Testing Normal Means: The Reconcilability of the Value and the Bayesian Evidence

Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 381539, 7 pages http://dx.doi.org/10.1155/2013/381539 Research Article Testing Normal Means: The Reconcilability of the 𝑃 Value and the Bayesian Evidence Yuliang Yin1 and Junlong Zhao2 1 2 School of Economics, Beijing Technology and Business University, Beijing 100048, China School of Mathematics and System Science, Beihang University, LMIB of the Ministry of Education, Beijing 100083, China Correspondence should be addressed to Yuliang Yin; Received 8 August 2013; Accepted 9 September 2013 Academic Editors: M. Guillén and S. Umarov Copyright © 2013 Y. Yin and J. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 𝐻0 in terms of the posterior probability of 𝐻0 being true and the frequentist evidence against 𝐻0 in terms of the P value in testing normal means where the nuisance parameters are present. The reconcilability of evidence can be obtained both for testing a normal mean and for the Behrens-Fisher problem. 1. Introduction In the problem of testing a statistical hypothesis 𝐻0 , a frequentist may give evidence against 𝐻0 by the observed significance level, the 𝑃 value, while a Bayesian may give it by the posterior probability that 𝐻0 is true. Lindley [1] illustrated the possible discrepancy between the Bayesian and the frequentist evidence. The relationship of these two measures of evidence is then extensively studied in the literature. Pratt [2] revealed that the 𝑃 values are usually approximately equal to the posterior probabilities in the onesided testing problems. Casella and Berger [3] considered testing the one-sided hypothesis for a location parameter and showed that the lower bounds of the posterior probability over some reasonable classes of priors are exactly equal to the corresponding 𝑃 values in many cases. Some important papers which deal with the reconcilability of the Bayesian and frequentist evidence are Bartlett [4], 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 form 𝐻0 : 𝜃 ≤ 𝑐 versus 𝐻1 : 𝜃 > 𝑐, (1) where 𝜃 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 2 The Scientific World Journal 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. where 𝑥 = (𝑥1 , . . . , 𝑥𝑛 )󸀠 , 𝜅𝑛 = 𝜅0 + 𝑛, 𝜇𝑛 (𝑥) = (𝜅0 𝜇0 + 𝑛𝑥)/𝜅𝑛 , ]𝑛 = ]0 +𝑛, and 𝜎𝑛2 (𝑥) = []0 𝜎02 +(𝑛−1)𝑠2 +𝜅0 𝑛(𝑥−𝜇0 )2 /𝜅𝑛 ]/]𝑛 . Therefore, we can give the posterior density for (𝜇, 𝜎2 ) as 𝜋 (𝜇, 𝜎2 | 𝑥) = × 2. Main Results ]𝑛 /2 𝐻0 : 𝜇 ≤ 𝑐 versus 𝐻1 : 𝜇 > 𝑐, ]𝑛 /2 𝑝 (𝑥) = 𝑃 (𝑇𝑛−1 ≤ √𝑛 (𝑐 − 𝑥) ), 𝑠 ]0 ]0 𝜎02 1 ∼ Gamma , ( ), 𝜎2 2 2 (4) where the prior parameters (𝜇0 , 𝜅0 ) can be interpreted as the mean and sample size of the normal prior observations and (𝜎02 , ]0 ) the sample variance and sample size of the Gamma prior observations. Under (4) we have 𝜇 | 𝑥, 𝜎2 ∼ 𝑁 (𝜇𝑛 (𝑥) , 2 𝜎 ), 𝜅𝑛 ]𝑛 ]𝑛 𝜎𝑛2 (𝑥) 1 | 𝑥 ∼ Gamma ( , ), 𝜎2 2 2 2 Then the marginal posterior density for 𝜇 can be obtained by integrating out 𝜎2 as −1/2 2 𝜋 (𝜇 | 𝑥) = √𝜅𝑛 ((]𝑛 /2) 𝜎𝑛 (𝑥)) Γ ((]𝑛 + 1) /2) Γ (]𝑛 /2) 2 (7) −(]𝑛 +1)/2 𝜅 (𝜇 − 𝜇𝑛 (𝑥)) ] × [1 + 𝑛 ]𝑛 𝜎𝑛2 (𝑥) , from which we know that √𝜅𝑛 (𝜇 − 𝜇𝑛 (𝑥)) ∼ 𝑡 (]𝑛 ) . 𝜎𝑛 (𝑥) (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. To derive the Bayesian evidence, we need a prior for the parameters. One reasonable and conventional class of priors for 𝜇 and 𝜎2 is the following conjugate class of prior distributions 𝐺𝑐1 : (6) 𝜅 (𝜇 − 𝜇𝑛 (𝑥)) + ]𝑛 𝜎𝑛2 (𝑥) × exp [− 𝑛 ]. 2𝜎2 (2) where 𝑐 is a fixed constant. For this testing problem, where the nuisance parameter is present, we can still obtain the classical 𝑃 value as ] 𝜎2 (𝑥) exp [− 𝑛 𝑛 2 ] 2𝜎 Γ (]𝑛 /2) 𝜎]𝑛 +2 2 2.1. One-Sample Normal Mean. Let 𝑋1 , . . . , 𝑋𝑛 be a random sample from a normal population 𝑁(𝜇, 𝜎2 ), where both the mean 𝜇 and the variance 𝜎2 are unknown. Consider now the following problem of testing the mean of a normal distribution 𝜎2 ), 𝜅0 ((]𝑛 /2) 𝜎𝑛2 (𝑥)) √𝜅𝑛 ((]𝑛 /2) 𝜎𝑛 (𝑥)) = √2𝜋Γ (]𝑛 /2) 𝜎]𝑛 +3 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 𝑃 value. 𝜇 | 𝜎2 ∼ 𝑁 (𝜇0 , 2 𝜅 (𝜇 − 𝜇𝑛 (𝑥)) √𝜅𝑛 exp [− 𝑛 ] √2𝜋𝜎 2 (8) Consequently, the posterior probability of 𝐻0 being true is 𝑃 (𝐻0 | 𝑥) = 𝑃 (𝑇]𝑛 ≤ √ 𝜅𝑛 (𝑐 − 𝜇𝑛 (𝑥))) , 2 𝜎𝑛 (𝑥) (9) where 𝑇]𝑛 is a 𝑡-variable with ]𝑛 degrees of freedom. Notice that if 𝜇0 = 𝑐, we have lim 𝑃 (𝐻0 | 𝑥) = 𝑃 (𝑇]𝑛 ≤ √ 𝜅0 ,𝜎0 → 0 ]𝑛 √𝑛 (𝑐 − 𝑥) ). 𝑛−1 (...truncated)