Some important papers which deal with the reconcilability of the Bayesian and frequentist evidence are Bartlett [4], Erlotinib solubility Cox [5], Shafer [6], Berger and Delampady [7], and Berger and Sellke [8].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 location-scale 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 P values are typically not available.

Tsui and Weerahandi [9], considering testing the one-sided hypothesis of the H1:��?versus?formH0:�ȡ�c>c,(1)where �� is the parameter of interest and c is a fixed constant, introduced the concept of the generalized P value, which appears to be useful in situations where conventional frequentist approaches do not provide useful solutions.Tsui and Weerahandi [9] and some later relevant works formulated the generalized P values for many specific examples. Hannig et al. [10] provided a general method for constructing the generalized P value via fiducial inference.In this paper, for the one-sided testing situations about normal means where the nuisance parameters are present, we study the reconcilability of the Bayesian evidence and the generalized P value.

It is shown that, under the conjugate class of prior distributions, the Bayesian evidence and the generalized P value are reconcilable both for the problem of testing a normal mean and for the Behrens-Fisher problem.This paper is organized as follows. In Section 2, we give the main results of the reconcilability of the P value and the Bayesian evidence in testing normal means. Some conclusions and discussions are given in Section 3.2. Main ResultsIn 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 P value.2.1.

One-Sample Normal Mean Let X1,��, Xn be a random sample from a normal population N(��, ��2), where both the mean �� and the variance ��2 are unknown. Consider now the following problem of testing the mean of a normal H1:��?versus?distributionH0:�̡�c>c,(2)where c is a fixed constant.For this testing problem, where the nuisance parameter is present, we can still obtain the classical Anacetrapib P value asp(x)=P(Tn?1��n(c?x��)s),(3)where Tn?1 is a t-variable with n ? 1 degrees of freedom and x�� and s2 stand for the observed sample mean and sample variance, respectively.