Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances. (English) Zbl 1397.62201
Summary: We present a unified approach to goodness-of-fit testing in \(\mathbb{R}^d\) and on lower-dimensional manifolds embedded in \(\mathbb{R}^d\) based on sums of powers of weighted volumes of \(k\)th nearest neighbor spheres. We prove asymptotic normality of a class of test statistics under the null hypothesis and under fixed alternatives. Under such alternatives, scaled versions of the test statistics converge to the \(\alpha\)-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests. The tests are applied to two data sets of gamma-ray bursts in astronomy.
MSC:
| 62H15 | Hypothesis testing in multivariate analysis |
| 60F05 | Central limit and other weak theorems |
| 60D05 | Geometric probability and stochastic geometry |
Keywords:
multivariate goodness-of-fit test; nearest neighbors; \(\alpha\)-entropy; manifold; test for uniformity on a circle or a sphere; gamma-ray burst dataReferences:
| [1] | Baryshnikov, Yu.; Penrose, M.; Yukich, J. E., Gaussian limits for generalized spacings, Ann. Appl. Probab., 19, 158-185 (2009) · Zbl 1159.60315 |
| [2] | Berrendero, J. R.; Cuevas, A.; Pateiro-López, B., A multivariate uniformity test for the case of unknown support, Stat. Comput., 22, 259-271 (2012) · Zbl 1322.62142 |
| [3] | Berrendero, J. R.; Cuevas, A.; Pateiro-López, B., Testing uniformity for the case of a planar unknown support, Canad. J. Statist., 40, 378-395 (2012) · Zbl 1348.62157 |
| [4] | Berrendero, J. R.; Cuevas, A.; Vázquez-Grande, F., Testing multivariate uniformity: The distance-to-boundary method, Canad. J. Statist., 34, 693-707 (2006) · Zbl 1115.62046 |
| [5] | Bhattacharya, R.; Patrangenaru, V., Statistics on manifolds and landmarks based image analysis: A nonparametric theory with applications, J. Statist. Plann. Inference, 145, 1-22 (2014) · Zbl 1278.62068 |
| [6] | Biau, G.; Devroye, L.; Dujmovic, V.; Krzẏzak, A., An affine invariant \(k\)-nearest neighbor regression estimate, J. Multivariate Anal., 112, 24-34 (2012) · Zbl 1274.62275 |
| [7] | Bickel, P. J.; Breiman, L., Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test, Ann. Probab., 11, 185-214 (1983) · Zbl 0502.62045 |
| [8] | Coleman Miller, M., Gravitational waves: A golden binary, Nature, 551, 36-37 (2017) |
| [9] | Connaughton, V.; Briggs, M. S.; Goldstein, A.; Meegan, C. A.; Paciesas, W. S.; Preece, R. D.; Wilson-Hodge, C. A.; Gibby, M. H.; Greiner, J.; Gruber, D.; Jenke, P.; Kippen, R. M.; Pelassa, V.; Xiong, S.; Yu, H.-F.; Bhat, P. N.; Burgess, J. M.; Byrne, D.; Fitzpatrick, G.; Foley, S.; Giles, M. M.; Guiriec, S.; van der Horst, A. J.; von Kienlin, A.; McBreen, S.; McGlynn, S.; Tierney, D.; Zhang, B.-B., Localization of gamma-ray bursts using the Fermi gamma-ray burst monitor, Astrophys. J. Suppl. Ser., 216, 1-32 (2015) |
| [10] | Dette, H.; Henze, N., Some peculiar boundary phenomena for extremes of \(r\) th nearest neighbor links, Statist. Probab. Lett., 10, 381-390 (1990) · Zbl 0719.60022 |
| [11] | Devroye, L.; Wagner, T., The strong uniform consistency of nearest neighbor density estimates, Ann. Statist., 5, 536-540 (1977) · Zbl 0367.62061 |
| [12] | Diaconis, P.; Holmes, S.; Shahshahani, M., Sampling from a manifold, (Advances in Modern Statistical Theory and Applications. a Festschrift in Honor of Morris L. Eaton (2013), IMS: IMS Beachwood, OH), 102-125 · Zbl 1356.62015 |
| [13] | B. Ebner, N. Henze, J.E. Yukich, Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances, ArXiv e-prints, arXiv:1612.06601; B. Ebner, N. Henze, J.E. Yukich, Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances, ArXiv e-prints, arXiv:1612.06601 · Zbl 1397.62201 |
| [14] | Fisher, N. I.; Lewis, T.; Embleton, B. J.J., Statistical Analysis of Spherical Data (1987), Cambridge University Press · Zbl 0651.62045 |
| [15] | Gadat, S.; Klein, T.; Marteau, C., Classification in general finite-dimensional spaces with the \(k\)-nearest neighbor rule, Ann. Statist., 44, 982-1009 (2016) · Zbl 1338.62082 |
| [16] | Giné, E. M., Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms, Ann. Statist., 3, 1243-1266 (1975) · Zbl 0322.62058 |
| [17] | Henze, N., The limit distribution of maxima of “weighted” \(r\) th nearest neighbour distances, J. Appl. Probab., 19, 344-354 (1982) · Zbl 0484.62034 |
| [18] | Henze, N., A multivariate two-sample test based on the number of nearest neighbor type coincidences, Ann. Statist., 16, 772-783 (1988) · Zbl 0645.62062 |
| [19] | N. Henze, On the consistency of the spacings test for multivariate uniformity, arXiv:1708.09211; N. Henze, On the consistency of the spacings test for multivariate uniformity, arXiv:1708.09211 |
| [20] | Jammalamadaka, S. R.; SenGupta, A., Topics in Circular Statistics (2001), World Scientific Publishing · Zbl 1006.62050 |
| [21] | Jammalamadaka, S. R.; Zhou, S., Goodness of fit in multidimensions based on nearest neighbor distances, J. Nonparametr. Stat., 2, 271-284 (1993) · Zbl 1360.62230 |
| [22] | Janson, S., Maximal spacings in several dimensions, Ann. Probab., 15, 274-280 (1987) · Zbl 0626.60017 |
| [23] | Jupp, P. E., Modifications of the Rayleigh and Bingham tests for uniformity of directions, J. Multivariate Anal., 77, 1-20 (2001) · Zbl 1044.62058 |
| [24] | Jupp, P. E., Data-driven Sobolev tests of uniformity on compact Riemannian manifolds, Ann. Statist., 36, 1246-1260 (2008) · Zbl 1360.62218 |
| [25] | Kimball, B. F., Some basic theorems for developing tests of fit for the case of the non-parametric distribution function, I, Ann. Math. Statist., 18, 540-548 (1947) · Zbl 0029.15402 |
| [26] | Kuiper, N. H., Tests concerning random points on a circle, Nederl. Akad. Wetensch. Proc., 63, 38-47 (1960) · Zbl 0096.12504 |
| [27] | R. Lachièze-Rey, M. Schulte, J.E. Yukich, Normal approximation for sums of stabilizing functionals, arXiv:1702.00726; R. Lachièze-Rey, M. Schulte, J.E. Yukich, Normal approximation for sums of stabilizing functionals, arXiv:1702.00726 |
| [28] | L. Lenkic, P. Tzanavaris, S.C. Gallagher, T.D. Desjardins, L.M. Walker, K.E. Johnson, K. Fedotov, J. Charlton, A.E. Hornschemeier, P.R. Durrell, C. Gronwall, VizieR Online Data Catalog: Compact Group Galaxies UV and IR SFR (Lenkic+, 2016). VizieR Online Data Catalog, 745, 2017.; L. Lenkic, P. Tzanavaris, S.C. Gallagher, T.D. Desjardins, L.M. Walker, K.E. Johnson, K. Fedotov, J. Charlton, A.E. Hornschemeier, P.R. Durrell, C. Gronwall, VizieR Online Data Catalog: Compact Group Galaxies UV and IR SFR (Lenkic+, 2016). VizieR Online Data Catalog, 745, 2017. |
| [29] | Ley, C.; Verdebout, T., Modern Directional Statistics (2017), CRC Press: CRC Press London · Zbl 1448.62005 |
| [30] | Fermi Gamma-ray Burst Monitor, INTEGRAL, Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. Lett., 848, 1-27 (2017), L13 |
| [31] | Mardia, K. V.; Jupp, P. E., Directional Statistics (2000), Wiley: Wiley New York · Zbl 0707.62095 |
| [32] | Mészáros, A.; Bagoly, Z.; Balázs, L. G.; Horváth, I.; Vavrek, R., Probing the isotropy in the sky distribution of the gamma-ray bursts, (Costa, E.; Frontera, F.; Hjorth, J., Gamma-Ray Bursts in the Afterglow Era. ESO ASTROPHYSICS SYMPOSIA (European Southern Observatory) (2003), Springer: Springer New York) |
| [33] | Mondal, P. K.; Biswas, M.; Ghosh, A. K., On high dimensional two-sample tests based on nearest neighbors, J. Multivariate Anal., 141, 168-178 (2015) · Zbl 1323.62037 |
| [34] | Penrose, M. D.; Yukich, J. E., Central limit theorems for some graphs in computational geometry, Ann. Appl. Probab., 11, 1005-1041 (2001) · Zbl 1044.60016 |
| [35] | Penrose, M. D.; Yukich, J. E., Weak laws of large numbers in geometric probability, Ann. Appl. Probab., 13, 277-303 (2003) · Zbl 1029.60008 |
| [36] | Penrose, M. D.; Yukich, J. E., Limit theory for point processes in manifolds, Ann. Appl. Probab., 23, 2161-2211 (2013) · Zbl 1285.60021 |
| [37] | Petrie, A.; Willemain, Th. R., An empirical study of tests for uniformity in multidimensional data, Comput. Statist. Data Anal., 64, 253-268 (2013) · Zbl 1468.62159 |
| [38] | RR; RR |
| [39] | Schilling, M. F., Multivariate two-sample tests based on nearest neighbors, J. Amer. Statist. Assoc., 81, 799-806 (1986) · Zbl 0612.62081 |
| [40] | M. Tsagris, G. Athineou, A. Sajib, Directional: Directional Statistics, R; M. Tsagris, G. Athineou, A. Sajib, Directional: Directional Statistics, R |
| [41] | Vajda, I., On the amount of information contained in a sequence of independent observations, Kybernetika, 6, 306-323 (1970) · Zbl 0202.17802 |
| [42] | Wade, A. R., Explicit laws of large numbers for random nearest neighbor type graphs, Adv. Appl. Probab., 39, 326-342 (2007) · Zbl 1122.60012 |
| [43] | Watson, G. S., Goodness-of-fit tests on the circle, Biometrika, 48, 109-114 (1961) · Zbl 0212.21905 |
| [44] | Weiss, L., The asymptotic power of certain tests of fit based on sample spacings, Ann. Math. Statist., 28, 783-786 (1957) · Zbl 0087.14801 |
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