The problem of reconciling the frequentist and Bayesian evidence in testing statistical hypotheses has been extensively studied in the literature. Most of the existing work considers cases without the nuisance parameters which is not the frequently encountered situation since the presence of the nuisance parameters is very common in practice. In this paper, we consider the reconcilability of the Bayesian evidence against the null hypothesis
In the problem of testing a statistical hypothesis
Although many researches have been carried out to deal with the problem of reconciling the Bayesian and frequentist evidence and some of them show that evidence is reconcilable in several specific situations, most of the existing work assumes that no other unknown parameters are present except the parameters of interest. In fact, we may be confronted with the nuisance parameters in various situations. In the locationscale settings, for example, when the location parameter is unknown, so is the scale parameter, in general.
However, in significance testing of hypotheses with the nuisance parameters, the classical
Tsui and Weerahandi [
In this paper, for the onesided testing situations about normal means where the nuisance parameters are present, we study the reconcilability of the Bayesian evidence and the generalized
This paper is organized as follows. In Section
In this section, we consider two testing problems in which the nuisance parameters are present. When no efficient classical frequentist evidence is available because of the presence of the nuisance parameters, we formulate the frequentist evidence by the generalized
Let
For this testing problem, where the nuisance parameter is present, we can still obtain the classical
To derive the Bayesian evidence, we need a prior for the parameters. One reasonable and conventional class of priors for
Under (
Let
Suppose that
Now take
For testing the hypothesis of the form (
Now we turn to consider the BehrensFisher problem. It is a classical testing situation in which the nuisance parameters are present and no useful pivotal quantities are available. Suppose that
In situations where the traditional frequentist approaches fail to provide useful solutions, the conception of the generalized
In this problem, we consider the reconcilability of evidence under the following conjugate class of prior distributions
Now we prove an interesting result that, when
As
Let
(I) We first prove that, given
Similarly, as
(II) We now show that the final conclusion holds. In fact, if we let
Similarly, for (
Note that for each
Consequently, we have
In addition, by the symmetry of the
The following theorem shows that, even for fixed
As
We still adopt the notations of Theorem
Let
Therefore, for any fixed
On the other hand, for any fixed
The following simulation results show that even for small and fixed values of
For fixed




−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.1165  0.1405  0.1785  0.1880  0.2330  0.3110  0.3850  0.4155  0.4745 

0.0195  0.0420  0.0505  0.0705  0.1280  0.1945  0.3140  0.3770  0.4675 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0385  0.0520  0.0760  0.0995  0.1405  0.2250  0.3460  0.3845  0.4705 

0.0060  0.0195  0.0400  0.0465  0.0765  0.1630  0.2840  0.3640  0.4630 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.1165  0.1405  0.1785  0.1880  0.2330  0.3110  0.3850  0.4155  0.4745 

0.0195  0.0420  0.0505  0.0705  0.1280  0.1945  0.3140  0.3770  0.4675 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0600  0.0755  0.1030  0.1360  0.1765  0.2615  0.3495  0.4355  0.4595 

0.0070  0.0205  0.0405  0.0545  0.0845  0.1715  0.3005  0.3760  0.4670 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0835  0.1130  0.1260  0.1585  0.2075  0.2975  0.3700  0.4245  0.4995 

0.0185  0.0315  0.0515  0.0730  0.1025  0.1810  0.2905  0.3635  0.4585 
For fixed




−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0030  0.0080  0.0130  0.0240  0.0490  0.1310  0.2720  0.3495  0.4565 

0.0010  0.0035  0.0060  0.0135  0.0280  0.1030  0.2245  0.3375  0.4410 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0025  0.0045  0.0165  0.0270  0.0545  0.1130  0.2550  0.3480  0.4595 

0.0005  0.0015  0.0050  0.0080  0.0245  0.1000  0.2125  0.3155  0.4345 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0025  0.0085  0.0145  0.0295  0.0470  0.1225  0.2550  0.3400  0.4610 

0.0015  0.0030  0.0045  0.0130  0.0275  0.1055  0.2215  0.3245  0.4455 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0055  0.0070  0.0170  0.0265  0.0545  0.1215  0.2485  0.3535  0.4650 

0.0010  0.0055  0.0070  0.0125  0.0440  0.1085  0.2500  0.3285  0.4300 
 


 

−2.5000  −2.1000  −1.8000  −1.6000  −1.3000  −0.9000  −0.5000  −0.3000  −0.1000 

0.0025  0.0060  0.0150  0.0230  0.0570  0.1145  0.2645  0.3685  0.4460 

0.0020  0.0025  0.0065  0.0095  0.0330  0.0890  0.2205  0.3105  0.4305 
In the presence of the nuisance parameters, we study the reconcilability of the
This provides another illustration of testing situation where the Bayesian and frequentist evidence can be reconciled and may therefore to some extent prevent people from debasing or even dismissing
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).