Inference for the Behrens-Fisher problem after outlier removal

doi:10.7523/j.ucas.2025.018

Abstract

Abstract: Selective inference aims at reducing the bias caused by the detect-and-forget strategy for handling statistical inferential problems in the presence of outliers. In this paper, we study selective inference for the classical Behrens-Fisher problem. Several selective confidence intervals for the difference between the means of two normal populations with unequal variances are presented, including the selective modifications of the traditional Welch’s interval and several generalized pivotal quantities-based intervals. These methods are compared via Monte Carlo simulations and a real data analysis. Suggestions for practical use are also discussed.

Key words: selective inference, generalized pivotal quantity, truncated distribution

CLC Number:

O212.7

MU Weiyan, WANG Junhao, XIONG Shifeng. Inference for the Behrens-Fisher problem after outlier removal[J]. Journal of University of Chinese Academy of Sciences, DOI: 10.7523/j.ucas.2025.018.

References

[1] Huber P J, Ronchetti E M.Robust statistics[M]. New York: John Wiley & Sons, 2009. DOI: 10.1002/9780470434697.
[2] Grubbs F E.Procedures for detecting outlying observations in samples[J]. Technometrics, 1969, 11(1):1-21. DOI:10.1080/00401706.1969.10490657.
[3] Hawkins D M.Identification of Outliers[M]. London: Chapman and Hall, 1980.
[4] Barnett V, Lewis T.Outliers in statistical data[M]. 3rd ed. Chichester: Wiley & Sons, 1994.
[5] John G H.Robust decision trees: removing outliers from databases[C]//Proceedings of the First International Conference on KDD. Montréal, Québec, Canada: AAAI Press, 1995: 174-179. DOI:10.5555/3001335.3001364.
[6] Aggarwal C C, Yu P S.Outlier detection for high dimensional data[C]//Proceedings of the 2001 ACM SIGMOD international conference on Management of data. Santa Barbara California USA. ACM, 2001:37-46. DOI:10.1145/375663.375668.
[7] Taylor J, Tibshirani R J.Statistical learning and selective inference[J]. Proceedings of the National Academy of Sciences of the United States of America, 2015, 112(25):7629-7634. DOI:10.1073/pnas.1507583112.
[8] Benjamini Y.Simultaneous and selective inference: Current successes and future challenges[J]. Biometrical Journal, 2010, 52(6):708-721. DOI:10.1002/bimj.200900299.
[9] Benjamini Y, Bogomolov M.Selective inference on multiple families of hypotheses[J]. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2014, 76(1):297-318. DOI:10.1111/rssb.12028.
[10] Tibshirani R J, Taylor J, Lockhart R, et al.Exact post-selection inference for sequential regression procedures[J]. Journal of the American Statistical Association, 2016, 111(514):600-620. DOI:10.1080/01621459.2015.1108848.
[11] Tian X Y, Taylor J.Selective inference with a randomized response[J]. The Annals of Statistics, 2018, 46(2):679-710. DOI:10.1214/17-aos1564.
[12] Chen S X, Bien J.Valid inference corrected for outlier removal[J]. Journal of Computational and Graphical Statistics, 2020, 29(2):323-334. DOI:10.1080/10618600.2019.1660180.
[13] Loftus J R, Taylor J E. Selective inference in regression models with groups of variables[EB/OL].2015: 1511.01478. https://arxiv.org/abs/1511.01478v1.
[14] Lee J D, Sun D L, Sun Y K, et al.Exact post-selection inference, with application to the lasso[J]. The Annals of Statistics, 2016, 44(3):907-927. DOI:10.1214/15-aos1371.
[15] Behrens W U.A contribution to error estimation with few observations[J]. Journal of Agriculture Scientific Archives of the Royal Prussian State College-Economy, 1929, 68: 807-837.
[16] Fisher R A.The fiducial argument in statistical inference[J]. Annals of Eugenics, 1935, 6(4):391-398. DOI:10.1111/j.1469-1809.1935.tb02120.x.
[17] Welch B L.The significance of the difference between two means when the population variances are unequal[J]. Biometrika, 1938, 29(3/4):350-362. DOI:10.1093/biomet/29.3-4.350.
[18] Cochran W G, Cox G M.Experimental designs[M]. New York: Wiley, 1950.
[19] Aspin A A.An examination and further development of a formula arising in the problem of comparing two mean values[J]. Biometrika, 1948, 35(1/2):88. DOI:10.2307/2332631.
[20] Lee A F S, Gurland J. Size and power of tests for equality of means of two normal populations with unequal variances[J]. Journal of the American Statistical Association, 1975, 70(352):933-941. DOI:10.1080/01621459.1975.10480326.
[21] Jeffreys H.The theory of probability[M]. Oxford, UK: Oxford University Press, 1998.
[22] Kim S H, Cohen A S.On the Behrens-Fisher problem: A review[J]. Journal of Educational and Behavioral Statistics, 1998, 23(4):356-377. DOI:10.3102/10769986023004356.
[23] Yin Y L, Li B R.Analysis of the Behrens‐Fisher Problem Based on Bayesian Evidence[J]. Journal of Applied Mathematics, 2014, 2014(1): 978691. DOI:10.1155/2014/978691.
[24] 仇丽莎, 韦来生. 正态总体均值和误差方差同时的经验Bayes估计[J]. 中国科学院大学学报, 2013, 30(4):454-461. DOI:10.7523/j.issn.2095-6134.2013.04.005.
[25] Weerahandi S .Generalized Confidence Intervals[J]. Journal of the American Statistical Association, 1993, 88(423): 899-905. DOI: 10.1080/01621459.1993.10476355.
[26] Tsui K W, Weerahandi S.Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters[J]. Journal of the American Statistical Association, 1989, 84(406):602-607. DOI:10. 1080/01621459.1989.10478810.
[27] 扈慧敏, 杨荣, 徐兴忠. 单因素方差分析模型中的广义p-值[J]. 中国科学院研究生院学报, 2007, 24(4):408-418. DOI:10.3969/j.issn.1002-1175.2007.04.002.
[28] 牟唯嫣, 熊世峰. 混合模型中的广义置信域[J]. 中国科学院研究生院学报, 2008, 25(3):297-304. DOI:10.7523/j.issn.2095-6134.2008.3.002.
[29] Xu L W, Wang S G.A new generalized p-value and its upper bound for ANOVA under unequal error variances[J]. Communications in Statistics—Theory and Methods, 2008, 37(7):1002-1010. DOI:10. 1080/03610920701713252.
[30] Ananda M M A, Weerahandi S. Two-way ANOVA with unequal cell frequencies and unequal variances[J]. Statistica Sinica, 1997, 7(3):631-646.
[31] Mu W Y, Xiong S F.Robust generalized confidence intervals[J]. Communications in Statistics-Simulation and Computation, 2017, 46(8):6049-6060. DOI:10.1080/03610918.2016.1189566.
[32] 刘琰, 李仕明, 张三国. 基于符号秩的高维均值检验[J]. 中国科学院大学学报, 2022, 39(5):586-592. DOI: 10.7523/j.ucas.2020.0059.
[33] Cook R D.Detection of influential observation in linear regression[J]. Technometrics, 1977, 19(1):15-18. DOI:10.1080/00401706.1977.10489493.
[34] Xie M G, Singh K.Confidence distribution, the frequentist distribution estimator of a parameter: A review[J]. International Statistical Review, 2013, 81(1):3-39. DOI:10.1111/insr.12000.
[35] Hannig J.On generalized fiducial inference[J]. Statistica Sinica, 2009, 19(2):491-544. DOI:10.1016/j.probengmech.2008.06.005.
[36] Hannig J, Iyer H, Patterson P.Fiducial generalized confidence intervals[J]. Journal of the American Statistical Association, 2006, 101(473):254-269. DOI:10.1198/016214505000000736.
[37] Xiong S F, Mu W Y.On construction of asymptotically correct confidence intervals[J]. Journal of Statistical Planning and Inference, 2009, 139(4):1394-1404. DOI:10.1016/j.jspi.2008.08.014.
[38] Singh P, Saxena K K, Srivastava O P.Power comparisons of solutions to the Behrens-Fisher problem[J]. American Journal of Mathematical and Management Sciences, 2002, 22(3/4):233-250. DOI:10.1080/01966324.2002.10737589 .
[39] Chen C, Liu H S, Wu C X, et al.A simple approximation solution for the Behrens-Fisher problem[J]. Communications in Statistics-Simulation and Computation, 2023:1-14. DOI:10.1080/03610918.2023.2280451.
[40] Güven G, Acıtaş S, Şamkar H, et al. RobustBF: An R package for robust solution to the Behrens-Fisher problem[J]. The R Journal, 2021, 13(2): 642. DOI: 10.32614/rj-2021-107.
[41] Kang G L, Mirzaei S S, Zhang H, et al.Robust Behrens-Fisher Statistic Based on Trimmed Means and Its Usefulness in Analyzing High-Throughput Data[J]. Frontiers in Systems Biology, 2022, 2:877601. DOI:10.3389/fsysb.2022.877601.
[42] Shapiro S S, Wilk M B.An analysis of variance test for normality (complete samples)[J]. Biometrika, 1965, 52(3/4):591-611. DOI:10. 1093/biomet/52.3-4.591.
[43] Bartlett M S.Properties of sufficiency and statistical tests[J]. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 1937, 160(901):268-282. DOI:10. 1098/rspa.1937.0109.
[44] Wang R, Xu W L.An approximate randomization test for the high-dimensional two-sample Behrens-Fisher problem under arbitrary covariances[J]. Biometrika, 2022, 109(4):1117-1132. DOI:10.1093/biomet/asac014.
[45] Mu W Y, Xiong S F, Xu X Z.Generalized confidence regions of fixed effects in the two-way ANOVA[J]. Journal of Systems Science and Complexity, 2008, 21(2):276-282. DOI:10.1007/s11424-008-9111-0.
[46] Berenguer-Rico V, Wilms I.Heteroscedasticity testing after outlier removal[J]. Econometric Reviews, 2021, 40(1):51-85. DOI:10.1080/07474938.2020.1735749.
[47] Mu W Y, Xiong S F.On Huber’s contaminated model[J]. Journal of Complexity, 2023, 77:101745. DOI:10.1016/j.jco.2023.101745.
[48] Thériault R, Ben-Shachar M S, Patil I, et al. Check your outliers?! An introduction to identifying statistical outliers in R with easystats[J]. Behavior Research Methods, 2024, 56(4):4162-4172. DOI:10.3758/s13428-024-02356-w.
[49] 牟唯嫣, 贾晓芳, 熊世峰. 逆瑞利分布和对数逻辑分布下过程能力指数的区间估计[J]. 中国科学院大学学报(中英文), 2024, 41(6):728-735.