On Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence

In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the di erence of means, the new Bayesian measure of evidence given that prior will solve Lindleys' paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong and Doguwa (2018, Open Journal of Statistics, 8: 902-914). Using the Pareto prior as an example, we have shown by the use of simulation results that the new Bayesian measure of evidence solves Lindley's paradox.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Goltong, N. and Doguwa, S. (2019) On Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Open Journal of Statistics, 9, 1-14. doi: 10.4236/ojs.2019.91001.

 [1] Lindley, D.-V. (1957) A Statistical Paradox. Biometrika, 44, 187-192. https://doi.org/10.1093/biomet/44.1-2.187 [2] Robert, C.-P. (2014) On the Je_reys-Lindley Paradox. Philosophy of Science, 81, 216-232. https://doi.org/10.1086/675729 [3] Sprenger, J. (2013) Testing a Precise Null Hypothesis: The Case of Lindleys Paradox. Philosophy of Science, 80, 733-744. https://doi.org/10.1086/673730 [4] Spanos, A. (2013) Who Should Be Afraid of the Je_reys-Lindley Paradox? Philosophy of Science, 80, 73-93. https://doi.org/10.1086/668875 [5] Yin, Y. (2012) A New Bayesian Procedure for Testing Point Null Hypotheses. Computational Statistics, 27, 237-249. https://doi.org/10.1007/s00180-011-0252-6 [6] Yin, Y. and Li, B. (2014) Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Journal of Applied Mathematics, 2014, Article ID: 978691. [7] Goltong, N.-E. and Doguwa, S.-I. (2018) Bayesian Analysis of the Behrens-Fisher Problem under a Gamma Prior. Open Journal of Statistics, 8, 902-914. https://doi.org/10.4236/ojs.2018.86060 [8] Tsui, K.-W. and Weerahandi, S. (1989) Generalized p-Values in Signi_cance Testing of Hypotheses in the Presence of Nuisance Parameters. Journal of the American Statistical Association, 84, 602-607. https://doi.org/10.2307/2289949 [9] Iyer, H.-K. and Paterson, P.-D. (2002) A Recipe for Constructing Generalized Pivotal Quantities and Generalized Con_dence Intervals. Technical Report 2002/10, Department of Statistics, Colorado State University, Fort Collins. [10] Mitra, P.-K. and Sinha, B.-K. (2007) A Generalized p-Value Approach to Inference on Common Mean. Technical Report, Department of Mathematics and Statistics, University of Maryland, Baltimore. [11] Zheng, S., Shi, N.-Z. and Ma, W. (2010) Statistical Inference on Di_erence or Ratio of Means from Heteroscedastic Normal Populations. Journal of Statistical Planning and Inference, 140, 1236-1242. [12] Ozkip, E., Yazici, B. and Sezer, A. (2014) A Simulation Study on Tests for the Behrens-Fisher Problem. Turkiye Klinikleri Journal of Biostatistics, 6, 59-66. [13] Degroot, M.-H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339. https://doi.org/10.1080/01621459.1982.10477811 [14] Berger, J.-O. and Selke, T. (1987) Testing a Point Null Hypothesis: The Irreconcilability of p Values and Evidence. Journal of the American Statistical Association, 82, 112-122. https://doi.org/10.2307/2289131 [15] Casella, G. and Berger, R.-L. (1987) Reconciling Bayesian and Frequentist Evidence in One-Sided Testing Problem. Journal of the American Statistical Association, 82, 106-111. https://doi.org/10.1080/01621459.1987.10478396 [16] Berger, J.-O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-335. https://doi.org/10.1214/ss/1177013238 [17] Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160. https://doi.org/10.1214/aos/1176325622 [18] Yin, Y. and Zhao, J. (2013) Testing Normal Means: The Reconcilability of the p Value and the Bayesian Evidence. The Scienti_c World Journal, 2013, Article ID: 381539. [19] Pestman, W., Tuerlinckx, F. and Vanpaemel, W. (2018) A Reverse to the Je_reys-Lindley Paradox. Probability and Mathematical Statistics, 38, 243-247.