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.

Bayesian Inference under Small Sample Sizes Using General Noninformative Priors

This paper proposes a Bayesian inference method for problems with small sample sizes. A general type of noninformative prior is proposed to formulate the Bayesian posterior. It is shown that this type of prior can represent a broad range of priors such as classical noninformative priors and asymptotically locally invariant priors and can be derived as the limiting states of normal-inverse-Gamma conjugate priors, allowing for analytical evaluations of Bayesian posteriors and predictors. The performance of different noninformative priors under small sample sizes is compared using the likelihood combining both fitting and prediction performances. Laplace approximation is used to evaluate the likelihood. A realistic fatigue reliability problem was used to illustrate the method. Following that, an actual aeroengine disk lifing application with two test samples is presented, and the results are compared with the existing method.

Genotype-Guided P2Y 12 Inhibitor Therapy After Percutaneous Coronary Intervention: A Bayesian Analysi

Background: Among patients receiving percutaneous coronary intervention (PCI), the role of a genotype-guided approach for antiplatelet therapy compared with usual care is unclear. We conducted a Bayesian analysis of the entire TAILOR-PCI (Tailored Antiplatelet Initiation to Lessen Outcomes Due to Decreased Clopidogrel Response After Percutaneous Coronary Intervention) randomized clinical trial population to evaluate the effect of the genotype-guided antiplatelet therapy post-PCI compared with the usual care on the risk of major adverse cardiovascular events (MACE). Methods: The primary outcome for our study was the composite of MACE (myocardial infarction, stroke, and cardiovascular death). Secondary outcomes included cardiovascular death, stroke, myocardial infarction, stent thrombosis, and major/minor bleeding. Bayesian modeling was used to estimate the probability of clinical benefit of genotype-guided therapy using (1) noninformative priors (ie, analyzing the TAILOR-PCI trial) and (2) informative priors derived from the ADAPT, POPular Genetics, IAC-PCI, and PHARMCLO trials (ie, analyzing TAILOR-PCI trial in the context of prior evidence). Risk ratio (RR: ratio of cumulative outcome incidence between genotype-guided and conventional therapy group) and 95% credible interval (CrI) were estimated for the study outcomes, and probability estimates for RR <1 were computed. Results: Using noninformative priors, in TAILOR-PCI the RR for MACE was 0.78 (95% CrI, 0.55–1.07) in genotype-guided therapy after PCI, and the probability of RR <1 was 94%. Using noninformative priors, the probability of RR <1 for cardiovascular death (RR, 0.95 [95% CrI, 0.52–1.74]), stroke (RR, 0.68 [95% CrI, 0.44–1.06]), myocardial infarction (RR, 0.84 [95% CrI, 0.37–1.89]), stent thrombosis (RR, 0.75 [95% CrI, 0.37–1.45]), and major or minor bleeding (RR, 1.22 [95% CrI, 0.84–1.77]) were 57%, 96%, 67%, 94%, and 15%, respectively. Using informative priors, the posterior probability of RR <1 for MACE, from genotype-guided therapy, was 99% (RR, 0.69 [95% CrI, 0.57–0.84]). Using informative priors, the posterior probability of RR <1 for cardiovascular death (RR, 0.86 [95% CrI, 0.61–1.19]), stroke (RR, 0.69 [95% CrI, 0.48–0.99]), myocardial infarction (RR:0.56 [95% CrI, 0.40–0.78]), stent thrombosis (RR, 0.59 [95% CrI, 0.38–0.94]), and major or minor bleeding (RR, 0.84 [95% CrI, 0.70–0.99]) were 81%, 99%, 99%, 99%, and 99%, respectively. Conclusions: Bayesian analysis of the TAILOR-PCI trial provides clinically meaningful data on the posterior probability of reducing MACE using genotype-guided P2Y 12 inhibitor therapy after PCI.

Robust Assessing the Lifetime Performance of Products with Inverse Gaussian Distribution in Bayesian and Classical Setup

The inverse Gaussian (Wald) distribution belongs to the two-parameter family of continuous distributions having a range from 0 to ∞ and is considered as a potential candidate to model diffusion processes and lifetime datasets. Bayesian analysis is a modern inferential technique in which we estimate the parameters of the posterior distribution obtained by formally combining a prior distribution with an observed data distribution. In this article, we have attempted to perform the Bayesian and classical analyses of the Wald distribution and compare the results. Jeffreys' and uniform priors are used as noninformative priors, while the exponential distribution is used as an informative prior. The analysis comprises finding joint posterior distributions, the posterior means, predictive distributions, and credible intervals. To illustrate the entire estimation procedure, we have used real and simulated datasets, and the results thus obtained are discussed and compared. We have used the Bayesian specialized Open BUGS software to perform Markov Chain Monte Carlo (MCMC) simulations using a real dataset.

On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study.

Statistical Analysis of Location Parameter of Inverse Gaussian Distribution Under Noninformative Priors

Bayesian estimation for location parameter of the inverse Gaussian distribution is presented in this paper. Noninformative priors (Uniform and Jeffreys) are assumed to be the prior distributions for the location parameter as the shape parameter of the distribution is considered to be known. Four loss functions: Squared error, Trigonometric, Squared logarithmic and Linex are used for estimation. Bayes risks are obtained to find the best Bayes estimator through simulation study and real life data