The Bayes Estimators of the Variance and Scale Parameters of the Normal Model With a Known Mean for the Conjugate and Noninformative Priors Under Stein's Loss.

For the normal model with a known mean, the Bayes estimation of the variance parameter under the conjugate prior is studied in Lehmann and Casella (1998) and Mao and Tang (2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss function. Zhang (2017) calculates the Bayes estimator of the variance parameter of the normal model with a known mean with respect to the conjugate prior under Stein's loss function which penalizes gross overestimation and gross underestimation equally, and the corresponding Posterior Expected Stein's Loss (PESL). Motivated by their works, we have calculated the Bayes estimators of the variance parameter with respect to the noninformative (Jeffreys's, reference, and matching) priors under Stein's loss function, and the corresponding PESLs. Moreover, we have calculated the Bayes estimators of the scale parameter with respect to the conjugate and noninformative priors under Stein's loss function, and the corresponding PESLs. The quantities (prior, posterior, three posterior expectations, two Bayes estimators, and two PESLs) and expressions of the variance and scale parameters of the model for the conjugate and noninformative priors are summarized in two tables. After that, the numerical simulations are carried out to exemplify the theoretical findings. Finally, we calculate the Bayes estimators and the PESLs of the variance and scale parameters of the S&P 500 monthly simple returns for the conjugate and noninformative priors.

The contemplated average success probability for normally distributed models with an application to optimal sample sizes selection.

We analytically obtain the average success probability (ASP) and the contemplated average success probability (CASP) for normally distributed observed differences in the treatment group and the placebo group means of the early trial and the confirmatory trial, assuming a uniform noninformative prior for the population treatment effect and a common known variance of the observations from both groups. For the CASP optimization problem with a fixed subtotal sample size of the early trial and the confirmatory trial of one arm larger than a threshold, we obtain the optimal plan of the sample sizes in a theorem. Moreover, in the theorem, we obtain the analytical formula of the optimal CASP as an increasing function of the subtotal sample size. After that, we calculate and compare the numerical values of the ASP with those in Table 1 of Chuang-Stein (2006). Finally, we investigate the numerical features of the CASP and find the optimal plan of the sample sizes for a given subtotal sample size.