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A multivariate uniformity test for the case of unknown support. (English) Zbl 1322.62142

Summary: A test for the hypothesis of uniformity on a support \(S\subset\mathbb R^d\) is proposed. It is based on the use of multivariate spacings as those studied in [S. Janson, Ann. Probab. 15, 274–280 (1987; Zbl 0626.60017)]. As a novel aspect, this test can be adapted to the case that the support \(S\) is unknown, provided that it fulfils the shape condition of \(\lambda\)-convexity. The consistency properties of this test are analyzed and its performance is checked through a small simulation study. The numerical problems involved in the practical calculation of the maximal spacing (which is required to obtain the test statistic) are also discussed in some detail.

MSC:

62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis

Citations:

Zbl 0626.60017
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References:

[1] Aurenhammer, F.: Voronoi diagrams. A survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)
[2] Baddeley, A., Turner, R.: Spatstat: an R package for analyzing spatial point patterns. J. Stat. Softw. 6, 1–42 (2005)
[3] Barber, B.C., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469–483 (1996) · Zbl 0884.65145
[4] Baringhaus, L., Henze, N.: A test for uniformity with unknown limits based on D’Agostino’s D. Stat. Probab. Lett. 9, 299–304 (1990) · Zbl 0698.62045
[5] Berrendero, J.R., Cuevas, A., Vázquez-Grande, F.: Testing multivariate uniformity: The distance-to-boundary method. Can. J. Stat. 34, 693–707 (2006) · Zbl 1115.62046
[6] Cuevas, A., Febrero, M., Fraiman, R.: Estimating the number of clusters. Can. J. Stat. 28, 367–382 (2000) · Zbl 0981.62054
[7] Cuevas, A., Fraiman, R.F.: Set estimation. In: Kendall, W.S., Molchanov, I. (ed.) New Perspectives on Stochastic Geometry, pp. 374–397. Oxford University Press, London (2009) · Zbl 1192.62164
[8] Deheuvels, P.: Strong bounds for multidimensional spacings. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 411–424 (1983) · Zbl 0506.60006
[9] Delaunay, B.: Sur la sphere vide. Bull. Acad. Sci. USSR 7, 793–800 (1934) · JFM 60.0946.06
[10] Diggle, P.J.: Statistical Analysis of Spatial Point Patterns, 2nd edn. Edward Arnold, Sevenoaks (2003) · Zbl 1021.62076
[11] Edelsbrunner, H., Kirkpatrick, D.G., Seidel, R.: On the shape of a set of points in the plane. IEEE Trans. Inf. Theory 29, 551–559 (1983) · Zbl 0512.52001
[12] Friedman, J.H., Rafsky, L.C.: Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Ann. Stat. 7, 697–717 (1979) · Zbl 0423.62034
[13] Gerrard, D.: Competition quotient: a new measure of the competition affecting individual forest trees. Research Bulletin 20, Agricultural Experiment Station, Michigan State University (1969)
[14] Grasman, R., Gramacy, R.B.: Geometry: Mesh generation and surface tesselation. R package version 0.1-7. http://CRAN.R-project.org/package=geometry (2010)
[15] Jain, A., Xu, X., Ho, T., Xiao, F.: Uniformity testing using minimal spanning tree. In: Proceedings of the 16th International Conference on Pattern Recognition, vol. 4, pp. 281–284 (2002)
[16] Jammalamadaka, S.R., Goria, M.N.: A test of goodness-of-fit based on Gini’s index of spacings. Stat. Probab. Lett. 68, 177–187 (2004) · Zbl 1058.62039
[17] Janson, S.: Maximal spacings in several dimensions. Ann. Probab. 15, 274–280 (1987) · Zbl 0626.60017
[18] Liang, J.J., Fang, K.T., Hickernell, F.J., Li, R.: Testing multivariate uniformity and its applications. Math. Comput. 70, 337–355 (2001) · Zbl 0958.65016
[19] Marhuenda, Y., Morales, D., Pardo, M.C.: A comparison of uniformity tests. Statistics 39, 315–328 (2005) · Zbl 1084.62041
[20] Møller, J., Waagepetersen, R.P.: Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, London/Boca Raton (2004) · Zbl 1044.62101
[21] Pateiro-López, B.: Set estimation under convexity-type restrictions. Ph.D. Thesis, Universidad de Santiago de Compostela (2008) · Zbl 1416.62201
[22] Pateiro-López, B., Rodríguez-Casal, A.: Generalizing the convex hull of a sample: The R package alphahull. J. Stat. Softw. 5, 1–28 (2010)
[23] R Development Core Team.: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org (2008)
[24] Ripley, B.D.: Tests of ’randomness’ for spatial point patterns. J. R. Stat. Soc. Ser. A 39, 172–212 (1979) · Zbl 0427.62065
[25] Ripley, B.D., Rasson, J.P.: Finding the edge of a Poisson forest. J. Appl. Probab. 14, 483–491 (1977) · Zbl 0373.62058
[26] Rodríguez-Casal, A.: Set estimation under convexity-type assumptions. Ann. Inst. Henri Poincaré Probab. Stat. 43, 763–774 (2007) · Zbl 1169.62317
[27] Smith, S.P., Jain, A.K.: Testing for uniformity in multidimensional data. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 73–81 (1984)
[28] Tenreiro, C.: On the finite sample behavior of fixed bandwidth Bickel-Rosenblatt test for univariate and multivariate uniformity. Commun. Stat., Simul. Comput. 36, 827–846 (2007) · Zbl 1126.62036
[29] Tibshirani, R., Guenther, W., Hastie, T.: Estimating the number of clusters in a data set via the gap statistic. J. R. Stat. Soc. B 63, 411–423 (2001) · Zbl 0979.62046
[30] Voronoi, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J. Reine Angew. Math. 133, 97–178 (1907)
[31] Walther, G.: Granulometric smoothing. Ann. Stat. 25, 2273–2299 (1997) · Zbl 0919.62026
[32] Walther, G.: On a generalization of Blaschke’s Rolling Theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22, 301–316 (1999) · Zbl 0933.52003
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