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 CALCULATIONS OF THE QUANTITIES AND EXPRESSIONS OF THE SCALE PARAMETER FOR THE CONJUGATE PRIOR
Since X|σ ∼ N µ, σ 2 , we have Therefore, we guess that the conjugate prior of σ is π c (σ) ∼ SRIG (α, β), that is, Consequently, where the integrand is the pdf of the IG (α, β) distribution, we have for α > 0 and β > 0. Consequently, , for α > 1 2 and β > 0.

Frontiers
Finally, let us calculate E (log σ). We have The calculations are complete.

THE PROOF OF PROPOSITION 2
The "only if" part is correct by Definition 1. Now we prove the "if" part.
The proposition is proved.

THE PROOF OF THE INEQUALITY FOR THE PESLS OF THE VARIANCE PARAMETER
In this section, we will prove forα > 1.
The inequality (S1) is equivalent to The Taylor series expansion for e x with x = 1 α−1 gives Therefore, (S2) is correct and (S1) is proved.

THE PROOF OF THE INEQUALITY FOR THE BAYES ESTIMATORS OF THE SCALE PARAMETER
In this section, we will prove forα > 1 2 . The inequality (S3) is equivalent to We claim that ψ (x) = d 2 dx 2 log Γ (x) = trigamma (x) > 0, for x > 0, which can be checked in R software by plotting the trigamma function over the intervals [0.001, 0.1], [0.1, 10], and [10, 1000]. Alternatively, we can prove that ψ (x) > 0 by exploiting the formula (6.4.1) Abramowitz and Stegun (1970). Let n = 1 and z = x > 0 in the above formula, we obtain since the integrand is positive for all t > 0. Therefore, ψ (x) is an increasing function, and thus ψ (x) < ψ x + 1 2 and G (x) < 0. Consequently, G (x) is a decreasing function for x > 0. Therefore, (S4) is correct and (S3) is proved.
The p-value of the S&P 500 monthly simple returns during the period by the Shapiro-Wilk normality test is 0.6431 > 0.05, which means that the data can be regarded as normal.
The S&P 500 monthly close prices during the period are depicted in Figure S1. From the figure, we see that the S&P 500 monthly close prices increase a lot during the period.  Figure S1. The S&P 500 monthly close prices during the period.
The S&P 500 monthly simple returns during the period are depicted in Figure S2. From the figure, we see that the S&P 500 monthly simple returns fluctuate, and the returns are positive except two occasions.  Figure S2. The S&P 500 monthly simple returns during the period.
The histogram of the S&P 500 monthly simple returns during the period are depicted in Figure S3. From the figure, we see that the density estimation curve of the S&P 500 monthly simple returns are roughly bell shaped, and density can be reasonably approximated by a normal distribution.  Figure S3. The histogram of the S&P 500 monthly simple returns during the period.