[19] Geller, D.; Marinucci, D., Spin wavelets on the sphere, J. Fourier Anal. Appl., 16, 6, 840-884, (2010) · Zbl 1206.42039
[20] Grafakos, L., Classical Fourier Analysis, (2008), Springer · Zbl 1220.42001
[21] Gretton, A., A kernel method for the two-sample problem, J. Mach. Learn. Res., 1, 1-10, (2008)
[22] Hall, P.; Tajvidi, N., Permutation teests for equality of distributions in high-dimensional settings, Biometrika, 89, 2, 359-374, (2002) · Zbl 1017.62040
[23] Iuppa, R., Cosmic-ray anisotropies observed by the ARGO-YBJ experiment, Nucl. Instrum. Methods Phys. Res. A, 160-164, (2012)
[24] Ledoux, M., Chaos of a Markov operator and the fourth moment condition, Ann. Probab., 40, 6, 2439-2459, (2012) · Zbl 1266.60042
[25] Lee, A. J., (U-statistics: Theory and Practice, Mathematics and its Applications, vol. 465, (1990), Marcel Dekker, Inc) · Zbl 0771.62001
[26] Mardia, K. W.; Jupp, P. E., Directional Statistics, (2009), John Wiley and Sons
[27] Marinucci, D.; Peccati, G., Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications, (2011), Cambridge University Press · Zbl 1260.60004
[28] Narcowich, F. J.; Petrushev, P.; Ward, J. D., Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, J. Funct. Anal., 238, 2, 530-564, (2006) · Zbl 1114.46026
[29] Narcowich, F. J.; Petrushev, P.; Ward, J. D., Localized tight frames on spheres, SIAM J. Math. Anal., 38, 574-594, (2006) · Zbl 1143.42034
[30] Nourdin, I.; Peccati, G., Stein’s method on Wiener chaos, Probab. Theory Related Fields, 145, 1-2, 75-118, (2009) · Zbl 1175.60053
[31] Nourdin, I.; Peccati, G., Normal Approximations Using Malliavin Calculus: From Stein’s Method to Universality, (2012), Cambridge University Press · Zbl 1266.60001
[32] Peccati, G.; Taqqu, M. S., Wiener Chaos: Moments, Cumulants and Diagrams, (2011), Springer-Verlag · Zbl 1231.60003
[33] Peccati, G.; Zheng, C., Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab., 15, 48, 1487-1527, (2010) · Zbl 1228.60031
[34] Peccati, G., Stein’s method and normal approximation of Poisson functionals, Ann. Probab., 38, 2, 443-478, (2010) · Zbl 1195.60037
[35] Przyborowski, J.; Wilenski, H., Homogeneity of results in testing samples from Poisson series with an application to testing clover seed for dodder, Biometrika, 31, 313-323, (1940) · JFM 66.0640.02
[36] Rao Jammalamadaka, S.; SenGupta, A., Topics in Circular Statistics, (2001), World Scientific · Zbl 1006.62050
[37] Scodeller, S., Introducing Mexican needlets for CMB analysis: issues for practical applications and comparison with standard needlets, Appl. J., 733, 121, (2011)
[38] van der Vaart, A. W., Asymptotic Statistics, (1998), Cambridge · Zbl 0910.62001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.