On some tests-based projection pursuit for elliptical symmetry of a high-dimensional distribution. (English) Zbl 0981.62036
Summary: Some test statistics of Kolmogorov type and Cramér-von Mises type based on projection pursuit techniques are proposed for the testing problem of sphericity of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis and under any fixed alternative. The asymptotic properties of bootstrap approximation are investigated. Furthermore, for computational reasons, an approximation for the statistics, based on the number theoretic method, is suggested.
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62H15 | Hypothesis testing in multivariate analysis |
| 62G09 | Nonparametric statistical resampling methods |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
projection pursuit type statistic; elliptical symmetry; bootstrap approximation; number theoretic methodReferences:
| [1] | Butler, C. C., A test for symmetry using the sample distribution function, Ann. Math. Statist., 40, 2209-2209 (1969) · Zbl 0214.46002 · doi:10.1214/aoms/1177697302 |
| [2] | Rothman, E. D.; Woodrofe, M., A Cramér von Mises type statistics for testing symmetry, Ann. Math. Statist., 43, 2035-2035 (1972) · Zbl 0276.62045 · doi:10.1214/aoms/1177690879 |
| [3] | Baringhaus, L., Testing for spherical symmetry of a multivariate distribution, Ann. Statist., 19, 899-899 (1991) · Zbl 0725.62053 · doi:10.1214/aos/1176348127 |
| [4] | Huber, P., Projection pursuit (with discussion), Ann. Statist., 13, 435-435 (1985) · Zbl 0595.62059 · doi:10.1214/aos/1176349519 |
| [5] | Fang, K. T.; Zhu, L. X.; Bentler, P. M., A necessary test of goodness of fit for sphericity, J. of Multivariate Anal., 44, 34-34 (1993) · Zbl 0792.62043 · doi:10.1006/jmva.1993.1025 |
| [6] | Watson, G. S., Statistics in Sphere (1983), Beijing: John Wiley and Sons, Beijing |
| [7] | Beran, R.; Millar, P. W., Confidence sets for a multivariate distribution, Ann. Statist., 14, 431-431 (1986) · Zbl 0599.62057 · doi:10.1214/aos/1176349931 |
| [8] | Giné, E.; Zinn, J., Bootstrapping general empirical measures, Ann. Probab., 18, 851-851 (1990) · Zbl 0706.62017 · doi:10.1214/aop/1176990862 |
| [9] | Hua, L. G.; Wang, Y., Remarks concerning numerical integration, Sci. Record (N. S.), 4, 8-8 (1960) · Zbl 0090.34503 |
| [10] | Niederreiter, H., Existence of good lattice methods in the sense of Hlawka, Monatsh. Math., 86, 203-203 (1978) · Zbl 0395.10053 · doi:10.1007/BF01659720 |
| [11] | Fang, K. T.; Wang, Y., Number-theoretic Methods and Applications in Statistics (1993), London/New York: Chapman and Hall, London/New York |
| [12] | Hua, L. G.; Wang, Y., The Applications of Number Theory to Approximate Analysis (1981), Beijing: Science Press, Beijing |
| [13] | Zhang, J.; Zhu, L. X.; Cheng, P., Exponential bounds for the uniform deviation of a kind of empirical processes (II), J. of Multivariate Anal., 37, 250-250 (1993) · Zbl 0790.62046 · doi:10.1006/jmva.1993.1082 |
| [14] | Zhu, L. X.; Cheng, P., The optimal lower bounds of tail probability for the supreme of Gaussian processes indexed by half spaces, Sankya, A, 56, 265-265 (1994) · Zbl 0838.60015 |
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.
