Some high-dimensional tests for a one-way MANOVA. (English) Zbl 1130.62058
Summary: A statistic is proposed for testing the equality of the mean vectors in a one-way multivariate analysis of variance. The asymptotic null distribution of this statistic, as both the sample size and the number of variables go to infinity, is shown to be normal. Thus, this test can be used when the number of variables is not small relative to the sample size. In particular, it can be used when the number of variables exceeds the degrees of freedom for error, a situation in which standard MANOVA tests are invalid. A related statistic, also having an asymptotic normal distribution, is developed for tests concerning the dimensionality of the hyperplane formed by the population mean vectors. The finite sample size performances of the normal approximations are evaluated in a simulation study.
MSC:
| 62H15 | Hypothesis testing in multivariate analysis |
| 62E20 | Asymptotic distribution theory in statistics |
| 62J10 | Analysis of variance and covariance (ANOVA) |
References:
| [1] | Anderson, T. W.; Das Gupta, S., Some inequalities on characteristic roots of matrices, Biometrika, 50, 522-524 (1963) · Zbl 0133.41602 |
| [2] | Bai, Z. D.; Saranadasa, H., Effect of high dimension: by an example of a two sample problem, Statist. Sinica, 6, 311-329 (1996) · Zbl 0848.62030 |
| [3] | Bai, Z. D.; Yin, Y. Q., Limit of the smallest eigenvalue of a large dimensional sample covariance-matrix, Ann. Probab., 21, 1275-1294 (1993) · Zbl 0779.60026 |
| [4] | Bartlett, M. S., Further aspects of the theory of multiple regression, Proc. Cambridge Philos. Soc., 34, 33-40 (1938) · Zbl 0018.15803 |
| [5] | Chung, J. H.; Fraser, D. A.S., Randomization tests for a multivariate two-sample problem, J. Amer. Statist. Assoc., 53, 729-735 (1958) · Zbl 0087.14701 |
| [6] | Davis, A. W., On the differential equation for Meijer’s \(G_{p, p}^{p, 0}\) function, and further tables of Wilk’s likelihood ratio criterion, Biometrika, 66, 519-531 (1979) · Zbl 0417.62010 |
| [7] | Dempster, A. P., A high dimensional two sample significance test, Ann. Math. Statist., 29, 995-1010 (1958) · Zbl 0226.62014 |
| [8] | Dempster, A. P., A significance test for the separation of two highly multivariate small samples, Biometrics, 16, 41-50 (1960) · Zbl 0218.62065 |
| [9] | Fujikoshi, Y., Multivariate analysis for the case when the dimension is large compared to the sample size, J. Korean Statist. Soc., 33, 1-24 (2004) |
| [10] | Fujikoshi, Y.; Himeno, T.; Wakaki, H., Asymptotic results of a high dimensional MANOVA test and power comparisons when the dimension is large compared to the sample size, J. Japan Statist. Soc., 34, 19-26 (2004) · Zbl 1061.62085 |
| [11] | Hall, P.; Heyde, C. C., Martingale Limit Theory and its Applications (1980), Academic Press: Academic Press New York · Zbl 0462.60045 |
| [12] | Lee, Y. S., Some results on the distribution of Wilk’s likelihood ratio criterion, Biometrika, 59, 649-664 (1972) · Zbl 0295.62052 |
| [13] | Mathai, A. M., On the distribution of the likelihood ratio criterion for testing linear hypotheses on regression coefficients, Ann. Inst. Statist. Math., 23, 181-197 (1971) · Zbl 0256.62062 |
| [14] | Pillai, K. C.S.; Gupta, A. K., On the exact distribution of Wilk’s criterion, Biometrika, 56, 109-118 (1969) · Zbl 0181.46202 |
| [15] | Schatzoff, M., Exact distributions of Wilk’s likelihood ratio criterion, Biometrika, 53, 347-358 (1966) · Zbl 0139.37203 |
| [16] | Schott, J. R., Matrix Analysis for Statistics (2005), Wiley: Wiley New York · Zbl 1076.15002 |
| [17] | Schott, J. R., A high dimensional test for the equality of the smallest eigenvalues of a covariance matrix, J. Multivariate Anal., 97, 827-843 (2006) · Zbl 1086.62072 |
| [18] | M.S. Srivastava, Multivariate theory for analyzing high dimensional data, J. Japan Statist. Soc. (2007), to appear.; M.S. Srivastava, Multivariate theory for analyzing high dimensional data, J. Japan Statist. Soc. (2007), to appear. · Zbl 1140.62047 |
| [19] | Srivastava, M. S.; Fujikoshi, Y., Multivariate analysis of variance with fewer observations than the dimension, J. Multivariate Anal., 97, 1927-1940 (2006) · Zbl 1101.62051 |
| [20] | Tonda, T.; Fujikoshi, Y., Asymptotic expansion of the null distribution of LR statistic for multivariate linear hypothesis when the dimension is large, Comm. Statist. Theory Methods, 33, 1205-1220 (2004) · Zbl 1114.62315 |
| [21] | Yin, Y. Q.; Bai, Z. D.; Krishnaiah, P. R., On the limit of the largest eigenvalue of the large dimensional sample covariance matrix, Probab. Theory Related Fields, 78, 509-521 (1988) · Zbl 0627.62022 |
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.
