Yin [1] has developed a new Bayesian measureof 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.
References
[1]
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
[2]
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
[3]
Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192.
https://doi.org/10.1093/biomet/44.1-2.187
[4]
Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93. https://doi.org/10.1086/668875
[5]
Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232. https://doi.org/10.1086/675729
[6]
Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432. https://doi.org/10.1214/aoms/1177731214
[7]
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
[8]
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
[9]
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
[10]
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
[11]
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.
[12]
Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339. https://doi.org/10.1080/01621459.1982.10477811
[13]
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
[14]
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
[15]
Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352. https://doi.org/10.1214/ss/1177013238
[16]
Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160. https://doi.org/10.1214/aos/1176325622
[17]
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
[18]
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
[19]
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
[20]
Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192.
https://doi.org/10.1093/biomet/44.1-2.187
[21]
Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93. https://doi.org/10.1086/668875
[22]
Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232. https://doi.org/10.1086/675729
[23]
Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432. https://doi.org/10.1214/aoms/1177731214
[24]
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
[25]
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
[26]
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
[27]
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
[28]
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.
[29]
Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339. https://doi.org/10.1080/01621459.1982.10477811
[30]
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
[31]
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
[32]
Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352. https://doi.org/10.1214/ss/1177013238
[33]
Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160. https://doi.org/10.1214/aos/1176325622
[34]
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
