Statistical references#
Primary literature is cited in the method-specific validation ledgers. This site keeps references close to the formulas they support so readers can audit the path from publication to implementation.
Thomas J. Fisher. On testing for an identity covariance matrix when the dimensionality equals or exceeds the sample size. Journal of Statistical Planning and Inference, 142(1):312–326, 2012.
Tung-Lung Wu and Ping Li. Tests for high-dimensional covariance matrices using random matrix projection. arXiv:1511.01611, 2015.
Jun Li and Song Xi Chen. Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940, 2012. doi:10.1214/12-AOS993.
Tony Cai, Weidong Liu, and Yin Xia. Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108(501):265–277, 2013. doi:10.1080/01621459.2012.758041.
Kyoungjae Lee, Kisung You, and Lizhen Lin. Bayesian optimal two-sample tests for high-dimensional gaussian populations. Bayesian Analysis, 19(3):869–893, 2024. doi:10.1214/23-BA1373.
James R. Schott. Some tests for the equality of covariance matrices. Journal of Statistical Planning and Inference, 94(1):25–36, 2001.
James R. Schott. A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Computational Statistics & Data Analysis, 51(12):6535–6542, 2007. doi:10.1016/j.csda.2007.03.004.
Munmun Biswas and Anil K. Ghosh. A nonparametric two-sample test applicable to high dimensional data. Journal of Multivariate Analysis, 123:160–171, 2014. doi:10.1016/j.jmva.2013.09.004.
Harold Hotelling. The generalization of student's ratio. The Annals of Mathematical Statistics, 2(3):360–378, 1931. doi:10.1214/aoms/1177732979.
B. L. Welch. The generalization of student's problem when several different population variances are involved. Biometrika, 34(1/2):28–35, 1947. doi:10.2307/2332510.
Howard Levene. Robust tests for equality of variances. In Ingram Olkin, Sudhish G. Ghurye, Wassily Hoeffding, William G. Madow, and Henry B. Mann, editors, Contributions to Probability and Statistics, pages 278–292. Stanford University Press, 1960.
Morton B. Brown and Alan B. Forsythe. Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346):364–367, 1974. doi:10.1080/01621459.1974.10482955.
A. P. Dempster. A high dimensional two sample significance test. The Annals of Mathematical Statistics, 29(4):995–1010, 1958.
Zhidong Bai and Hewa Saranadasa. Effect of high dimension: by an example of a two sample problem. Statistica Sinica, 6(2):311–329, 1996.
Muni S. Srivastava and Meng Du. A test for the mean vector with fewer observations than the dimension. Journal of Multivariate Analysis, 99(3):386–402, 2008. doi:10.1016/j.jmva.2006.11.002.
Ying Yao. An approximate degrees of freedom solution to the multivariate behrens–fisher problem. Biometrika, 52(1/2):139–147, 1965.
Soren Johansen. The welch–james approximation to the distribution of the residual sum of squares in a weighted linear regression. Biometrika, 67(1):85–92, 1980.
D. G. Nel and C. A. Van Der Merwe. A solution to the multivariate behrens–fisher problem. Communications in Statistics - Theory and Methods, 15(12):3719–3735, 1986.
K. Krishnamoorthy and Jianqi Yu. Modified nel and van der merwe test for the multivariate behrens–fisher problem. Statistics & Probability Letters, 66(2):161–169, 2004.
Miles E. Lopes, Laurent Jacob, and Martin J. Wainwright. A more powerful two-sample test in high dimensions using random projection. In Advances in Neural Information Processing Systems 24, 1206–1214. 2011.
T. Tony Cai, Weidong Liu, and Yin Xia. Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B, 76(2):349–372, 2014. doi:10.1111/rssb.12034.
Måns Thulin. A high-dimensional two-sample test for the mean using random subspaces. Computational Statistics & Data Analysis, 74:26–38, 2014.
James R. Schott. Some high-dimensional tests for a one-way manova. Journal of Multivariate Analysis, 98(9):1825–1839, 2007.
JinTing Zhang and JinFeng Xu. On the k-sample behrens–fisher problem for high-dimensional data. Science in China Series A: Mathematics, 52(6):1285–1304, 2009.
Ming-Xiang Cao, Junyong Park, and Dao-Jiang He. A test for the k sample behrens–fisher problem in high dimensional data. Journal of Statistical Planning and Inference, 201:86–102, 2019.
Barry C. Arnold and Robert M. Shavelle. Joint confidence sets for the mean and variance of a normal distribution. The American Statistician, 52(2):133–140, 1998.
S. K. Perng and Ramon C. Littell. A test of equality of two normal population means and variances. Journal of the American Statistical Association, 71(356):968–971, 1976.
Robb J. Muirhead. Aspects of Multivariate Statistical Theory. Wiley, New York, 1982.
Lingyun Zhang, Xinzhong Xu, and Gemai Chen. The exact likelihood ratio test for equality of two normal populations. The American Statistician, 66(3):180–184, 2012.
Zhongying Liu, Baisen Liu, Shurong Zheng, and Ning-Zhong Shi. Simultaneous testing of mean vector and covariance matrix for high-dimensional data. Journal of Statistical Planning and Inference, 188:82–93, 2017.
Masashi Hyodo and Takahiro Nishiyama. A simultaneous testing of the mean vector and the covariance matrix among two populations for high-dimensional data. TEST, 27(3):680–699, 2018.
S. S. Shapiro and M. B. Wilk. An analysis of variance test for normality (complete samples). Biometrika, 52(3/4):591–611, 1965.
S. S. Shapiro and R. S. Francia. An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67(337):215–216, 1972.
Carlos M. Jarque and Anil K. Bera. Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6(3):255–259, 1980.
Carlos M. Urzúa. On the correct use of omnibus tests for normality. Economics Letters, 53(3):247–251, 1996.
Yulia R. Gel and Joseph L. Gastwirth. A robust modification of the jarque–bera test of normality. Economics Letters, 99(1):30–32, 2008.
Mengta Yang and Reza Modarres. Multivariate tests of uniformity. Statistical Papers, 58(3):627–639, 2017.
Software and comparison sources#
Agreement with another package is supporting evidence, not the sole correctness criterion. Each pySHT validation record states exactly what an external comparison establishes.