The BehrensFisher problem concerns the inferences for the difference between the means of two normal populations without making any assumption about the variances. Although the problem has been extensively studied in the literature, researchers cannot agree on its solution at present. In this paper, we propose a new method for dealing with the BehrensFisher problem in the Bayesian framework. The Bayesian evidence for testing the equality of two normal means and a credible interval at a specified level for the difference between the means are derived. Simulation studies are carried out to evaluate the performance of the provided Bayesian evidence.
The BehrensFisher problem may arise in the comparison of two treatments, products, and so forth. It concerns comparing the means of two normal distributions whose variances are unknown. Suppose that
The difficulty with the BehrensFisher problem is that the standard classical frequentist evidence is not available because nuisance parameters are present. Tsui and Weerahandi [
Behrens [
For more discussions of the BehrensFisher problem see Wilks [
In this paper, we derive the Bayesian evidence for the BehrensFisher problem using the procedure in Yin [
This paper is organized as follows. In Section
Yin [
Now consider the BehrensFisher problem of testing hypotheses
Now we carry out a simulation study to illustrate the performance of the proposed Bayesian evidence. The simulation results listed in Table














2.00  2.00  0.7823  0.7810  0.7816  2.00  2.00  0.7635  0.7606  0.7615 
2.00  2.01  0.3133  0.3104  0.3102  2.00  2.01  0.3028  0.3031  0.3017 
2.00  2.02  0.1722  0.1713  0.1711  2.00  2.02  0.1949  0.1950  0.1938 
2.00  2.03  0.0854  0.0863  0.0868  2.00  2.03  0.0628  0.0627  0.0630 
2.00  2.04  0.0418  0.0420  0.0428  2.00  2.04  0.0188  0.0185  0.0190 
2.00  2.05  0.0117  0.0122  0.0121  2.00  2.05  0.0018  0.0017  0.0016 

















2.000  2.000  0.8597  0.8603  0.8596  2.000  2.000  0.8525  0.8522  0.8511 
2.000  2.001  0.1329  0.1319  0.1336  2.000  2.001  0.4979  0.4998  0.4961 
2.000  2.002  0.0688  0.0690  0.0696  2.000  2.002  0.1090  0.1070  0.1085 
2.000  2.003  0.0384  0.0383  0.0379  2.000  2.003  0.0239  0.0239  0.0237 
2.000  2.004  0.0313  0.0317  0.0318  2.000  2.004  0.0014  0.0014  0.0011 
2.000  2.005  0.0033  0.0032  0.0035  2.000  2.005  0.0007  0.0008  0.0006 

















2.00  2.00  0.5239  0.5245  0.5279  2.00  2.00  0.9963  0.9963  0.9962 
2.00  2.01  0.3327  0.3346  0.3357  2.00  2.01  0.2523  0.2507  0.2500 
2.00  2.02  0.0224  0.0228  0.0218  2.00  2.02  0.0996  0.1006  0.0991 
2.00  2.03  0.0032  0.0033  0.0034  2.00  2.03  0.0366  0.0366  0.0368 
2.00  2.04  0.0017  0.0017  0.0018  2.00  2.04  0.0119  0.0122  0.0122 
2.00  2.05  0.0001  0.0001  0.0001  2.00  2.05  0.0036  0.0040  0.0040 
By this Bayesian evidence for the BehrensFisher problem, we consider two examples. One is included in Lehmann [
Measures of driving times from following two different routes.
Route  Times  

I  6.5  6.8  7.1  7.3  10.2  
II  5.8  5.8  5.9  6.0  6.0  6.0  6.3  6.3  6.4  6.5  6.5 
Scores of surgical and nonsurgical treatments.
Treatment  Scores  

Surgical  15  9  12  16  14  15  18  13  12  11  15  9  16  9  
Nonsurgical  6  8  7  4  4  6  8  3  7  8  9  6  3  6  4 
Based on the proposed Bayesian evidence, a credible interval for the difference of means
For the BehrensFisher problem, let
On one hand,
By Theorem
In fact, we have another interesting result about the interval estimation of
For the BehrensFisher problem,
We first prove that the Bayesian evidence for testing (
It then follows that
Theorem
Now we return to the examples of comparing means of driving time and comparing improvement scores of treatments discussed above. We recommend the
We carry out Bayesian analysis of the BehrensFisher problem in this paper. The Bayesian evidence for testing the hypothesis
By this method of analyzing the BehrensFisher problem, we give an efficient way of dealing with nuisance parameters which are the source of the difficulty with this problem. This is because our inferences about
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the editors and reviewers for their kind help and valuable comments that lead to significant improvement of this paper. The work was supported by the Foundation for Training Talents of Beijing (Grant no. 19000532377), the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (Grant no. IDHT20130505), and the Research Foundation for Youth Scholars of Beijing Technology and Business University (Grant no. QNJJ201203).