Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley’s paradox. Yin and Li [2] extended the methodology of Yin [1] to the case of the Behrens-Fisher problem by assigning Jeffreys’ independent prior to the nuisance parameters. In this paper, we were able to show both analytically and through the results from simulation studies that the methodology of Yin [1] solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when a Gamma prior is assigned to the nuisance parameters.
Yin, Y. (2012) A New Bayesian Procedure for Testing Point Null Hypothesis. Computational Statistics, 27, 237-249. https://doi.org/10.1007/s00180-011-0252-6
Yin, Y. and Li, B. (2014) Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Journal of Applied Mathematics, 2014, Article ID: 978691. https://doi.org/10.1155/2014/978691
Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192. https://doi.org/10.1093/biomet/44.1-2.187
Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93. https://doi.org/10.1086/668875
Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232. https://doi.org/10.1086/675729
Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432. https://doi.org/10.1214/aoms/1177731214
Fraser, D.A.S. and Streit, F. (1972) On the Behrens-Fisher Problem. Australian Journal of Statistics, 14, 167-171. https://doi.org/10.1111/j.1467-842X.1972.tb00354.x
Robinson, G.K. (1976) Properties of Students t and of the Behrens-Fisher Solution to the Two Means Problem. The Annals of Statistics, 4, 963-971. https://doi.org/10.1214/aos/1176343594
Tsui, K.-W. and Weerahandi, S. (1989) Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters. Journal of American Statistical Association, 84, 602-607. https://doi.org/10.2307/2289949
Zheng, S., Shi, N.-Z. and Ma, W. (2009) Statistical Inference on the Difference or Ratio of Means from Heteroscedastic Normal Populations. Journal of Statistical Planning and Inference, 140, 1236-1242. https://doi.org/10.1016/j.jspi.2009.11.010
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.
Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339. https://doi.org/10.1080/01621459.1982.10477811
Berger, J.O. and Sellke, 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
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
Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352. https://doi.org/10.1214/ss/1177013238
Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160. https://doi.org/10.1214/aos/1176325622
Ghosh, M. and Kim, Y.-H. (2001) The Behrens-Fisher Problem Revisited: A Bayes-Frequentist Synthesis. The Canadian Journal of Statistics, 29, 5-17. https://doi.org/10.2307/3316047
Yin, Y. (2012) A New Bayesian Procedure for Testing Point Null Hypothesis. Computational Statistics, 27, 237-249. https://doi.org/10.1007/s00180-011-0252-6
Yin, Y. and Li, B. (2014) Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Journal of Applied Mathematics, 2014, Article ID: 978691. https://doi.org/10.1155/2014/978691
Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192. https://doi.org/10.1093/biomet/44.1-2.187
Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93. https://doi.org/10.1086/668875
Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232. https://doi.org/10.1086/675729
Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432. https://doi.org/10.1214/aoms/1177731214
Fraser, D.A.S. and Streit, F. (1972) On the Behrens-Fisher Problem. Australian Journal of Statistics, 14, 167-171. https://doi.org/10.1111/j.1467-842X.1972.tb00354.x
Robinson, G.K. (1976) Properties of Students t and of the Behrens-Fisher Solution to the Two Means Problem. The Annals of Statistics, 4, 963-971. https://doi.org/10.1214/aos/1176343594
Tsui, K.-W. and Weerahandi, S. (1989) Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters. Journal of American Statistical Association, 84, 602-607. https://doi.org/10.2307/2289949
Zheng, S., Shi, N.-Z. and Ma, W. (2009) Statistical Inference on the Difference or Ratio of Means from Heteroscedastic Normal Populations. Journal of Statistical Planning and Inference, 140, 1236-1242. https://doi.org/10.1016/j.jspi.2009.11.010
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.
Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339. https://doi.org/10.1080/01621459.1982.10477811
Berger, J.O. and Sellke, 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
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
Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352. https://doi.org/10.1214/ss/1177013238
Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160. https://doi.org/10.1214/aos/1176325622
Ghosh, M. and Kim, Y.-H. (2001) The Behrens-Fisher Problem Revisited: A Bayes-Frequentist Synthesis. The Canadian Journal of Statistics, 29, 5-17. https://doi.org/10.2307/3316047