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.
1 Introduction
There are four basic elements in Bayesian decision theory and specifically in Bayesian point estimation: The data, the model, the prior, and the loss function. In this paper, 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. We will analytically calculate the Bayes estimators of the variance and scale parameters of the normal model with a known mean, with respect to the conjugate and noninformative priors under Stein’s and the squared error loss functions.
The squared error loss function has been used by many authors for the problem of estimating the variance, σ2, based on a random sample from a normal distribution (see for instance (Maatta and Casella, 1990)). As pointed out by (Casella and Berger, 2002), the squared error loss function penalizes overestimation and underestimation equally, which is fine for the location parameter with parameter space . For a variance or scale parameter, the parameter space is where 0 is a natural lower bound and the estimation problem is not symmetric. In these cases, we should not choose the squared error loss function, but choose a loss function which penalizes gross overestimation and gross underestimation equally, that is, an action a will incur an infinite loss when it tends to 0 or ∞. Stein’s loss function has this property, and thus it is recommended to use for the positive restricted parameter space by many authors (see for example (James and Stein, 1961; Petropoulos and Kourouklis, 2005; Oono and Shinozaki, 2006; Bobotas and Kourouklis, 2010; Zhang, 2017; Xie et al., 2018; Zhang et al., 2019; Sun et al., 2021)). In the normal model with a known mean μ, our parameters of interest are θ = σ2 (a variance parameter) and θ = σ (a scale parameter). Therefore, we will select Stein’s loss function.
The motivation and contributions of our paper are summarized as follows. For the normal model with a known mean μ, the Bayes estimation of the variance parameter θ = σ2 under the conjugate prior which is an Inverse Gamma distribution is studied in Example 4.2.5 (p.236) of (Lehmann and Casella, 1998) and Example 1.3.5 (p.15) of (Mao and Tang, 2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss. (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 equally, 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 and scale parameters of the normal model with a known mean for the conjugate and noninformative priors under Stein’s loss function. The contributions of our paper are summarized as follows. In this paper, we have calculated the Bayes estimators of the variance parameter θ = σ2 with respect to the noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function, and the corresponding Posterior Expected Stein’s Losses (PESLs). Moreover, we have calculated 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. For more literature on Bayesian estimation and inference, we refer readers to (Sindhu and Aslam, 2013a; Sindhu and Aslam, 2013b; Sindhu et al., 2013; Sindhu et al., 2016a; Sindhu et al., 2016b; Sindhu et al., 2016c; Sindhu et al., 2017; Sindhu et al., 2018; Sindhu and Hussain, 2018)
The rest of the paper is organized as follows. In the next Section 2, we analytically calculate the Bayes estimators of the variance and scale parameters of the normal model with a known mean, with respect to the conjugate and noninformative priors under Stein’s loss function, and the corresponding PESLs. We also analytically calculate the Bayes estimators under the squared error 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 for the conjugate and noninformative priors are summarized in two tables. Section 3 reports vast amount of numerical simulation results of the combination of the noninformative prior and the scale parameter to support the theoretical studies of two inequalities of the Bayes estimators and the PESLs, and that the PESLs depend only on the number of observations, but do not depend on the mean and the sample. In Section 4, 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. Some conclusions and discussions are provided in Section 5.
2 Bayes Estimator, PESL, IRSL, and BRSL
In this section, we will analytically calculate the Bayes estimator of the variance parameter under Stein’s loss function, the PESL at , , and the Integrated Risk under Stein’s Loss (IRSL) at , , which is also the Bayes Risk under Stein’s Loss (BRSL) for π, θ. See (Robert, 2007) for the definitions of the posterior expected loss, the integrated risk, and the Bayes risk. We will also analytically calculate the Bayes estimator of the scale parameter under Stein’s loss function, the PESL at , , and the IRSL at , , which is also the BRSL for π, σ.
Suppose that we observe X1, X2, …, Xn from the hierarchical normal model with a mixing variance parameter θ = σ2:where − ∞ < μ < ∞ is a known constant, is the normal distribution with a known mean μ and an unknown variance θ, and is the prior distribution of θ. For the normal model with a known mean μ, the Bayes estimation of the variance parameter θ = σ2 under the conjugate prior which is an Inverse Gamma distribution is studied in Example 4.2.5 (p.236) of (Lehmann and Casella, 1998) and Example 1.3.5 (p.15) of (Mao and Tang, 2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss. (Zhang, 2017) calculates the Bayes estimator of the variance parameter θ = σ2 with respect to the conjugate prior under Stein’s loss function, and the corresponding 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. The usual Bayes estimator with respect to a prior is to calculate under the squared error loss function. As pointed out in the introduction, we should calculate and use the Bayes estimator of the variance parameter θ with respect to a prior under Stein’s loss function, that is, .
Alternatively, we may be interested in the scale parameter θ = σ. Motivated by the works of (Lehmann and Casella, 1998; Mao and Tang, 2012; Zhang, 2017), we also 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, …, Xn from the hierarchical normal model with a mixing scale parameter θ = σ:where − ∞ < μ < ∞ is a known constant, is the normal distribution with a known mean μ and an unknown variance σ2, and is the prior distribution of σ. The usual Bayes estimator with respect to a prior is to calculate under the squared error loss function. As pointed out in the introduction, we should calculate and use the Bayes estimator of the scale parameter σ with respect to a prior under Stein’s loss function, that is, .
Now let us explain why we choose Stein’s loss function on . Stein’s loss function is given bywhere θ > 0 is the unknown parameter of interest and a is an action or estimator. The squared error loss function is given by
The asymmetric Linear Exponential (LINEX) loss function ((Varian et al., 1975; Zellner, 1986; Robert, 2007)) is given bywhere c ≠ 0 serving to determine its shape. In particular, when c > 0, the LINEX loss function tends to ∞ exponentially, while when c < 0, the LINEX loss function tends to ∞ linearly. Note that on the positive restricted parameter space , Stein’s loss function penalizes gross overestimation and gross underestimation equally, that is, an action a will incur an infinite loss when it tends to 0 or ∞. Whereas, the squared error loss function does not penalize gross overestimation and gross underestimation equally, as an action a will incur a finite loss (in fact θ2) when it tends to 0 and incur an infinite loss when it tends to ∞. Similarly, the LINEX loss functions also do not penalize gross overestimation and gross underestimation equally, as an action a will incur a finite loss (in fact e−cθ + cθ − 1) when it tends to 0 and incur an infinite loss when it tends to ∞. Figure 1 shows the four loss functions on when θ = 2.
FIGURE 1
The four loss functions on when θ = 2.
As pointed out by (Zhang, 2017), the Bayes estimatorminimizes the PESL, that is,where is an action space, is an action (estimator), which is a function only of x, given by (Eq. 3) is Stein’s loss function, and θ > 0 is the unknown parameter of interest. Note that Stein’s loss function has a nice property that it penalizes gross overestimation and gross underestimation equally, that is, an action a will incur an infinite loss when it tends to 0 or ∞. Moreover, note that θ may be the variance parameter σ2 or the scale parameter σ.
The usual Bayes estimator of θ is which minimizes the Posterior Expected Squared Error Loss. It is interesting to note thatwhose proof exploits Jensen’s inequality and the proof can be found in (Zhang, 2017). Note that the inequality (Eq. 6) is a special inequality in (Zhang et al., 2018). As calculated in (Zhang, 2017), the PESL at isand the PESL at isAs observed in (Zhang, 2017),which is a direct consequence of the general methodology for finding a Bayes estimator or due to minimizes the PESL. The numerical simulations will exemplify (Eqs 6,7) later. Note that the calculations of , , , and depend only on the three expectations , , and .
2.1 Conjugate Prior
The problem of finding the Bayes estimator under a conjugate prior is a standard problem that is treated in almost every text on Mathematical Statistics.
The quantities and expressions of the variance and scale parameters of the normal models (Eqs 1,2) with a known mean μ for the conjugate prior are summarized in Table 1. In the table, α > 0 and β > 0 are known constants,is the digamma function, and is the gamma function. In R software (R Core Team. R, 2021), the function digamma(z) calculates . The quantities and expressions of the variance parameter θ = σ2 for the conjugate prior are calculated in and quoted from (Zhang, 2017). The calculations of the quantities and expressions of the scale parameter θ = σ for the conjugate prior can be found in the Supplementary Material. We remark that the calculations of the quantities and expressions in Table 1 are not trivial, especially .
TABLE 1
| Quantities | ||
|---|---|---|
| α*β* | ||
The quantities and expressions for the conjugate prior.
2.2 Noninformative Priors
Famous noninformative priors include the Jeffreys’s ( (Jeffreys, 1961)), reference ( (Bernardo, 1979; Berger and Bernardo, 1992)), and matching ( (Tibshirani, 1989; Datta and Mukerjee, 2004)) priors. See also (Berger, 2006; Berger et al., 2015) and the references therein.
The Jeffreys’s noninformative prior for θ = σ2 is
See Part I (p.66) of (Chen, 2014), where μ is assumed known in the normal model . The Jeffreys’s noninformative prior for θ = σ is
See Example 3.5.6 (p.131) of (Robert, 2007), where μ is assumed known in the normal model .
Since μ is assumed known in the normal models, there is only one unknown parameter. Therefore, the reference prior is equal to the Jeffreys’s prior, and the matching prior is also equal to the Jeffreys’s prior (see pp.130–131 of (Ghosh et al., 2006)). In summary, when μ is assumed known in the normal models, the three noninformative priors equal, that is,andwhere stands for the noninformative prior.
Note that as in many statistics textbooks, the probability density function (pdf) of is given by
The conjugate prior of the scale parameter θ = σ is a Square Root of the Inverse Gamma (SRIG) distribution that we define below.
DEFINITION 1Letwithα > 0 andβ > 0. Thenand the pdf ofσis given byDefinition 1 gives the definition of the SRIG distribution, which is the conjugate prior of the scale parameter θ = σ of the normal distribution. Because the SRIG distribution can not be found in standard textbooks, so we give its definition here. Moreover, Definition 1 is reasonable, sinceWe have the following proposition which gives the three expectations of the distribution. The calculations needed in the proposition can be found in the Supplementary Material. We remark that the calculations of and are straightforward by utilizing a simple transformation of θ = σ2 and the integration of an distribution. However, the calculations of is skillful by first a transformation of and then a change of the order of integration and differentiation.
PROPOSITION 1Letwithα > 0 andβ > 0. ThenThe relationship between the two distributions and are given in the following proposition whose proof can be found in the Supplementary Material. We remark that the proof of the proposition is straightforward by utilizing monotone transformations θ = σ2 and .
PROPOSITION 2if and only if, whereα > 0 andβ > 0.The posterior distributions of θ and σ for the noninformative priors are given in the following theorem whose proof can be found in the Supplementary Material.
Letandwhereμis known andθ = σ2is unknown,, and. Thenwhere
We have the following two remarks for Theorem 1.
Remark 1Let θ = σ2. In the derivation of , if we derive it in this way,wherethen by Proposition 2, , which is different from . In fact, the above practice is equivalent to the derivation of the pdf of θ in terms of the pdf of σ by , ignoring the term, which is obviously wrong. Therefore, the above derivation which is a pitfall for incautious users is wrong. ‖
Remark 2The two posterior distributions in Theorem 1, and , follow Proposition 2 by accident. We haveandNote that , and thuswhich is the reason why and follow Proposition 2. Note that the posterior distributions depend on the prior distributions. If the prior distributions and are selected different from and , then the relationship (Eq. 9) may not be satisfied, and thus and may not follow Proposition 2. ‖
2.2.1 The Quantities and Expressions of the Variance Parameter
In this subsubsection, we will calculate the expressions of the quantities (three posterior expectations, two Bayes estimators, and two PESLs) of the variance parameter θ = σ2.
Now we calculate the three expectations , , and for the variance parameter θ = σ2. By Theorem 1, , and thus
From (Zhang, 2017), we know that
It is easy to see that, for ,which exemplifies (Eq. 6). From (Zhang, 2017), we find thatand
It can be directly proved that for , which exemplifies (Eq. 7), and its proof which exploits the Taylor series expansion for ex can be found in the Supplementary Material. Note that and depend only on . Therefore, they depend only on n, but do not depend on μ and x. Numerical simulations will exemplify this result.
The IRSL at or the BRSL for θ = σ2 is (similar to (Robert, 2007))since does not depend on x, whereis the marginal density of x with prior .
2.2.2 The Quantities and Expressions of the Scale Parameter
In this subsubsection, we will calculate the expressions of the quantities (three posterior expectations, two Bayes estimators, and two PESLs) of the scale parameter θ = σ.
Now let us calculate , , , and for the scale parameter σ. To calculate these quantities, we need to calculate the three expectations , , and . Since by Theorem 1, from Proposition 1, we have
It can be proved that, for ,which exemplifies (Eq. 6), and the proof which exploits the positivity of can be found in the Supplementary Material.
Now we calculate and for the scale parameter σ. From (Zhang, 2017), we know that the PESL at isand the PESL at is
Substituting (Eqs 10,11,12), into the above expressions, we obtainfor and , andfor and . It can be directly proved that for and , which exemplifies (Eq. 7), and its proof which exploits the Taylor series expansion for log u with u near 1 can be found in the Supplementary Material. Note that and depend only on . Therefore, they depend only on n, but do not depend on μ and x. Numerical simulations will exemplify this result.
The IRSL at or the BRSL for θ = σ is (similar to (Robert, 2007))since does not depend on x, whereis the marginal density of x with prior .
The quantities and expressions of the variance and scale parameters for the noninformative priors are summarized in Table 2. In the table, and are given by (Eq. 8).
TABLE 2
| Quantities | ||
|---|---|---|
The quantities and expressions for the noninformative priors.
From Tables 1, 2, we find that there are four combinations of the expressions of the quantities: conjugate prior and variance parameter, conjugate prior and scale parameter, noninformative prior and variance parameter, and noninformative prior and scale parameter. The forms of the expressions of the quantities are the same for the variance parameter under the conjugate and noninformative priors, since they have the same Inverse Gamma posterior distributions. Similarly, the forms of the expressions of the quantities are the same for the scale parameter under the conjugate and noninformative priors, since they have the same Square Root of the Inverse Gamma posterior distributions.
The inequalities (Eqs 6,7) exist in Tables 1, 2. In fact, there are 8 inequalities in Tables 1, 2 and 4 inequalities in each table. Since the forms of the expressions of the quantities are the same in Tables 1, 2, with the only difference of the parameters, there are actually 4 different inequalities which are in Table 2. One inequality of the four inequalities about the Bayes estimators is obvious, and the proofs of the other three inequalities can be found in the Supplementary Material.
3 Numerical Simulations
In this section, we will numerically exemplify the theoretical studies of (Eqs 6,7), and that the PESLs depend only on n, but do not depend on μ and x. The numerical simulation results are similar for the four combinations of the expressions of the quantities, and thus we only present the results for the combination of the noninformative prior and the scale parameter.
First, we fix μ = 0 and n = 10, and assume that σ = 1 is drawn from the improper prior distribution. After that, we draw a random samplefrom N(μ, σ2).
To generate a random sample with k = 1000 fromwe will adopt the following algorithm. First, compute and from (Eq. 8). Second, generate a random sampleThird, compute
Fourth, compute
Hence, σ is a random sample from the distribution. Figure 2 shows the histogram of σ|x and the density estimation curve of πn(σ|x). It is πn(σ|x) that we find to minimize the PESL. From the figure, we see that the distribution is left peaked, right skewed, and continuous.
FIGURE 2
The histogram of σ|x and the density estimation curve of πn(σ|x).
The Bayes estimators ( and ) and the PESLs ( and ) are computed by the following algorithm. First, compute and from (Eq. 8). Second, computeThird, computeNumerical results show thatandwhich exemplify the theoretical studies of (6) and (7).
In Figure 3, we fix μ = 0 and n = 10, but allow the seed number to change from 1 to 10 (i.e., we change x). From the figure we see that the estimators and PESLs are functions of x. We see from the left plot of the figure that the estimators depend on x in an unpredictable manner, and are unanimously smaller than , and thus (Eq. 6) is exemplified. The two Bayes estimators are distinguishable since we fix n = 10 to be a small number. The right plot of the figure exhibits that the PESLs do not depend on x, and are unanimously smaller than , and thus (Eq. 7) is exemplified.
FIGURE 3
The estimators are functions of x(left) and the PESLs are also functions of x(right).
Now we allow one of the two parameters μ and n to change, holding other parameters fixed. Moreover, we also assume that the sample x is fixed, as it is the case for the real data. Figure 4 shows the estimators and PESLs as functions of μ and n. We see from the left plots of the figure that the estimators depend on μ and n, and (Eq. 6) is exemplified. More specifically, the estimators are first decreasing and then increasing functions of μ, and the estimators attain the minimum when μ = 0. However, the estimators fluctuate around some value when n increases. The right plots of the figure exhibit that the PESLs depend only on n, but do not depend on μ , and (Eq. 7) is exemplified. More specifically, the PESLs are decreasing functions of n. Furthermore, the two PESLs as functions of n are indistinguishable, as the two PESLs are very close. In summary, the results of the figure exemplify the theoretical studies of (Eqs 6,7).
FIGURE 4
Left: The estimators as functions of μ and n. Right: The PESLs as functions of μ and n.
Since the estimators and and the PESLs and depend on and , where and , we can plot the surfaces of the estimators and the PESLs on the domain via the R function persp3d() in the R package rgl (see (Adler and Murdoch, 2017; Zhang et al., 2017; Zhang et al., 2019; Sun et al., 2021)). We remark that the R function persp() in the R package graphics can not add another surface to the existing surface, but persp3d() can. Moreover, persp3d() allows one to rotate the perspective plots of the surface according to one’s wishes. Figure 5 plots the surfaces of the estimators and the PESLs, and the surfaces of the difference of the estimators and the difference of the PESLs. From the left two plots of the figure, we see that for all on D, which exemplifies (Eq. 6). From the right two plots of the figure, we see that for all on D, which exemplifies (Eq. 7). In summary, the results of the figure exemplify the theoretical studies of (Eqs 6,7).
FIGURE 5
The domain for is D = (0.5, 10] × (0, 10] for all the plots. a is for and b is for in the axes of all the plots. The red surface is for and the blue surface is for in the upper two plots. (upper left) The estimators as functions of and . for all on D. (upper right) The PESLs as functions of and . for all on D. (lower left) The surface of which is positive for all on D. (lower right) The surface of which is also positive for all on D.
4 A Real Data Example
In this section, we exploit the data from finance. The R package quantmod ( (Ryan and Ulrich, 2017)) is exploited to download the data ˆGSPC (the S&P 500) during 2020-04-24 and 2021-07-02 from “finance.yahoo.com.” It is commonly believed that the monthly simple returns of the index data or the stock data are normally distributed. It is simple to check that the S&P 500 monthly simple returns follow the normal model. Usually, the data from real examples can be regarded as iid from the normal model with an unknown mean μ. However, the mean μ could be estimated by prior information or historical information. Alternatively, the mean μ could be estimated by the sample mean. Therefore, for simplicity, we assume that the mean μ is known. Assume thatfor the S&P 500 monthly simple returns.
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 are summarized in
Table 3. From the table, we observe the following facts.
• Given the prior (conjugate or noninformative), the Bayes estimators are similar across different loss functions (Stein’s or squared error).
• Given the loss function, the Bayes estimators are quite different across different priors. Therefore, the prior has a larger influence than the loss function in calculating the Bayes estimators.
TABLE 3
| Conjugate prior | Noninformative prior | |||
|---|---|---|---|---|
| θ = σ2 | θ = σ | θ = σ2 | θ = σ | |
| 0.111474 | 0.338545 | 0.000408 | 0.020528 | |
| 0.125408 | 0.348644 | 0.000467 | 0.021224 | |
| 0.056583 | 0.014410 | 0.063800 | 0.016285 | |
| 0.063800 | 0.014846 | 0.073126 | 0.016846 | |
The Bayes estimators and the PESLs of the S&P 500 monthly simple returns.
More results (the data of the S&P 500 monthly simple returns, the plot of the S&P 500 monthly close prices, the plot of the S&P 500 monthly simple returns, the histogram of the S&P 500 monthly simple returns) for the real data example can be found in the Supplementary Material due to space limitations.
5 Conclusions and Discussions
For the variance (θ = σ2) and scale (θ = σ) parameters of the normal model with a known mean μ, we recommend and analytically calculate the Bayes estimators, , with respect to the conjugate and noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function which penalizes gross overestimation and gross underestimation equally. These estimators minimize the PESLs. We also analytically calculate the Bayes estimators, , with respect to the conjugate and noninformative priors under the squared error loss function, and the corresponding PESLs. The quantities (, , , , , , , , ) and expressions of the variance and scale parameters for the conjugate and noninformative priors are summarized in Tables 1, 2, respectively. Note that , which is essential for the calculation of and , depends on the digamma function.
Proposition 1 gives the three expectations of the distribution. Moreover, Proposition 2 gives the relationship between the two distributions and .
For the conjugate and noninformative priors, the posterior distribution of θ = σ2, , follows an Inverse Gamma distribution, and the posterior distribution of σ, , follows an SRIG distribution which is defined in Definition 1.
We find that the IRSL at or the BRSL for θ = σ2 is
In addition, the IRSL at or the BRSL for θ = σ is
The numerical simulations of the combination of the noninformative prior and the scale parameter exemplify the theoretical studies of (Eqs 6,7), and that the PESLs depend only on n, but do not depend on μ and x. Moreover, in the real data example, we have calculated 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.
Unlike in frequentist paradigm, if is the Maximum Likelihood Estimator (MLE) of σ, then is the MLE of σ2. In Bayesian paradigm, we usually should estimate the variance parameter σ2 and the scale parameter σ separately. In Table 2, we find that
It is easy to see thatSimilarly,
When there is no prior information about the unknown parameter of interest, we prefer the noninformative prior, as the hyperparameters α and β are somewhat arbitrary for the conjugate prior.
We remark that the Bayes estimator under Stein’s loss function is more appropriate than that under the squared error loss function, not because the former is smaller, but because Stein’s loss function which penalizes gross overestimation and gross underestimation equally is more appropriate for the positive restricted parameter.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
This work was carried out in collaboration among all authors. Author YYZ wrote the first draft of the article. Author TZR did literature searches and revised the article. Author MML revised the article. All authors read and approved the final article.
Funding
The research was supported by the Ministry of Education (MOE) project of Humanities and Social Sciences on the west and the border area (20XJC910001), the National Social Science Fund of China (21XTJ001), the National Natural Science Foundation of China (12001068; 72071019), and the Fundamental Research Funds for the Central Universities (2020CDJQY-Z001; 2021CDJQY-047).
Acknowledgments
The authors are extremely grateful to the editor, the guest associate editor, and the reviewers for their insightful comments that led to significant improvement of the article.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fdata.2021.763925/full#supplementary-material
References
1
AdlerD.MurdochD. (2017). Rgl: 3D Visualization Using OpenGL. OthersR package version 0.98.1.
2
BergerJ. O.BernardoJ. M. (1992). On the Development of the Reference Prior Method. Bayesian Statistics 4. London: Oxford University Press.
3
BergerJ. O.BernardoJ. M.SunD. C. (2015). Overall Objective Priors. Bayesian Anal.10, 189–221. 10.1214/14-ba915
4
BergerJ. O. (2006). The Case for Objective Bayesian Analysis. Bayesian Anal.1, 385–402. 10.1214/06-ba115
5
BernardoJ. M. (1979). Reference Posterior Distributions for Bayesian Inference. J. R. Stat. Soc. Ser. B (Methodological)41, 113–128. 10.1111/j.2517-6161.1979.tb01066.x
6
BobotasP.KourouklisS. (2010). On the Estimation of a Normal Precision and a Normal Variance Ratio. Stat. Methodol.7, 445–463. 10.1016/j.stamet.2010.01.001
7
CasellaG.BergerR. L. (2002). Statistical Inference (USA: Duxbury). 2nd edition.
8
ChenM. H. (2014). Bayesian Statistics Lecture. Changchun, China: Statistics Graduate Summer SchoolSchool of Mathematics and Statistics, Northeast Normal University.
9
DattaG. S.MukerjeeR. (2004). Probability Matching Priors: Higher Order Asymptotics. New York: Springer.
10
GhoshJ. K.DelampadyM.SamantaT. (2006). An Introduction to Bayesian Analysis. New York: Springer.
11
JamesW.SteinC. (1961). Estimation with Quadratic Loss. Proc. Fourth Berkeley Symp. Math. Stat. Probab.1, 361–380.
12
JeffreysH. (1961). Theory of Probability. 3rd edition. Oxford: Clarendon Press.
13
LehmannE. L.CasellaG. (1998). Theory of Point Estimation. 2nd edition. New York: Springer.
14
MaattaJ. M.CasellaG. (1990). Developments in Decision-Theoretic Variance Estimation. Stat. Sci.5, 90–120. 10.1214/ss/1177012263
15
MaoS. S.TangY. C. (2012). Bayesian Statistics. 2nd edition. Beijing, China: Statistics Press.
16
OonoY.ShinozakiN. (2006). On a Class of Improved Estimators of Variance and Estimation under Order Restriction. J. Stat. Plann. Inference136, 2584–2605. 10.1016/j.jspi.2004.10.023
17
PetropoulosC.KourouklisS. (2005). Estimation of a Scale Parameter in Mixture Models with Unknown Location. J. Stat. Plann. Inference128, 191–218. 10.1016/j.jspi.2003.09.028
18
R Core Team. R (2021). A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
19
RobertC. P. (2007). The Bayesian Choice: From Decision-Theoretic Motivations to Computational Implementation. 2nd paperback edition. New York: Springer.
20
RyanJ. A.UlrichJ. M. (2017). R Package Version 0, 4–10.Quantmod: Quantitative Financial Modelling Framework.
21
SindhuT. N.AslamM. (2013). Bayesian Estimation on the Proportional Inverse Weibull Distribution under Different Loss Functions. Adv. Agric. Sci. Eng. Res.3, 641–655.
22
SindhuT. N.AslamM.HussainZ. (2016). Bayesian Estimation on the Generalized Logistic Distribution under Left Type-II Censoring. Thailand Statistician14, 181–195.
23
SindhuT. N.AslamM. (2013). Objective Bayesian Analysis for the Gompertz Distribution under Doudly Type II Cesored Data. Scientific J. Rev.2, 194–208.
24
SindhuT. N.HussainZ. (2018). Mixture of Two Generalized Inverted Exponential Distributions with Censored Sample: Properties and Estimation. Stat. Applicata-Italian J. Appl. Stat.30, 373–391.
25
SindhuT. N.SaleemM.AslamM. (2013). Bayesian Estimation for Topp Leone Distribution under Trimmed Samples. J. Basic Appl. Scientific Res.3, 347–360.
26
SindhuT. N.AslamM.HussainZ. (2016). A Simulation Study of Parameters for the Censored Shifted Gompertz Mixture Distribution: A Bayesian Approach. J. Stat. Manage. Syst.19, 423–450. 10.1080/09720510.2015.1103462
27
SindhuT. N.FerozeN.AslamM. (2017). A Class of Improved Informative Priors for Bayesian Analysis of Two-Component Mixture of Failure Time Distributions from Doubly Censored Data. J. Stat. Manage. Syst.20, 871–900. 10.1080/09720510.2015.1121597
28
SindhuT. N.KhanH. M.HussainZ.Al-ZahraniB. (2018). Bayesian Inference from the Mixture of Half-Normal Distributions under Censoring. J. Natn. Sci. Found. Sri Lanka46, 587–600. 10.4038/jnsfsr.v46i4.8633
29
SindhuT. N.RiazM.AslamM.AhmedZ. (2016). Bayes Estimation of Gumbel Mixture Models with Industrial Applications. Trans. Inst. Meas. Control.38, 201–214. 10.1177/0142331215578690
30
SunJ.ZhangY.-Y.SunY. (2021). The Empirical Bayes Estimators of the Rate Parameter of the Inverse Gamma Distribution with a Conjugate Inverse Gamma Prior under Stein's Loss Function. J. Stat. Comput. Simulation91, 1504–1523. 10.1080/00949655.2020.1858299
31
TibshiraniR. (1989). Noninformative Priors for One Parameter of Many. Biometrika76, 604–608. 10.1093/biomet/76.3.604
32
VarianH. R. (1975). “A Bayesian Approach to Real Estate Assessment,” in Studies in Bayesian Econometrics and Statistics. Editors FienbergS. E.ZellnerA. (Amsterdam: North Holland), 195–208.
33
XieY.-H.SongW.-H.ZhouM.-Q.ZhangY.-Y. (2018). The Bayes Posterior Estimator of the Variance Parameter of the Normal Distribution with a Normal-Inverse-Gamma Prior Under Stein’s Loss. Chin. J. Appl. Probab. Stat.34, 551–564.
34
ZellnerA. (1986). Bayesian Estimation and Prediction Using Asymmetric Loss Functions. J. Am. Stat. Assoc.81, 446–451. 10.1080/01621459.1986.10478289
35
ZhangY.-Y. (2017). The Bayes Rule of the Variance Parameter of the Hierarchical Normal and Inverse Gamma Model under Stein's Loss. Commun. Stat. - Theor. Methods46, 7125–7133. 10.1080/03610926.2016.1148733
36
ZhangY.-Y.WangZ.-Y.DuanZ.-M.MiW. (2019). The Empirical Bayes Estimators of the Parameter of the Poisson Distribution with a Conjugate Gamma Prior under Stein's Loss Function. J. Stat. Comput. Simulation89, 3061–3074. 10.1080/00949655.2019.1652606
37
ZhangY.-Y.XieY.-H.SongW.-H.ZhouM.-Q. (2018). Three Strings of Inequalities Among Six Bayes Estimators. Commun. Stat. - Theor. Methods47, 1953–1961. 10.1080/03610926.2017.1335411
38
ZhangY.-Y.ZhouM.-Q.XieY.-H.SongW.-H. (2017). The Bayes Rule of the Parameter in (0,1) under the Power-Log Loss Function with an Application to the Beta-Binomial Model. J. Stat. Comput. Simulation87, 2724–2737. 10.1080/00949655.2017.1343332
Summary
Keywords
Bayes estimator, variance and scale parameters, normal model, conjugate and noninformative priors, Stein’s loss
Citation
Zhang Y-Y, Rong T-Z and Li M-M (2022) 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. Front. Big Data 4:763925. doi: 10.3389/fdata.2021.763925
Received
24 August 2021
Accepted
01 November 2021
Published
03 January 2022
Volume
4 - 2021
Edited by
Niansheng Tang, Yunnan University, China
Reviewed by
Guikai Hu, East China University of Technology, China
Akio Namba, Kobe University, Japan
Tabassum Sindhu, Quaid-i-Azam University, Pakistan
Updates
Copyright
© 2022 Zhang, Rong and Li.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ying-Ying Zhang, robertzhangyying@qq.com
This article was submitted to Data Science, a section of the journal Frontiers in Big Data
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