Abstract
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.
Highlights
We are interested in the data from the normal model with a known mean, with respect to the conjugate and noninformative (Jeffreys’s, reference, and matching) priors, under Stein’s and the squared error loss functions
In the normal model with a known mean μ, our parameters of interest are θ σ2 and θ σ
The quantities and expressions of the variance and scale parameters for the conjugate and noninformative priors are summarized in two tables
Summary
There are four basic elements in Bayesian decision theory and in Bayesian point estimation: The data, the model, the prior, and the loss function. (Zhang, 2017) calculates the Bayes estimator of the variance parameter θ σ2 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 and the corresponding Posterior Expected Stein’s Loss (PESL). Motivated by the works of (Lehmann and Casella, 1998; Mao and Tang, 2012; Zhang, 2017), we want to calculate the Bayes estimators of the variance parameter of the normal model with a known mean for the noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function. Motivated by the works of (Lehmann and Casella, 1998; Mao and Tang, 2012; Zhang, 2017), we want to calculate the Bayes estimators of the scale parameter θ σ with respect to the conjugate and noninformative priors under Stein’s loss function, and the corresponding PESLs. Suppose that we observe X1, X2, . Note that the calculations of δπs ,θ(x), δ2π,θ(x), PESLπs ,θ(x), and PESLπ2,θ(x) depend only on the three expectations E(θ|x), E(θ−1|x), and E(log θ|x)
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