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 Estimation ofWarner’s Randomized Response Technique with Transmuted Kumaraswamy Prior

In recent time, the Bayesian approach to randomized response technique has been used for estimating the population proportion especially of respondents possessing sensitive attributes such as induced abortion, tax evasion and shoplifting. This is done by combining suitable prior information about an unknown parameter of the population with the sample information for the estimation of the unknown parameter. In this study, possibility of using a transmuted Kumaraswamy prior is raised, yielding a new Bayes estimator for estimating population proportion of sensitive attribute for Warner’s randomized response technique. Consequently, the proposed Bayes estimator with transmuted Kumaraswamy prior is compared with existing Bayes estimators developed with a simple beta and Kumaraswamy priors in terms of their mean square error. The proposed estimator competes well with the existing estimators for some values of population proportion. The performances of Bayes estimators were also compared using some benchmark data.

Hierarchical Bayesian Modeling and Randomized Response Method for Inferring the Sensitive-Nature Proportion

In this study, a new three-statement randomized response estimation method is proposed to improve the drawback that the maximum likelihood estimation method could generate a negative value to estimate the sensitive-nature proportion (SNP) when its true value is small. The Bayes estimator of the SNP is obtained via using a hierarchical Bayesian modeling procedure. Moreover, a hybrid algorithm using Gibbs sampling in Metropolis–Hastings algorithms is used to obtain the Bayes estimator of the SNP. The highest posterior density interval of the SNP is obtained based on the empirical distribution of Markov chains. We use the term 3RR-HB to denote the proposed method here. Monte Carlo simulations show that the quality of 3RR-HB procedure is good and that it can improve the drawback of the maximum likelihood estimation method. The proposed 3RR-HB procedure is simple for use. An example regarding the homosexual proportion of college freshmen is used for illustration.

Estimation of the Rayleigh Distribution under Unified Hybrid Censoring

We derive some estimators of the scale parameter of the Rayleigh distribution under the unified hybrid censoring scheme. We also derive some estimators of the reliability function and the entropy of the Rayleigh distribution. First, we obtain the maximum likelihood estimator of the scale parameter. Second, we obtain the Bayes estimator using the mean of the posterior distribution. Lastly, we obtain the Bayes estimator using the mode of the posterior distribution. We also derive the interval estimation (confidence interval, credible interval, and HPD credible interval) for the scale parameter under the unified hybrid censoring scheme. We compare the proposed estimators in the sense of the mean squared error through Monte Carlo simulation. Coverage probability and average lengths of 95 % and 90% intervals are obtained.