Distribution Sampling of Vector Variance without Duplications

Authors: Erna T. Herdiani, Maman A. Djauhari

Abstract:

In recent years, the use of vector variance as a measure of multivariate variability has received much attention in wide range of statistics. This paper deals with a more economic measure of multivariate variability, defined as vector variance minus all duplication elements. For high dimensional data, this will increase the computational efficiency almost 50 % compared to the original vector variance. Its sampling distribution will be investigated to make its applications possible.

Keywords: Asymptotic distribution, covariance matrix, likelihood ratio test, vector variance.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057275

References:

[1] F.B. Alt, and N.D. Smith Multivariate process control. In: Krishnaiah, P.R., Rao, C.R., eds. Handbook of Statistics, Vo. 7 Elsevier Sciences Publishers, 1988, pp. 333-351.
[2] M.J. Anderson, Distance based test for homogeneity of multivariate dispersion, Biometrics, 62, 2006. 245 -253.
[3] T.W. Anderson, An Introduction to multivariate statistical analysis. John Wiley & Sons, Inc., New York. 1958.
[4] J. Ng. R Chilson., A. Wagner, and R. Zamar, Parallel Computation of high dimensional robust correlation and covariance matrices, algorithmica, 45(3), 2006 pp. 403 - 431.
[5] R. Cleroux, Multivariate Association and Inference Problems in Data Analysis, proceedings of the fifth international symposium on data analysis and informatics, vol. 1, Versailles, France. 1987.
[6] Jr, N. Da costa,. S. Nunes, P.Ceretta and S. Da Silva, Stock market comovements revisited, Economics Bulletin, 7(3), 2005, pp. 1-9.
[7] M. A. Djauhari, Improved Monitoring of Multivariate Variability, Journal of Quality Technology, 37(1), 2005, pp. 32-39.
[8] M. A. Djauhari, A Measure of Data Concentration, Journal of Probability and Statistics, 2(2), 2007, pp. 139-155.
[9] M. A. Djauhari, M. Mashuri, and D. E. Herwindiati, Multivariate Process Variability Monitoring, Communications in statistics - Theory and Methods, 37(1), 2008, pp. 1742 - 1754.
[10] H. El Maache, and Y. Lepage, Measures d-Association Vectorielle Basées sur une Matrice de Corrélation. Revue de Statistique Appliquée, 46(4), 1998, pp. 27-43.
[11] Y. Escoufier, Le traitement des variables vectorielles. Biometrics, 29, 1973, pp.751-760.
[12] A. K. Gupta, and J. Tang, Distribution of likelihood ratio statistic for testing equality of covariance matrices of multivariate Gaussian models, Biometrika, 71(3), 1984, pp. 555-559.
[13] D. E., Herwindiati, M. A., Djauhari, and M., Mashuri, Robust Multivariate Outlier Labeling, Communications in statistics - Computation and Simulation, 36(6), 2007, pp. 1287 - 1294.
[14] M., Hubert, P.J., Rouddeeuw, and S. van Aelst, Multivariate outlier detection and robustness, in handbook of statistics, vol. 24, Elsevier B.V., 2005, pp. 263-302.
[15] M. B. C., Khoo, and S. H. Quah, Multivariate control chart for process dispersion based on individual observations. Quality Engineering, 15(4), 2003, pp. 639-643.
[16] M. B. C., Khoo, and S. H. Quah, Alternatives to the multivariate control chart for process dispersion. Quality Engineering, 16(3), 2004, pp. 423- 435.
[17] O., Ledoit and M. Wolf, Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. The Annals of Statistics, 30(4), 2002, pp. 1081-1102.
[18] K.V., Mardia, J.M. Kent, Multivariate analysis, seventh printing, Academic press, London, 2000.
[19] D.C. Montgomery, Introduction to statistical quality control, fourth edition, John Willey & Sons, Inc., New York, 2001.
[20] R. J. Muirhead, Aspects of multivariate statistical theory, John Willey & Sons, Inc., New York, 1982.
[21] V. Ragea, Testing correlation stability during hectic financial markets, financial market and Portfolio Management, 17(3), 2003, pp. 289-308.
[22] P.J. Rosseeuw, Multivariate estimation with high breakdown point. In mathematical statistics and applications, B, Grossman W., Pflug G., Vincze I and Wertz, W., editors, D. Reidel Publishing Company, 1985, pp. 283-297.
[23] P.J., Rosseeuw, and A.M. Leroy, Robust regression and outlier detection, John Willey & Sons, Inc., New York, 1987.
[24] P.J., Rosseeuw, and M. Hubert, Regression Depth, Journal of the American Statistical Association, 94, 1999, pp.388-402.
[25] P.J., Rosseeuw, and K. van Driessen, A fast algorithm for the minimum covariance determinant estimatior, Technometrics, 41, 1999, pp. 212 - 223.
[26] J., Schafer, and K.,Strimmer, A Shrinkage Approach to large scale covariance matrix estimation and implications for functional genomics. Statistical Applications in genetics and molecular biology, 4, 2005, pp 1- 30.
[27] J. R., Schott, Matrix analysis for statistics, John Wiley & Sons, New York, 1997.
[28] J. R., Schott, Some tests for the equality of covariance matrices, journal of statistical planning and inference, 94, 2001, pp 25 - 36.
[29] G.A.F., Seber, Multivariate Observations, John Wiley & Sons, New York, 1984.
[30] R.J. Serfling, Approximation Theorems of mathematical statistics, John Wiley & Sons, New York, 1980.
[31] J.H. Sullivan, and W.H. Woodall, A Comparison of multivariate control charts for individual observations, Journal of Quality Technology, 28(4), 1996, pp 398-408.
[32] J.H., Sullivan, Z.G., Stoumbos, R.L. Mason, and J.C. Young, Step down analysis for changes in the covariance matrix and other parameters, Journal of Quality Technology, 39(1), 2007, pp 66-84.
[33] G.Y.N. Tang, The Intertemporal stability of the covariance and correlation matrices of Hongkong Stock Returns, Applied financial economics, 8, 1998, pp 359-365.
[34] M. Werner, Identification of multivariate outliers in large data sets, PhD dissertation, University of Colorado at Denver, 2003
[35] S.J., Wierda, Multivariate statistical process control, Recent results and directions for future research, statistica neerlandica, 48(2), 1994 pp. 147- 168.
[36] W.H. Woodall, and D.C. Montgomery, (1999). Research issues and ideas in statistical process control, Journal of Quality Technology, 31(4), 1999, pp 376-386.