A Normality Test for High-dimensional Data Based on the Nearest Neighbor Approach

References

• Anderson, T. W. (2003), An Introduction To Multivariate Statistical Analysis (3rd ed.), New York: Wiley-Intersceince.
   Google Scholar
• Baringhaus, L., and Henze, N. (1988), “A Consistent Test for Multivariate Normality Based on the Empirical Characteristic Function,” Metrika, 35, 339–348. DOI: 10.1007/BF02613322.
   Google Scholar
• Bartoszyński, R., Pearl, D. K., and Lawrence, J. (1997), “A Multidimensional Goodness-of-Fit Test Based on Interpoint Distances,” Journal of the American Statistical Association, 92, 577–586. DOI: 10.1080/01621459.1997.10474010.
   Web of Science ®Google Scholar
• Berk, R., Brown, L., Buja, A., Zhang, K., Zhao, L. (2013), “Valid Post-Selection Inference,” The Annals of Statistics, 41, 802–837. DOI: 10.1214/12-AOS1077.
   Web of Science ®Google Scholar
• Bickel, P. J., and Levina, E. (2004), “Some Theory for Fisher’s Linear Discriminant Function, Naive Bayes’, and Some Alternatives When There are Many More Variables Than Observations,” Bernoulli, 10, 989–1010. DOI: 10.3150/bj/1106314847.
   Web of Science ®Google Scholar
• Bickel, P. J., and Levina, E. (2008), “Regularized Estimation of Large Covariance Matrices,” The Annals of Statistics, 36, 199–227. DOI: 10.1214/009053607000000758.
   Web of Science ®Google Scholar
• Cai, T., and Liu, W. (2011), “A Direct Estimation Approach to Sparse Linear Discriminant Analysis,” Journal of the American Statistical Association, 106, 1566–1577. DOI: 10.1198/jasa.2011.tm11199.
   Web of Science ®Google Scholar
• Cai, T. T., Liu, W., and Xia, Y. (2014), “Two-Sample Test of High Dimensional Means Under Dependency,” Journal of Royal Statistical Society, Series B, 76, 349–372. DOI: 10.1111/rssb.12034.
   Google Scholar
• Chan, H. P., and Walther, G. (2015), “Optimal Detection of Multi-Sample Aligned Sparse Signals,” The Annals of Statistics, 43, 1865–1895. DOI: 10.1214/15-AOS1328.
   Web of Science ®Google Scholar
• Chen, H. (2019), “Sequential Change-Point Detection Based on Nearest Neighbors,” The Annals of Statistics, 47, 1381–1407. DOI: 10.1214/18-AOS1718.
   Web of Science ®Google Scholar
• Chen, H., Chen, X., and Su, Y. (2018), “A Weighted Edge-Count Two-Sample Test for Multivariate and Object Data,” Journal of the American Statistical Association, 113, 1146–1155. DOI: 10.1080/01621459.2017.1307757.
   Web of Science ®Google Scholar
• Chen, H., and Friedman, J. H. (2017), “A New Graph-Based Two-Sample Test for Multivariate and Object Data,” Journal of the American statistical Association, 112, 397–409. DOI: 10.1080/01621459.2016.1147356.
   Web of Science ®Google Scholar
• Chen, H., and Zhang, N. (2015), “Graph-Based Change-Point Detection,” The Annals of Statistics, 43, 139–176. DOI: 10.1214/14-AOS1269.
   Web of Science ®Google Scholar
• Dettling, M. (2004), “Bagboosting for Tumor Classification With Gene Expression Data,” Bioinformatics, 20, 3583–3593. DOI: 10.1093/bioinformatics/bth447.
   PubMed Web of Science ®Google Scholar
• Doornik, J. A., and Hansen, H. (2008), “An Omnibus Test for Univariate and Multivariate Normality,” Oxford Bulletin of Economics and Statistics, 70, 927–939. DOI: 10.1111/j.1468-0084.2008.00537.x.
   Web of Science ®Google Scholar
• Fan, J., and Fan, Y. (2008), “High Dimensional Classification Using Features Annealed Independence Rules,” Annals of Statistics, 36, 2605.
   PubMed Web of Science ®Google Scholar
• Fan, J., Feng, Y., and Wu, Y. (2009), “Network Exploration Via the Adaptive Lasso and Scad Penalties,” The Annals of Applied Statistics, 3, 521. DOI: 10.1214/08-AOAS215.
   PubMed Web of Science ®Google Scholar
• Fattorini, L. (1986), “Remarks on the Use of Shapiro–Wilk Statistic for Testing Multivariate Normality,” Statistica, 46, 209–217.
   Google Scholar
• Friedman, J., Hastie, T., and Tibshirani, R. (2008), “Sparse Inverse Covariance Estimation With the Graphical Lasso,” Biostatistics, 9, 432–441. DOI: 10.1093/biostatistics/kxm045.
   PubMed Web of Science ®Google Scholar
• Friedman, J. H., and Rafsky, L. C. (1979), “Multivariate Generalizations of the Wald-Wolfowitz and Smirnov Two-Sample Tests,” The Annals of Statistics, 697–717. DOI: 10.1214/aos/1176344722.
   Web of Science ®Google Scholar
• Gordon, G. J., Jensen, R. V., Hsiao, L.-L., Gullans, S. R., Blumenstock, J. E., Ramaswamy, S., Richards, W. G., Sugarbaker, D. J., and Bueno, R. (2002), “Translation of Microarray Data Into Clinically Relevant Cancer Diagnostic Tests Using Gene Expression Ratios in Lung Cancer and Mesothelioma,” Cancer Research, 62, 4963–4967.
   PubMed Web of Science ®Google Scholar
• Henze, N. (1988), “A Multivariate Two-Sample Test Based on the Number of Nearest Neighbor Type Coincidences,” The Annals of Statistics, 16, 772–783. DOI: 10.1214/aos/1176350835.
   Web of Science ®Google Scholar
• Henze, N., and Wagner, T. (1997), “A New Approach to the Bhep Tests for Multivariate Normality,” Journal of Multivariate Analysis, 62, 1–23. DOI: 10.1006/jmva.1997.1684.
   Web of Science ®Google Scholar
• Henze, N., and Zirkler, B. (1990), “A Class of Invariant Consistent Tests for Multivariate Normality,” Communications in Statistics-Theory and Methods, 19, 3595–3617. DOI: 10.1080/03610929008830400.
   Web of Science ®Google Scholar
• Jin, J., and Wang, W. (2016), “Influential Features PCA for High Dimensional Clustering,” Annals of Statistics, 44, 2323–2359.
   Web of Science ®Google Scholar
• Jurečková, J., and Kalina, J. (2012), “Nonparametric Multivariate Rank Tests and Their Unbiasedness,” Bernoulli, 229–251. DOI: 10.3150/10-BEJ326.
   Web of Science ®Google Scholar
• Lee, J. D., Sun, D. L., Sun, Y., and Taylor, J. E. (2016), “Exact Post-Selection Inference, With Application to the Lasso,” The Annals of Statistics, 44, 907–927. DOI: 10.1214/15-AOS1371.
   Web of Science ®Google Scholar
• Liu, K., Zhang, R., and Mei, Y. (2019), “Scalable Sum-Shrinkage Schemes for Distributed Monitoring Large-Scale Data Streams,” Statistica Sinica, 29, 1–22.
   Web of Science ®Google Scholar
• Liu, Q., Lee, J., and Jordan, M. (2016), “A Kernelized Stein Discrepancy for Goodness-of-Fit Tests,” in International Conference on Machine Learning, pp. 276–284.
   Google Scholar
• Liu, W. (2013), “Gaussian Graphical Model Estimation With False Discovery Rate Control,” Annals of Statistics, 41, 2948–2978.
   Web of Science ®Google Scholar
• Ma, S., Gong, Q., and Bohnert, H. J. (2007), “An Arabidopsis Gene Network Based on the Graphical Gaussian Model,” Genome Research, 17. DOI: 10.1101/gr.6911207.
   Web of Science ®Google Scholar
• Mai, Q., Zou, H., and Yuan, M. (2012), “A Direct Approach to Sparse Discriminant Analysis in Ultra-High Dimensions,” Biometrika, 99, 29–42. DOI: 10.1093/biomet/asr066.
   Web of Science ®Google Scholar
• Mardia, K. V. (1970), “Measures of Multivariate Skewness and Kurtosis With Applications,” Biometrika, 57, 519–530. DOI: 10.1093/biomet/57.3.519.
   Web of Science ®Google Scholar
• Marozzi, M. (2015), “Multivariate Multidistance Tests for High-Dimensional Low Sample Size Case-Control Studies,” Statistics in Medicine, 34, 1511–1526. DOI: 10.1002/sim.6418.
   PubMed Web of Science ®Google Scholar
• Rothman, A. J., Bickel, P. J., Levina, E., Zhu, J. (2008), “Sparse Permutation Invariant Covariance Estimation,” Electronic Journal of Statistics, 2, 494–515. DOI: 10.1214/08-EJS176.
   Web of Science ®Google Scholar
• Royston, J. (1983), “Some Techniques for Assessing Multivarate Normality Based on the Shapiro–Wilk W,” Applied Statistics, 32, 121–133. DOI: 10.2307/2347291.
   Web of Science ®Google Scholar
• Schilling, M. F. (1986), “Multivariate Two-Sample Tests Based on Nearest Neighbors,” Journal of the American Statistical Association, 81, 799–806. DOI: 10.1080/01621459.1986.10478337.
   Web of Science ®Google Scholar
• Shapiro, S. S., and Wilk, M. B. (1965), “An Analysis of Variance Test for Normality (Complete Samples),” Biometrika, 52, 91–611. DOI: 10.1093/biomet/52.3-4.591.
   Web of Science ®Google Scholar
• Smith, S. P., and Jain, A. K. (1988), “A Test to Determine the Multivariate Normality of a Data Set,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 10, 757–761. DOI: 10.1109/34.6789.
   Web of Science ®Google Scholar
• Taylor, J., and Tibshirani, R. (2018), “Post-Selection Inference for-Penalized Likelihood Models,” Canadian Journal of Statistics, 46, 41–61. DOI: 10.1002/cjs.11313.
   PubMed Web of Science ®Google Scholar
• Villasenor Alva, J. A., and Estrada, E. G. (2009), “A Generalization of Shapiro–Wilk’s Test for Multivariate Normality,” Communications in Statistics Theory and Methods, 38, 1870–1883. DOI: 10.1080/03610920802474465.
   Web of Science ®Google Scholar
• Wang, T., and Samworth, R. J. (2018), “High Dimensional Change Point Estimation Via Sparse Projection,” Journal of the Royal Statistical Society, Series B, 80, 57–83. DOI: 10.1111/rssb.12243.
   Google Scholar
• Xia, Y., Cai, T., and Cai, T. T. (2015), “Testing Differential Networks With Applications to the Detection of Gene–Gene Interactions,” Biometrika, 102, 247–266. DOI: 10.1093/biomet/asu074.
   PubMed Web of Science ®Google Scholar
• Xie, Y., and Siegmund, D. (2013), “Sequential Multi-Sensor Change-Point Detection,” The Annals of Statistics, 41, 670–692. DOI: 10.1214/13-AOS1094.
   Web of Science ®Google Scholar
• Yuan, M. (2010), “High Dimensional Inverse Covariance Matrix Estimation Via Linear Programming,” Journal of Machine Learning Research, 11, 2261–2286.
   Web of Science ®Google Scholar
• Yuan, M., and Lin, Y. (2007), “Model Selection and Estimation in the Gaussian Graphical Model,” Biometrika, 94, 19–35. DOI: 10.1093/biomet/asm018.
   Web of Science ®Google Scholar
• Zhou, M., and Shao, Y. (2014), “A Powerful Test for Multivariate Normality,” Journal of Applied Statistics, 41, 351–363. DOI: 10.1080/02664763.2013.839637.
   PubMed Web of Science ®Google Scholar