The Bayesian Posterior and Marginal Densities of the Hierarchical Gamma–Gamma, Gamma–Inverse Gamma, Inverse Gamma–Gamma, and Inverse Gamma–Inverse Gamma Models with Conjugate Priors

Abstract

Positive, continuous, and right-skewed data are fit by a mixture of gamma and inverse gamma distributions. For 16 hierarchical models of gamma and inverse gamma distributions, there are only 8 of them that have conjugate priors. We first discuss some common typical problems for the eight hierarchical models that do not have conjugate priors. Then, we calculate the Bayesian posterior densities and marginal densities of the eight hierarchical models that have conjugate priors. After that, we discuss the relations among the eight analytical marginal densities. Furthermore, we find some relations among the random variables of the marginal densities and the beta densities. Moreover, we discuss random variable generations for the gamma and inverse gamma distributions by using the R software. In addition, some numerical simulations are performed to illustrate four aspects: the plots of marginal densities, the generations of random variables from the marginal density, the transformations of the moment estimators of the hyperparameters of a hierarchical model, and the conclusions about the properties of the eight marginal densities that do not have a closed form. Finally, we illustrate our method by a real data example, in which the original and transformed data are fit by the marginal density with different hyperparameters.

1. Introduction

Mixture distribution refers to a distribution arising from a hierarchical structure. According to [1], a random variable X is said to have a mixture distribution if the distribution of X depends on a quantity that also has a distribution. In general, a hierarchical model will lead to a mixture distribution. In Bayesian analysis, we have a likelihood and a prior, and they naturally assemble into a hierarchical model. Therefore, the likelihood and prior naturally lead to a mixture distribution, which is the marginal distribution of the hierarchical model. Important hierarchical models or mixture distributions include binomial Poisson (also known as the Poisson binomial distribution; see [2,3,4,5,6]), binomial–negative binomial ([1]), Poisson gamma ([7,8,9,10,11,12,13,14]), binomial beta (also known as the beta binomial distribution; see [15,16,17,18,19]), negative binomial beta (also known as the beta negative binomial distribution; see [20,21,22,23,24]), multinomial Dirichlet ([25,26,27,28,29]), Chi-squared Poisson ([1]), normal–normal ([1,30,31,32,33,34]), normal–inverse gamma ([18,30,35,36]), normal–normal inverse gamma ([30,37,38,39]), gamma–gamma ([35]), inverse gamma–inverse gamma ([40]), and many others. See also [1,30,35] and the references therein.

By introducing the new parameter(s), several researchers considered new generalizations of the two-parameter gamma distribution, including [41,42,43,44]. Using the generalized gamma function of [45], ref. [44] defined the generalized gamma-type distribution with four parameters, based on which [46] introduced a new type of three-parameter finite mixture of gamma distributions, which can be regarded as mixing the shape parameter of the gamma distribution by a discrete distribution.

However, the present paper is not in the same direction of the literature in the previous paragraph. This paper is in the line of [35,40]. Reference [35] mixed the rate parameter of a gamma distribution by a gamma distribution. Reference [40] mixed the rate parameter of an inverse gamma distribution by an inverse gamma distribution. In this paper, we mix the scale or rate parameter of a gamma or inverse gamma distribution by a gamma or inverse gamma distribution. It is worth mentioning that the mixture distributions are no longer gamma or inverse gamma distributions.

A positive continuous datum could be modeled by a gamma or inverse gamma distribution $G\left(\nu ,\theta \right)$, $\tilde{G}\left(\nu ,\theta \right)$, $IG\left(\nu ,\theta \right)$, and $\widetilde{IG}\left(\nu ,\theta \right)$, where $\nu$ is a shape parameter and $\theta$ is an unknown parameter of interest and a scale or rate parameter. Suppose that $\theta$ also has a gamma or inverse gamma prior $G\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$. The definitions of G, $\tilde{G}$, $IG$, and $\widetilde{IG}$ will be made clear in Section 2.1. Then, we have $16=4\times 4$ hierarchical models. The new and original things in this paper are described as follows. To the best of our knowledge, there is no literature that considers the 16 hierarchical models as a whole. For the 16 hierarchical models of the gamma and inverse gamma distributions, we found that there are only 8 hierarchical models that have conjugate priors, that is $G-IG$, $G-\widetilde{IG}$, $\tilde{G}-G$, $\tilde{G}-\tilde{G}$, $IG-IG$, $IG-\widetilde{IG}$, $\widetilde{IG}-G$, and $\widetilde{IG}-\tilde{G}$. We then calculated the posterior densities and marginal densities of these eight hierarchical models that have conjugate priors. Moreover, some numerical simulations were performed to illustrate four aspects: the plots of marginal densities, the generations of random variables from the marginal density, the transformations of the moment estimators of the hyperparameters of a hierarchical model, and the conclusions about the properties of the eight marginal densities that do not have a closed form. Finally, we investigated a real data example, in which the original and transformed data, the close prices of the Shanghai, Shenzhen, and Beijing (SSB) A shares, were fit by the marginal density with different hyperparameters.

The rest of the paper is organized as follows. In the next Section 2, we first provide some preliminaries. Then, we discuss some common typical problems for the eight hierarchical models that do not have conjugate priors. After that, we analytically calculate the Bayesian posterior and marginal densities of the eight hierarchical models that have conjugate priors. Furthermore, the relations among the marginal densities and the relations among the random variables of the marginal densities and the beta densities are also given in this section. Moreover, we discuss random variable generations for the gamma and inverse gamma distributions by using the R software. In Section 3, numerical simulations are performed to illustrate four aspects. In Section 4, we illustrate our method by a real data example, in which the original and transformed data are fit by the marginal density with different hyperparameters. Finally, some conclusions and discussions are provided in Section 5.

2. Main Results

In this section, we first provide some preliminaries. Then, we discuss some common typical problems for the eight hierarchical models that do not have conjugate priors. After that, we analytically calculate the Bayesian posterior and marginal densities of the eight hierarchical models that have conjugate priors. Furthermore, we find some relations among the marginal densities. Moreover, we discover some relations among the random variables of the marginal densities and the beta densities. Finally, we discuss random variable generations for the gamma and inverse gamma distributions by using the R software.

2.1. Preliminaries

In this subsection, we provide some preliminaries. We first give the probability density functions (pdfs) of the $G\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$ distributions. After that, we summarize the 16 hierarchical models of the gamma and inverse gamma distributions in a table. Note that there are only 8 hierarchical models that have conjugate priors, and the other 8 hierarchical models do not have conjugate priors.

Suppose that $X\sim G\left(\alpha ,\beta \right)$ and $Y=1/X\sim IG\left(\alpha ,\beta \right)$. The pdfs of X and Y are, respectively, given by

$$\begin{array}{cc}\hfill f_X\left(x|\alpha ,\beta \right)& =\frac{1}{\Gamma \left(\alpha \right){\beta }^{\alpha }}{x}^{\alpha -1}exp\left(-\frac{x}{\beta }\right),x>0,\alpha ,\beta >0,\hfill \\ \hfill f_Y\left(y|\alpha ,\beta \right)& =\frac{1}{\Gamma \left(\alpha \right){\beta }^{\alpha }}{\left(\frac{1}{y}\right)}^{\alpha +1}exp\left(-\frac{1}{\beta y}\right),y>0,\alpha ,\beta >0.\hfill \end{array}$$

Suppose that $X\sim \tilde{G}\left(\alpha ,\beta \right)$ and $Y=1/X\sim \widetilde{IG}\left(\alpha ,\beta \right)$. The pdfs of X and Y are, respectively, given by

$$\begin{array}{cc}\hfill f_X\left(x|\alpha ,\beta \right)& =\frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}{x}^{\alpha -1}exp\left(-\beta x\right),x>0,\alpha ,\beta >0,\hfill \\ \hfill f_Y\left(y|\alpha ,\beta \right)& =\frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}{\left(\frac{1}{y}\right)}^{\alpha +1}exp\left(-\frac{\beta }{y}\right),y>0,\alpha ,\beta >0.\hfill \end{array}$$

The 16 hierarchical models of the gamma and inverse gamma distributions are summarized in

Table 1. The 16 hierarchical models of the gamma and inverse gamma distributions. The 8 hierarchical models that have conjugate priors are highlighted in boxes.

	Prior
Likelihood	$\mathit{G}\left(\mathit{\alpha },\mathit{\beta }\right)$	$\tilde{\mathit{G}}\left(\mathit{\alpha },\mathit{\beta }\right)$	$\mathit{IG}\left(\mathit{\alpha },\mathit{\beta }\right)$	$\widetilde{\mathit{IG}}\left(\mathit{\alpha },\mathit{\beta }\right)$
$G\left(\nu ,\theta \right)$	$G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$	$G\left(\nu ,\theta \right)-\tilde{G}\left(\alpha ,\beta \right)$	$\boxed{G\left(\nu ,\theta \right)-IG\left(\alpha ,\beta \right)}$	$\boxed{G\left(\nu ,\theta \right)-\widetilde{IG}\left(\alpha ,\beta \right)}$
$\tilde{G}\left(\nu ,\theta \right)$	$\boxed{\tilde{G}\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)}$	$\boxed{\tilde{G}\left(\nu ,\theta \right)-\tilde{G}\left(\alpha ,\beta \right)}$	$\tilde{G}\left(\nu ,\theta \right)-IG\left(\alpha ,\beta \right)$	$\tilde{G}\left(\nu ,\theta \right)-\widetilde{IG}\left(\alpha ,\beta \right)$
$IG\left(\nu ,\theta \right)$	$IG\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$	$IG\left(\nu ,\theta \right)-\tilde{G}\left(\alpha ,\beta \right)$	$\boxed{IG\left(\nu ,\theta \right)-IG\left(\alpha ,\beta \right)}$	$\boxed{IG\left(\nu ,\theta \right)-\widetilde{IG}\left(\alpha ,\beta \right)}$
$\widetilde{IG}\left(\nu ,\theta \right)$	$\boxed{\widetilde{IG}\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)}$	$\boxed{\widetilde{IG}\left(\nu ,\theta \right)-\tilde{G}\left(\alpha ,\beta \right)}$	$\widetilde{IG}\left(\nu ,\theta \right)-IG\left(\alpha ,\beta \right)$	$\widetilde{IG}\left(\nu ,\theta \right)-\widetilde{IG}\left(\alpha ,\beta \right)$

. However, there are only eight hierarchical models that have conjugate priors, that is $G-IG$, $G-\widetilde{IG}$, $\tilde{G}-G$, $\tilde{G}-\tilde{G}$, $IG-IG$, $IG-\widetilde{IG}$, $\widetilde{IG}-G$, and $\widetilde{IG}-\tilde{G}$, where the former density represents the likelihood and the latter density represents the prior, that is the likelihood prior. Note that the eight hierarchical models that have conjugate priors are highlighted in boxes in the table. The posterior densities and marginal densities of these eight hierarchical models that have conjugate priors will be calculated later.

For the other eight hierarchical models in Table 1, that is $G-G$, $G-\tilde{G}$, $\tilde{G}-IG$, $\tilde{G}-\widetilde{IG}$, $IG-G$, $IG-\tilde{G}$, $\widetilde{IG}-IG$, and $\widetilde{IG}-\widetilde{IG}$, they do not have conjugate priors. Therefore, the posterior densities and marginal densities of these eight hierarchical models cannot be recognized. In particular, the posterior densities are not gamma distributions, nor the inverse gamma distributions. Moreover, the marginal densities of these eight hierarchical models cannot be obtained in analytical forms, although they are proper densities, that is they integrate to 1.

2.2. Common Typical Problems for the Eight Hierarchical Models That Do Not Have Conjugate Priors

In this subsection, we discuss some common typical problems for the eight hierarchical models that do not have conjugate priors in terms of the hierarchical $G-G$ model.

Let us calculate the posterior density and marginal density for the hierarchical $G-G$ model:

$$\left\{\begin{array}{c}X|\theta \sim G\left(\nu ,\theta \right),\hfill \\ \theta \sim G\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters. It is easy to obtain

$$\begin{array}{cc}\hfill f\left(x|\theta \right)& =\frac{1}{\Gamma \left(\nu \right){\theta }^{\nu }}{x}^{\nu -1}exp\left(-\frac{x}{\theta }\right),\hfill \\ \hfill \pi \left(\theta \right)& =\frac{1}{\Gamma \left(\alpha \right){\beta }^{\alpha }}{\theta }^{\alpha -1}exp\left(-\frac{\theta }{\beta }\right).\hfill \end{array}$$

By the Bayes theorem, we have

$$\begin{array}{cc}\hfill \pi \left(\theta |x\right)& \propto f\left(x|\theta \right)\pi \left(\theta \right)\hfill \\ \hfill & \propto \frac{1}{{\theta }^{\nu }}exp\left(-\frac{x}{\theta }\right){\theta }^{\alpha -1}exp\left(-\frac{\theta }{\beta }\right)\hfill \\ \hfill & ={\theta }^{\alpha -\nu -1}exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right)\hfill \\ \hfill & =exp\left(log{\theta }^{\alpha -\nu -1}\right)exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right)\hfill \\ \hfill & =exp\left[\left(\alpha -\nu -1\right)log\theta -\frac{x}{\theta }-\frac{\theta }{\beta }\right],\hfill \end{array}$$

which cannot be recognized as a familiar density. In particular, it is not a gamma distribution nor an inverse gamma distribution. However, $\pi \left(\theta |x\right)$ belongs to an exponential family (see [1]). Moreover, the marginal density of X is

$$\begin{array}{cc}\hfill m\left(x|\alpha ,\beta ,\nu \right)& ={\int }_0^{\infty }f\left(x,\theta \right)d\theta \hfill \\ \hfill & ={\int }_0^{\infty }f\left(x|\theta \right)\pi \left(\theta \right)d\theta \hfill \\ \hfill & ={\int }_0^{\infty }\frac{1}{\Gamma \left(\nu \right){\theta }^{\nu }}{x}^{\nu -1}exp\left(-\frac{x}{\theta }\right)\frac{1}{\Gamma \left(\alpha \right){\beta }^{\alpha }}{\theta }^{\alpha -1}exp\left(-\frac{\theta }{\beta }\right)d\theta \hfill \\ \hfill & ={\int }_0^{\infty }\frac{1}{\Gamma \left(\nu \right)\Gamma \left(\alpha \right)}\frac{1}{{\beta }^{\alpha }}{x}^{\nu -1}{\theta }^{\alpha -\nu -1}exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right)d\theta \hfill \\ \hfill & =\frac{1}{\Gamma \left(\nu \right)\Gamma \left(\alpha \right)}\frac{1}{{\beta }^{\alpha }}{x}^{\nu -1}{\int }_0^{\infty }{\theta }^{\alpha -\nu -1}exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right)d\theta .\hfill \end{array}$$

(1)

The above integral cannot be analytically calculated, and thus, the marginal density cannot be obtained in its analytical form. However, $m\left(x|\alpha ,\beta ,\nu \right)$ is a proper density, as

$$\begin{array}{cc}\hfill {\int }_0^{\infty }m\left(x|\alpha ,\beta ,\nu \right)dx& ={\int }_0^{\infty }{\int }_0^{\infty }f\left(x,\theta \right)d\theta dx\hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }f\left(x,\theta \right)dxd\theta \hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }f\left(x|\theta \right)\pi \left(\theta \right)dxd\theta \hfill \\ \hfill & ={\int }_0^{\infty }\left[{\int }_0^{\infty }f\left(x|\theta \right)dx\right]\pi \left(\theta \right)d\theta \hfill \\ \hfill & ={\int }_0^{\infty }\pi \left(\theta \right)d\theta \hfill \\ \hfill & =1,\hfill \end{array}$$

since $f\left(x|\theta \right)$ and $\pi \left(\theta \right)$ are proper densities.

In the above calculations, we encountered some common typical problems for the eight hierarchical models that do not have conjugate priors. The problem for $\pi \left(\theta |x\right)$ is that its kernel is

$${\theta }^{\alpha -\nu -1}exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right),$$

which is not a gamma distribution nor an inverse gamma distribution. The problem for $m\left(x|\alpha ,\beta ,\nu \right)$ is that the integral

$${\int }_0^{\infty }{\theta }^{\alpha -\nu -1}exp\left(-\frac{x}{\theta }-\frac{\theta }{\beta }\right)d\theta$$

cannot be analytically calculated, as the integrand cannot be recognized as a familiar kernel. We remark that the above integral and the marginal density given by (1) can be numerically evaluated by an R built-in function integrate(), which is an adaptive quadrature of functions of one variable over a finite or infinite interval.

2.3. The Bayesian Posterior and Marginal Densities of the Eight Hierarchical Models That Have Conjugate Priors

In this subsection, we calculate the Bayesian posterior and marginal densities of the eight hierarchical models that have conjugate priors, that is, $G-IG$, $G-\widetilde{IG}$, $\tilde{G}-G$, $\tilde{G}-\tilde{G}$, $IG-IG$, $IG-\widetilde{IG}$, $\widetilde{IG}-G$, and $\widetilde{IG}-\tilde{G}$. The straightforward, but lengthy calculations can be found in the Supplementary.

Model 1. Suppose that we observe X from the hierarchical $G-IG$ model:

$$\left\{\begin{array}{c}X|\theta \sim G\left(\nu ,\theta \right),\hfill \\ \theta \sim IG\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim IG\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=\frac{\beta }{x\beta +1}.$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\nu }{\left(\frac{1}{x\beta +1}\right)}^{\alpha },x>0,\alpha ,\beta ,\nu >0.$$

Model 2. Suppose that we observe X from the hierarchical $G-\widetilde{IG}$ model:

$$\left\{\begin{array}{c}X|\theta \sim G\left(\nu ,\theta \right),\hfill \\ \theta \sim \widetilde{IG}\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim \widetilde{IG}\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=x+\beta .$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\nu }{\left(\frac{\beta }{x+\beta }\right)}^{\alpha },x>0,\alpha ,\beta ,\nu >0.$$

Model 3. Suppose that we observe X from the hierarchical $\tilde{G}-G$ model:

$$\left\{\begin{array}{c}X|\theta \sim \tilde{G}\left(\nu ,\theta \right),\hfill \\ \theta \sim G\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim G\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=\frac{\beta }{x\beta +1}.$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\nu }{\left(\frac{1}{x\beta +1}\right)}^{\alpha },x>0,\alpha ,\beta ,\nu >0.$$

Model 4. Suppose that we observe X from the hierarchical $\tilde{G}-\tilde{G}$ model:

$$\left\{\begin{array}{c}X|\theta \sim \tilde{G}\left(\nu ,\theta \right),\hfill \\ \theta \sim \tilde{G}\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim \tilde{G}\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=x+\beta .$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\nu }{\left(\frac{\beta }{x+\beta }\right)}^{\alpha },x>0,\alpha ,\beta ,\nu >0.$$

Model 5. Suppose that we observe X from the hierarchical $IG-IG$ model:

$$\left\{\begin{array}{c}X|\theta \sim IG\left(\nu ,\theta \right),\hfill \\ \theta \sim IG\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim IG\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}={\left(\frac{1}{x}+\frac{1}{\beta }\right)}^{-1}.$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\alpha }{\left(\frac{\beta }{x+\beta }\right)}^{\nu },x>0,\alpha ,\beta ,\nu >0.$$

Model 6. Suppose that we observe X from the hierarchical $IG-\widetilde{IG}$ model:

$$\left\{\begin{array}{c}X|\theta \sim IG\left(\nu ,\theta \right),\hfill \\ \theta \sim \widetilde{IG}\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim \widetilde{IG}\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=\frac{1}{x}+\beta .$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\alpha }{\left(\frac{1}{x\beta +1}\right)}^{\nu },x>0,\alpha ,\beta ,\nu >0.$$

Model 7. Suppose that we observe X from the hierarchical $\widetilde{IG}-G$ model:

$$\left\{\begin{array}{c}X|\theta \sim \widetilde{IG}\left(\nu ,\theta \right),\hfill \\ \theta \sim G\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim G\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}={\left(\frac{1}{x}+\frac{1}{\beta }\right)}^{-1}.$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\alpha }{\left(\frac{\beta }{x+\beta }\right)}^{\nu },x>0,\alpha ,\beta ,\nu >0.$$

Model 8. Suppose that we observe X from the hierarchical $\widetilde{IG}-\tilde{G}$ model:

$$\left\{\begin{array}{c}X|\theta \sim \widetilde{IG}\left(\nu ,\theta \right),\hfill \\ \theta \sim \tilde{G}\left(\alpha ,\beta \right),\hfill \end{array}\right.$$

where $\alpha >0$, $\beta >0$, and $\nu >0$ are hyperparameters.

The posterior density of $\theta$ is

$$\pi \left(\theta |x\right)\sim \tilde{G}\left({\alpha }^{*},{\beta }^{*}\right),$$

where

$${\alpha }^{*}=\alpha +\nu \mathrm{and}{\beta }^{*}=\frac{1}{x}+\beta .$$

Moreover, the marginal density of X is

$$m\left(x|\alpha ,\beta ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\alpha }{\left(\frac{1}{x\beta +1}\right)}^{\nu },x>0,\alpha ,\beta ,\nu >0.$$

2.4. Relations among the Marginal Densities

In this subsection, we find some relations among the marginal densities.

It is not difficult to see that the marginal densities of the eight hierarchical models that have conjugate priors can be divided into four types, denoted as $m_i\left(x|\alpha ,\beta ,\nu \right)$$(i=1,2,3,4)$, where

$$\begin{array}{cc}\hfill m_1\left(x|\alpha ,\beta ,\nu \right)& =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\nu }{\left(\frac{1}{x\beta +1}\right)}^{\alpha },(G-IG,\tilde{G}-G),\hfill \\ \hfill m_2\left(x|\alpha ,\beta ,\nu \right)& =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\nu }{\left(\frac{\beta }{x+\beta }\right)}^{\alpha },(G-\widetilde{IG},\tilde{G}-\tilde{G}),\hfill \\ \hfill m_3\left(x|\alpha ,\beta ,\nu \right)& =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\beta }\right)}^{\alpha }{\left(\frac{\beta }{x+\beta }\right)}^{\nu },(IG-IG,\widetilde{IG}-G),\hfill \\ \hfill m_4\left(x|\alpha ,\beta ,\nu \right)& =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x\beta }{x\beta +1}\right)}^{\alpha }{\left(\frac{1}{x\beta +1}\right)}^{\nu },(IG-\widetilde{IG},\widetilde{IG}-\tilde{G}),\hfill \end{array}$$

(2)

for $x>0$ and $\alpha ,\beta ,\nu >0$.

There are some associations among the marginal densities of the hierarchical models with the same likelihood function and different prior distributions, for example $G-IG$ and $G-\widetilde{IG}$, $\tilde{G}-G$ and $\tilde{G}-\tilde{G}$, $IG-IG$ and $IG-\widetilde{IG}$, and $\widetilde{IG}-G$ and $\widetilde{IG}-\tilde{G}$. Suppose that $X\sim m_3\left(x|\alpha ,{\beta }_3,\nu \right)$ and ${\beta }_4=1/{\beta }_3$; it is easy to see that

$$\begin{array}{cc}\hfill m_3\left(x|\alpha ,{\beta }_3,\nu \right)& =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+{\beta }_3}\right)}^{\alpha }{\left(\frac{{\beta }_3}{x+{\beta }_3}\right)}^{\nu }\hfill \\ \hfill & =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x}{x+\frac{1}{{\beta }_4}}\right)}^{\alpha }{\left(\frac{\frac{1}{{\beta }_4}}{x+\frac{1}{{\beta }_4}}\right)}^{\nu }\hfill \\ \hfill & =\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}\left(\frac{1}{x}\right){\left(\frac{x{\beta }_4}{x{\beta }_4+1}\right)}^{\alpha }{\left(\frac{1}{x{\beta }_4+1}\right)}^{\nu }\hfill \\ \hfill & =m_4\left(x|\alpha ,{\beta }_4,\nu \right).\hfill \end{array}$$

Similarly,

$$m_1\left(x|\alpha ,{\beta }_1,\nu \right)=m_2\left(x|\alpha ,{\beta }_2,\nu \right),$$

where ${\beta }_2=1/{\beta }_1$.

Moreover, we can check that

$$m_1\left(x|\alpha ,\beta ,\nu \right)=m_4\left(x|\alpha =\nu ,\beta ,\nu =\alpha \right),$$

and

$$m_2\left(x|\alpha ,\beta ,\nu \right)=m_3\left(x|\alpha =\nu ,\beta ,\nu =\alpha \right).$$

In summary, we have

$$m_1\left(x|\alpha ,\beta ,\nu \right)=m_2\left(x|\alpha ,\beta =\frac{1}{\beta },\nu \right)=m_3\left(x|\alpha =\nu ,\beta =\frac{1}{\beta },\nu =\alpha \right)=m_4\left(x|\alpha =\nu ,\beta ,\nu =\alpha \right).$$

(3)

From (3), we find that the four families of marginal densities are the same, that is,

$$\begin{array}{cc}\hfill & \left\{m_1\left(x|\alpha ,\beta ,\nu \right)|x>0,\alpha ,\beta ,\nu >0\right\}\hfill \\ \hfill & =\left\{m_2\left(x|\alpha ,\beta ,\nu \right)|x>0,\alpha ,\beta ,\nu >0\right\}\hfill \\ \hfill & =\left\{m_3\left(x|\alpha ,\beta ,\nu \right)|x>0,\alpha ,\beta ,\nu >0\right\}\hfill \\ \hfill & =\left\{m_4\left(x|\alpha ,\beta ,\nu \right)|x>0,\alpha ,\beta ,\nu >0\right\}.\hfill \end{array}$$

(4)

In other words, for every $\alpha ,\beta ,\nu >0$, there exists a marginal density in each of the four families, such that the four marginal densities are the same. For example,

$$\begin{array}{cc}\hfill & m_1\left(x|\alpha =1,\beta =2,\nu =3\right)\hfill \\ \hfill & =m_2\left(x|\alpha =1,\beta =\frac{1}{2},\nu =3\right)\hfill \\ \hfill & =m_3\left(x|\alpha =3,\beta =\frac{1}{2},\nu =1\right)\hfill \\ \hfill & =m_4\left(x|\alpha =3,\beta =2,\nu =1\right)\hfill \\ \hfill & =\frac{\Gamma \left(4\right)}{\Gamma \left(1\right)\Gamma \left(3\right)}\left(\frac{1}{x}\right){\left(\frac{2x}{2x+1}\right)}^3{\left(\frac{1}{2x+1}\right)}^1,x>0.\hfill \end{array}$$

In summary, for the 16 hierarchical models of the gamma and inverse gamma distributions, only the 8 hierarchical models that have conjugate priors have analytical marginal densities. Moreover, the eight marginal densities can be divided into four types. Furthermore, in terms of families of marginal densities, there is only one.

2.5. Relations among the Random Variables of the Marginal Densities and the Beta Densities

In this subsection, we discover some relations among the random variables of the marginal densities and the beta densities.

Suppose that $X\sim m_2\left(x|\alpha ,\beta ,\nu \right)$ and $Y\sim Beta(\alpha ,\nu )$, where $\alpha ,\beta ,\nu >0$, and the probability density function (pdf) of Y is given by

$$f_Y\left(y|\alpha ,\nu \right)=\frac{\Gamma \left(\alpha +\nu \right)}{\Gamma \left(\alpha \right)\Gamma \left(\nu \right)}{y}^{\alpha -1}{\left(1-y\right)}^{\nu -1},0<y<1,\alpha ,\nu >0.$$

In the Supplementary, we provide two methods to prove that

$$X=\left(\frac{1}{Y}-1\right)\beta .$$

(5)

Let $X_i\sim m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$, $Y\sim Beta(\alpha ,\nu )$, and $Z\sim Beta(\nu ,\alpha )$. Then, the random variables $X_i\left(i=1,2,3,4\right)$, Y, and Z have the following relationships:

$$\begin{array}{cc}\hfill X_1& =\left(\frac{1}{Y}-1\right)\frac{1}{\beta },\hfill \\ \hfill X_2& =\left(\frac{1}{Y}-1\right)\beta ,\hfill \\ \hfill X_3& =\left(\frac{1}{Z}-1\right)\beta ,\hfill \\ \hfill X_4& =\left(\frac{1}{Z}-1\right)\frac{1}{\beta }.\hfill \end{array}$$

(6)

Alternatively, the marginal densities can be expressed as

$$\begin{array}{cc}\hfill m_1\left(x|\alpha ,\beta ,\nu \right)& =f_{X_1}\left(x\right)=f_{\left(\frac{1}{Beta(\alpha ,\nu )}-1\right)\frac{1}{\beta }}\left(x\right),\hfill \\ \hfill m_2\left(x|\alpha ,\beta ,\nu \right)& =f_{X_2}\left(x\right)=f_{\left(\frac{1}{Beta(\alpha ,\nu )}-1\right)\beta }\left(x\right),\hfill \\ \hfill m_3\left(x|\alpha ,\beta ,\nu \right)& =f_{X_3}\left(x\right)=f_{\left(\frac{1}{Beta(\nu ,\alpha )}-1\right)\beta }\left(x\right),\hfill \\ \hfill m_4\left(x|\alpha ,\beta ,\nu \right)& =f_{X_4}\left(x\right)=f_{\left(\frac{1}{Beta(\nu ,\alpha )}-1\right)\frac{1}{\beta }}\left(x\right),\hfill \end{array}$$

where $f_{X_i}\left(x\right)$ is the pdf of the random variable $X_i\left(i=1,2,3,4\right)$.

2.6. Random Variable Generations for the Gamma and Inverse Gamma Distributions

In this subsection, we discuss the random variable generations for the gamma and inverse gamma distributions $G\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$ by using the R software ([47]).

To generate random variables from the gamma and inverse gamma distributions $G\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$, we need to be careful about whether $\beta$ is a scale or rate parameter of the gamma and inverse gamma distributions.

The definition of the scale parameter can be found in Definition 3.5.4 of [1]. For convenience, we state it as follows. Let $f\left(x\right)$ be any pdf. Then, for any $\sigma >0$, the family of pdfs $\left(1/\sigma \right)f\left(x/\sigma \right)$, indexed by the parameter $\sigma$, is called the scale family with standard pdf $f\left(x\right)$ and $\sigma$ is called the scale parameter of the family.

Alternatively, we can equivalently define the scale parameter by a random variable. Let Z be a standard random variable having the standard pdf $f\left(z\right)$. Note that the standard random variable Z in this paper is a random variable corresponding to the standard pdf $f\left(z\right)$. It does not mean that Z is a random variable with unit variance. Then, $\left\{\sigma Z:\sigma >0\mathrm{and}Z\sim f\left(z\right)\right\}$ is a scale family and $\sigma$ is called the scale parameter of the family. It is easy to show that the pdf of $X=\sigma Z$ is $\left(1/\sigma \right)f\left(x/\sigma \right)$.

For convenience, now let $G\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$ be random variables having the corresponding distributions. For example, $G\left(\alpha ,\beta \right)$ is a random variable having the $G\left(\alpha ,\beta \right)$ distribution.

It is straightforward to show that

$$G\left(\alpha ,\beta \right)=\beta G\left(\alpha ,1\right),$$

by checking that the pdfs of the random variables of the both sides are equal. Therefore,

$$\left\{G\left(\alpha ,\beta \right):\alpha ,\beta >0\right\}=\left\{\beta G\left(\alpha ,1\right):\alpha ,\beta >0\right\}$$

is a scale family with a scale parameter $\beta$ and a standard random variable $G\left(\alpha ,1\right)$.

Moreover, we have

$$\widetilde{IG}\left(\alpha ,\beta \right)=\frac{1}{\tilde{G}\left(\alpha ,\beta \right)}=\frac{1}{G\left(\alpha ,\frac{1}{\beta }\right)}=\frac{1}{\frac{1}{\beta }G\left(\alpha ,1\right)}=\frac{1}{\frac{1}{\beta }\tilde{G}\left(\alpha ,1\right)}=\beta \widetilde{IG}\left(\alpha ,1\right).$$

Hence,

$$\left\{\widetilde{IG}\left(\alpha ,\beta \right):\alpha ,\beta >0\right\}=\left\{\beta \widetilde{IG}\left(\alpha ,1\right):\alpha ,\beta >0\right\}$$

is a scale family with a scale parameter $\beta$ and a standard random variable $\widetilde{IG}\left(\alpha ,1\right)$.

Furthermore, we have

$$\tilde{G}\left(\alpha ,\beta \right)=G\left(\alpha ,\frac{1}{\beta }\right)=\frac{1}{\beta }G\left(\alpha ,1\right)=\frac{1}{\beta }\tilde{G}\left(\alpha ,1\right).$$

Thus,

$$\left\{\tilde{G}\left(\alpha ,\beta \right):\alpha ,\beta >0\right\}=\left\{\frac{1}{\beta }\tilde{G}\left(\alpha ,1\right):\alpha ,\beta >0\right\}$$

is a scale family with

$$\mathrm{scale}=\frac{1}{\beta }\mathrm{or}\beta =\frac{1}{\mathrm{scale}}=\mathrm{rate}$$

and a standard random variable $\tilde{G}\left(\alpha ,1\right)$. Hence, $\beta$ is the rate parameter of the $\tilde{G}\left(\alpha ,\beta \right)$ distribution.

Finally, it is easy to obtain

$$IG\left(\alpha ,\beta \right)=\frac{1}{G\left(\alpha ,\beta \right)}=\frac{1}{\beta G\left(\alpha ,1\right)}=\frac{1}{\beta }IG\left(\alpha ,1\right).$$

Consequently,

$$\left\{IG\left(\alpha ,\beta \right):\alpha ,\beta >0\right\}=\left\{\frac{1}{\beta }IG\left(\alpha ,1\right):\alpha ,\beta >0\right\}$$

is a scale family with

$$\mathrm{scale}=\frac{1}{\beta }\mathrm{or}\beta =\frac{1}{\mathrm{scale}}=\mathrm{rate}$$

and a standard random variable $IG\left(\alpha ,1\right)$. Hence, $\beta$ is the rate parameter of the $IG\left(\alpha ,\beta \right)$ distribution.

In summary, $\beta$ is the scale parameter of the $G\left(\alpha ,\beta \right)$ and $\widetilde{IG}\left(\alpha ,\beta \right)$ distributions, while $\beta$ is the rate parameter of the $\tilde{G}\left(\alpha ,\beta \right)$ and $IG\left(\alpha ,\beta \right)$ distributions, as depicted in Figure 1.

Now, let us generate random variables from the gamma and inverse gamma distributions in the R software. Let $\alpha$ and $\beta$ be positive numbers. Then,

$$G=\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{scale}=\beta \right)$$

is a random variable from the $G\left(\alpha ,\beta \right)$ distribution, where $\mathrm{rgamma}\left(\right)$ is an R built-in function for random number generation for the gamma distribution with parameters shape and scale, and n is the number of observations. Moreover,

$$\tilde{G}=\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{rate}=\beta \right)$$

is a random variable from the $\tilde{G}\left(\alpha ,\beta \right)$ distribution, where $\mathrm{rate}$ is the rate parameter of the gamma distribution, and it is an alternative way to specify the scale. The two parameters have a relation $\mathrm{rate}=1/$scale. Unfortunately, there are no built-in R functions that can directly generate random variables from the $IG\left(\alpha ,\beta \right)$ and $\widetilde{IG}\left(\alpha ,\beta \right)$ distributions. To generate random variables from the $IG\left(\alpha ,\beta \right)$ and $\widetilde{IG}\left(\alpha ,\beta \right)$ distributions, we can utilize the reciprocal relationships of the gamma and inverse gamma distributions:

$$IG\left(\alpha ,\beta \right)=\frac{1}{G\left(\alpha ,\beta \right)}\mathrm{and}\widetilde{IG}\left(\alpha ,\beta \right)=\frac{1}{\tilde{G}\left(\alpha ,\beta \right)}.$$

Therefore,

$$IG=1/\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{scale}=\beta \right)$$

is a random variable from the $IG\left(\alpha ,\beta \right)$ distribution. Similarly,

$$\widetilde{IG}=1/\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{rate}=\beta \right)$$

is a random variable from the $\widetilde{IG}\left(\alpha ,\beta \right)$ distribution.

It is worth mentioning that we do not use

$$I{G}^{\prime }=1/\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{rate}=\beta \right)$$

to generate an $IG\left(\alpha ,\beta \right)$ random variable, although $\beta$ is the rate parameter of the $IG\left(\alpha ,\beta \right)$ distribution, as

$$I{G}^{\prime }=\frac{1}{\tilde{G}\left(\alpha ,\beta \right)}=\widetilde{IG}\left(\alpha ,\beta \right).$$

Similarly, we do not use

$${\widetilde{IG}}^{\prime }=1/\mathrm{rgamma}\left(\mathrm{n}=1,\mathrm{shape}=\alpha ,\mathrm{scale}=\beta \right)$$

to generate an $\widetilde{IG}\left(\alpha ,\beta \right)$ random variable, although $\beta$ is the scale parameter of the $\widetilde{IG}\left(\alpha ,\beta \right)$ distribution, as

$${\widetilde{IG}}^{\prime }=\frac{1}{G\left(\alpha ,\beta \right)}=IG\left(\alpha ,\beta \right).$$

3. Simulations

In this section, we conduct numerical simulations to illustrate four aspects. First, we plot the marginal densities for various hyperparameters. Second, we illustrate that the random variables can be generated from the marginal density by three methods. Third, we illustrate that the transformations of the moment estimators of the hyperparameters of one of the 8 hierarchical models that have conjugate priors can be used to obtain the moment estimators of the hyperparameters of another one of the 8 hierarchical models that have conjugate priors. Fourth, we obtain some conclusions about the properties of the eight marginal densities that do not have a closed form by simulation techniques and numerical integration.

3.1. Marginal Densities for Various Hyperparameters

In this subsection, we plot the marginal densities $m_1\left(x|\alpha ,\beta ,\nu \right)$ given by (2) for various hyperparameters $\alpha$, $\beta$, and $\nu$. We explore how the marginal densities change around $m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$. Other numerical values of the hyperparameters can also be specified. The goal of this subsection is to see what kind of data could be modeled by the marginal densities.

Figure 2 plots the marginal densities $m_1\left(x|\alpha ,\beta ,\nu \right)$ for varied $\alpha =1,2,3,4$, holding $\beta =2$ and $\nu =3$ fixed. From the figure, we can see that as $\alpha$ increases, the peak value of the curve increases and the position of the peak is gradually close to zero. Moreover, the marginal densities are right-skewed.

Figure 3 plots the marginal densities $m_1\left(x|\alpha ,\beta ,\nu \right)$ for varied $\beta =1,2,3,4$, holding $\alpha =1$ and $\nu =3$ fixed. From the figure, we can also see that as $\beta$ increases, the peak value of the curve increases and the position of the peak is gradually close to zero. The marginal densities are also right-skewed.

Figure 4 plots the marginal densities $m_1\left(x|\alpha ,\beta ,\nu \right)$ for varied $\nu =1,2,3,4$, holding $\alpha =1$ and $\beta =2$ fixed. It is observed from the figure that as $\nu$ increases, the peak value of the curve decreases, and the position of the peak is gradually away from zero. Moreover, all the marginal densities are right-skewed.

3.2. Random Variable Generations from the Marginal Density by Three Methods

In this subsection, we generate random variables from the marginal densities $m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$ by three methods. The goal of this subsection is to illustrate the theoretical result that the random variables can be generated from $m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$ by three methods.

Let $\alpha =1$, $\beta =2$, $\nu =3$, and $n=10,000$. Figure 5 shows the histograms of the samples generated from $m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$ by the three methods:

(a)

Method 1: $x_i\left(i=1,\dots ,n\right)$ are generated from the $G-IG$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $IG\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $G\left(\nu ,{\theta }_i\right)$.

(b)

Method 2: $x_i\left(i=1,\dots ,n\right)$ are generated from the $\tilde{G}-G$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $G\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $\tilde{G}\left(\nu ,{\theta }_i\right)$.

(c)

Method 3: $x_i\left(i=1,\dots ,n\right)$ are generated from the transformations of beta random variables. First, we draw $y_i\left(i=1,\dots ,n\right)$ iid from $Beta(\alpha ,\nu )$. After that, $x_i\left(i=1,\dots ,n\right)$ are obtained by the transformations $x_i=\left(1/y_i-1\right)/\beta$.

Let $\alpha =1$, $\beta =1/2$, $\nu =3$, and n = 10,000. Figure 6 shows the histograms of the samples generated from $m_2\left(x|\alpha =1,\beta =1/2,\nu =3\right)=m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$ by the three methods:

(a)

Method 1: $x_i\left(i=1,\dots ,n\right)$ are generated from the $G-\widetilde{IG}$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $\widetilde{IG}\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $G\left(\nu ,{\theta }_i\right)$.

(b)

Method 2: $x_i\left(i=1,\dots ,n\right)$ are generated from the $\tilde{G}-\tilde{G}$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $\tilde{G}\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $\tilde{G}\left(\nu ,{\theta }_i\right)$.

(c)

Method 3: $x_i\left(i=1,\dots ,n\right)$ are generated from the transformations of beta random variables. First, we draw $y_i\left(i=1,\dots ,n\right)$ iid from $Beta(\alpha ,\nu )$. After that, $x_i\left(i=1,\dots ,n\right)$ are obtained by the transformations $x_i=\left(1/y_i-1\right)\beta$.

Let $\alpha =3$, $\beta =1/2$, $\nu =1$, and $n=10,000$. Figure 7 shows the histograms of the samples generated from $m_3\left(x|\alpha =3,\beta =1/2,\nu =1\right)=m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$ by the three methods:

(a)

Method 1: $x_i\left(i=1,\dots ,n\right)$ are generated from the $IG-IG$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $IG\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $IG\left(\nu ,{\theta }_i\right)$.

(b)

Method 2: $x_i\left(i=1,\dots ,n\right)$ are generated from the $\widetilde{IG}-G$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $G\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $\widetilde{IG}\left(\nu ,{\theta }_i\right)$.

(c)

Method 3: $x_i\left(i=1,\dots ,n\right)$ are generated from the transformations of beta random variables. First, we draw $z_i\left(i=1,\dots ,n\right)$ iid from $Beta(\nu ,\alpha )$. After that, $x_i\left(i=1,\dots ,n\right)$ are obtained by the transformations $x_i=\left(1/z_i-1\right)\beta$.

Let $\alpha =3$, $\beta =2$, $\nu =1$, and $n=10,000$. Figure 8 shows the histograms of the samples generated from $m_4\left(x|\alpha =3,\beta =2,\nu =1\right)=m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$ by the three methods:

(a)

Method 1: $x_i\left(i=1,\dots ,n\right)$ are generated from the $IG-\widetilde{IG}$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $\widetilde{IG}\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $IG\left(\nu ,{\theta }_i\right)$.

(b)

Method 2: $x_i\left(i=1,\dots ,n\right)$ are generated from the $\widetilde{IG}-\tilde{G}$ model. First, we draw ${\theta }_i\left(i=1,\dots ,n\right)$ iid from $\tilde{G}\left(\alpha ,\beta \right)$. After that, we draw $x_i\left(i=1,\dots ,n\right)$ independently from $\widetilde{IG}\left(\nu ,{\theta }_i\right)$.

(c)

Method 3: $x_i\left(i=1,\dots ,n\right)$ are generated from the transformations of beta random variables. First, we draw $z_i\left(i=1,\dots ,n\right)$ iid from $Beta(\nu ,\alpha )$. After that, $x_i\left(i=1,\dots ,n\right)$ are obtained by the transformations $x_i=\left(1/z_i-1\right)/\beta$.

From the three plots in each of Figure 5, Figure 6, Figure 7 and Figure 8, we see that the histograms of the samples generated from $m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$ by the three methods are almost identical, which illustrates the theoretical result that the random variables can be generated from $m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$ by the three methods.

3.3. Transformations of the Moment Estimators of the Hyperparameters of One of the Eight Hierarchical Models That Have Conjugate Priors

In this subsection, we illustrate that the transformations of the moment estimators of the hyperparameters of one of the 8 hierarchical models that have conjugate priors can be used to obtain the moment estimators of the hyperparameters of another one of the 8 hierarchical models that have conjugate priors. For example, the transformations of the moment estimators of the hyperparameters of the $IG-IG$ model can be used to obtain the moment estimators of the hyperparameters of the other seven hierarchical models that have conjugate priors ($G-IG$, $G-\widetilde{IG}$, $\tilde{G}-G$, $\tilde{G}-\tilde{G}$, $IG-\widetilde{IG}$, $\widetilde{IG}-G$, and $\widetilde{IG}-\tilde{G}$). The goal of this subsection is to illustrate (3).

Let us illustrate the derivations of the moment estimators of the hyperparameters by Model 5 ($IG-IG$). The first three moments of X are, respectively, given by

$$\mathrm{E}X=\frac{\alpha \beta }{\nu -1},\mathrm{for}\nu >1,$$

$$\mathrm{E}{X}^2=\frac{\left(\alpha +1\right)\alpha {\beta }^2}{\left(\nu -1\right)\left(\nu -2\right)},\mathrm{for}\nu >2,$$

and

$$\mathrm{E}{X}^3=\frac{\left(\alpha +2\right)\left(\alpha +1\right)\alpha {\beta }^3}{\left(\nu -1\right)\left(\nu -2\right)\left(\nu -3\right)},\mathrm{for}\nu >3,$$

which can be obtained by iterated expectation. The moment estimators of $\alpha$, $\beta$, and $\nu$ are calculated by equating the population moments to the sample moments, that is,

$$\begin{array}{cc}\hfill \mathrm{E}X& =\frac{\alpha \beta }{\nu -1}=\frac{1}{n}\sum _{i=1}^n{X}_i=A_1,\hfill \\ \hfill \mathrm{E}{X}^2& =\frac{\left(\alpha +1\right)\alpha {\beta }^2}{\left(\nu -1\right)\left(\nu -2\right)}=\frac{1}{n}\sum _{i=1}^n{X}_i^2=A_2,\hfill \\ \hfill \mathrm{E}{X}^3& =\frac{\left(\alpha +2\right)\left(\alpha +1\right)\alpha {\beta }^3}{\left(\nu -1\right)\left(\nu -2\right)\left(\nu -3\right)}=\frac{1}{n}\sum _{i=1}^n{X}_i^3=A_3,\hfill \end{array}$$

where $A_k=\frac{1}{n}{\sum }_{i=1}^n{X}_i^k$, $k=1,2,3$, is the sample kth moment of X. After some tedious calculations, we obtain

$$\begin{array}{cc}\hfill {\alpha }_5\left(n\right)& =\frac{2A_1^2{A}_3-2A_1{A}_2^2}{-2A_1^2{A}_3+A_1{A}_2^2+A_2{A}_3},\hfill \\ \hfill {\beta }_5\left(n\right)& =\frac{-2A_1^2{A}_3+A_1{A}_2^2+A_2{A}_3}{A_1^2{A}_2+A_1{A}_3-2A_2^2},\hfill \\ \hfill {\nu }_5\left(n\right)& =\frac{A_1^2{A}_2+3A_1{A}_3-4A_2^2}{A_1^2{A}_2+A_1{A}_3-2A_2^2}.\hfill \end{array}$$

The calculations of the moment estimators of the hyperparameters of the eight hierarchical models that have conjugate priors can be found in the Supplementary Materials.

Let $\alpha =4$, $\beta =5$, $\nu =6$, and $n=1,000,000$. The reason why we chose these values is that $\alpha >3$, $\beta >0$, and $\nu >3$ are required in the moment estimations of the hyperparameters. Other numerical values of the hyperparameters can also be specified.

First, let us generate a random sample $\mathit{x}$ from the $IG-IG\left(\alpha =4,\beta =5,\nu =6\right)$ model. The histogram of the sample $\mathit{x}$ with the marginal density $m_3\left(x|\alpha =4,\beta =5,\nu =6\right)$ is plotted in Figure 9. From the figure, we see that the histogram fits the marginal density very well.

Second, let us compute the moment estimators of the hyperparameters of the eight hierarchical models that have conjugate priors from the sample $\mathit{x}$, and the numerical results are summarized in

Table 2. The moment estimators of the hyperparameters of the 8 hierarchical models that have conjugate priors and the transformations of the moment estimators of the hyperparameters of the $IG-IG$ model.

	Moment Estimators of the Hyperparameters			Transformations from the $\mathit{IG}-\mathit{IG}$ Model
	$\alpha$	$\beta$	$\nu$	$\alpha$	$\beta$	$\nu$
Model 1 ($G-IG$)	$5.779$	$0.218$	$4.178$	$5.779$	$0.218$	$4.178$
Model 2 ($G-\widetilde{IG}$)	$5.779$	$4.579$	$4.178$	$5.779$	$4.579$	$4.178$
Model 3 ($\tilde{G}-G$)	$5.779$	$0.218$	$4.178$	$5.779$	$0.218$	$4.178$
Model 4 ($\tilde{G}-\tilde{G}$)	$5.779$	$4.579$	$4.178$	$5.779$	$4.579$	$4.178$
Model 5 ($IG-IG$)	$4.178$	$4.579$	$5.779$	$4.178$	$4.579$	$5.779$
Model 6 ($IG-\widetilde{IG}$)	$4.178$	$0.218$	$5.779$	$4.178$	$0.218$	$5.779$
Model 7 ($\widetilde{IG}-G$)	$4.178$	$4.579$	$5.779$	$4.178$	$4.579$	$5.779$
Model 8 ($\widetilde{IG}-\tilde{G}$)	$4.178$	$0.218$	$5.779$	$4.178$	$0.218$	$5.779$

. Moreover, the transformations (3) of the moment estimators of the hyperparameters of the $IG-IG$ model are also summarized in this table. More precisely, for Model 5 ($IG-IG$) and Model 7 ($\widetilde{IG}-G$), the identical transformation was used ($\alpha$, $\beta$, and $\nu$ are unchanged); for Model 2 ($G-\widetilde{IG}$) and Model 4 ($\tilde{G}-\tilde{G}$), the transformation was the position change of $\alpha$ and $\nu$ ($\alpha \leftrightarrow \nu$); for Model 6 ($IG-\widetilde{IG}$) and Model 8 ($\widetilde{IG}-\tilde{G}$), the transformation was replacing $\beta$ by $1/\beta$ ($\beta \leftrightarrow 1/\beta$); for Model 1 ($G-IG$) and Model 3 ($\tilde{G}-G$), the transformation was both the position change of $\alpha$ and $\nu$ and replacing $\beta$ by $1/\beta$ ($\alpha \leftrightarrow \nu$, $\beta \leftrightarrow 1/\beta$). From the table, we see that the moment estimators of the hyperparameters of the eight hierarchical models that have conjugate priors and the transformations of the moment estimators of the hyperparameters of the $IG-IG$ model are the same.

3.4. The Marginal Densities of the Eight Hierarchical Models That Do Not Have Conjugate Priors

In this subsection, we obtain some conclusions about the properties of the eight marginal densities that do not have a closed form by simulation techniques and numerical integration. We take the hierarchical $G-G$ model as an example. The results for the other seven hierarchical models that do not have conjugate priors are similar, and thus, they are omitted. The results thus obtained are compatible with those in Section 2.2. The goal of this subsection is to obtain some conclusions about the properties of the eight marginal densities that do not have a closed form.

The histogram of the sample with sample size $n=5000$ from the hierarchical $G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$ model with $\alpha =1$, $\beta =2$, and $\nu =3$ and the fit marginal densities are plotted in Figure 10. In Plot (a), the histogram is fit by $m_i\left(x|\alpha =1,\beta =2,\nu =3\right)$,$i=1,2,3,4$. From this plot, we see that the histogram is poorly fit by the marginal densities. In other words, the four marginal densities $m_i\left(x|\alpha =1,\beta =2,\nu =3\right)$,$i=1,2,3,4$, are not the marginal densities for the hierarchical $G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$ model with $\alpha =1$, $\beta =2$, and $\nu =3$. In Plot (b), the histogram is fit by the four marginal densities $m_i\left(x|\alpha ={\widehat{\alpha }}_i,\beta ={\widehat{\beta }}_i,\nu ={\widehat{\nu }}_i\right)$,$i=1,2,3,4$, where ${\widehat{\alpha }}_i$, ${\widehat{\beta }}_i$, and ${\widehat{\nu }}_i$ are the moment estimators of the hyperparameters. Moreover, the histogram is also fit by $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$ given by (1). From this plot, we see that the four marginal densities $m_i\left(x|\alpha ={\widehat{\alpha }}_i,\beta ={\widehat{\beta }}_i,\nu ={\widehat{\nu }}_i\right)$, $i=1,2,3,4$, are the same, as the four curves overlap. It is easy to see that the four marginal densities do not fit the histogram well, especially when x is close to 0. It is worth noting that the marginal density $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$ fits the histogram very well. We would like to point out that the marginal density $m_{G-G}\left(x|\alpha ,\beta ,\nu \right)$ given by (1) was numerically evaluated by an R built-in function integrate().

The fit marginal densities, D-values, and p-values of the one-sample Kolmogorov–Smirnov (KS) test for a sample with sample size $n=5000$ from the hierarchical $G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$ model when $\alpha =1$, $\beta =2$, and $\nu =3$ are summarized in

Table 3. The fit marginal densities, D-values, and p-values of the one-sample KS test for a sample with sample size $n=5000$ from the hierarchical $G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$ model when $\alpha =1$, $\beta =2$, and $\nu =3$.

Fit Marginal Densities	D-Value	p-Value
$m_1\left(x|\alpha =1,\beta =2,\nu =3\right)$	$0.169$	<2.2 $\times {10}^{-16}$
$m_2\left(x|\alpha =1,\beta =2,\nu =3\right)$	$0.259$	<2.2 $\times {10}^{-16}$
$m_3\left(x|\alpha =1,\beta =2,\nu =3\right)$	$0.534$	<2.2 $\times {10}^{-16}$
$m_4\left(x|\alpha =1,\beta =2,\nu =3\right)$	$0.777$	<2.2 $\times {10}^{-16}$
$m_1\left(x|\alpha =11.887,\beta =0.011,\nu =0.724\right)$	$0.036$	4.7 $\times {10}^{-6}$
$m_2\left(x|\alpha =11.887,\beta =91.382,\nu =0.724\right)$	$0.036$	4.7 $\times {10}^{-6}$
$m_3\left(x|\alpha =0.724,\beta =91.382,\nu =11.887\right)$	$0.036$	4. $\times {10}^{-6}$
$m_4\left(x|\alpha =0.724,\beta =0.011,\nu =11.887\right)$	$0.036$	4.7 $\times {10}^{-6}$
$m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$	$0.009$	$0.792$

. In this table, it is worth noting that the D-values and p-values of the four marginal densities $m_i\left(x|\alpha ={\widehat{\alpha }}_i,\beta ={\widehat{\beta }}_i,\nu ={\widehat{\nu }}_i\right)$,$i=1,2,3,4$, are the same, where ${\widehat{\alpha }}_i$, ${\widehat{\beta }}_i$, and ${\widehat{\nu }}_i$ are the moment estimators of the hyperparameters. From the table, we see that the p-values for the first eight marginal densities are all less than $0.05$, which indicates that the sample is not from the first eight marginal densities. However, the p-value for the marginal density $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$ is $0.792>0.05$, which indicates that the sample can be regarded as generated from the marginal density $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$. Moreover, the D-value for the marginal density $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$ is $0.009$, which is the smallest among the D-values in this table. Therefore, the marginal density of the hierarchical $G\left(\nu ,\theta \right)-G\left(\alpha ,\beta \right)$ model when $\alpha =1$, $\beta =2$, and $\nu =3$, $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$, is not from the family of marginal densities (4). It is worth pointing out that the argument stop.on.error = FALSE is needed in integrate() to avoid the error that the “roundoff error is detected in the extrapolation table” when x is a small value, less than, say, $0.035$, when applying the one-sample KS test for $m_{G-G}\left(x|\alpha =1,\beta =2,\nu =3\right)$.

4. A Real Data Example

In this section, we illustrate our method by a real data example.

The data were the close prices of the Shanghai, Shenzhen, and Beijing (SSB) A shares on 6 May 2022. The official name of the A shares is Renminbi (RMB) ordinary shares, that is shares issued by Chinese-registered companies, listed in China, and marked with a face value in RMB for trading and subscription in RMB by individuals and institutions in China (excluding Hong Kong, Macao, and Taiwan). There are 4593 SSB A shares in total; however, 18 of them did not have close prices on that day because of the suspension of trading. Therefore, the sample size of the real data (henceforth, the original data $\mathit{x}$) was 4575 $(=4593-18)$. The range of the original data $\mathit{x}$ was $\left[1.21,1793.00\right]$, meaning the lowest close price was RMB $1.21$ and the highest close price was RMB $1793.00$. Some basic statistical indicators of the original data $\mathit{x}$ are summarized in

Table 4. Some basic statistical indicators of the original data $\mathit{x}$.

Sample Size	Mean	Variance	Standard Deviation	Skewness	Kurtosis
4575	$21.01$	$1907.64$	$43.68$	$18.12$	$621.40$

. From this table, we see that the original data $\mathit{x}$ are right-skewed (skewness = $18.12>0$) and fat-tailed (kurtosis = $621.40>0$).

Moreover, the histogram of the original data $\mathit{x}$ with a fit marginal density $m_1\left(x|\alpha =3.367,\beta =0.010,\nu =0.482\right)$ is plotted in Figure 11. From the figure, we see that the marginal density $m_1\left(x|\alpha =3.367,\beta =0.010,\nu =0.482\right)$ does not fit the original data $\mathit{x}$ well. Furthermore, the one-sample KS test shows that the D-value is $0.30204$ and the p-value (<2.2 $\times {10}^{-16}$) is less than $0.05$, indicating a bad fit of the original data $\mathit{x}$ by the marginal density.

Since the marginal density with the hyperparameters estimated by the moment estimators from the original data $\mathit{x}$ does not fit the original data $\mathit{x}$ well, we would like to transform the original data $\mathit{x}$ to see whether the marginal density with the hyperparameters estimated by the moment estimators from the transformed data fits the transformed data well.

In the following, we encounter some vector operations, which are defined componentwise. For example, let $\mathit{x}=\left(x_1,\dots ,x_n\right)$ and $\mathit{y}=\left(y_1,\dots ,y_n\right)$ be two vectors of the same length and c be a constant. Then,

$$\begin{array}{cc}\hfill \mathit{x}+\mathit{y}& =\left(x_1+y_1,\dots ,x_n+y_n\right),\hfill \\ \hfill \mathit{x}-\mathit{y}& =\left(x_1-y_1,\dots ,x_n-y_n\right),\hfill \\ \hfill \mathit{x}·\mathit{y}& =\left(x_1{y}_1,\dots ,x_n{y}_n\right),\hfill \\ \hfill \mathit{x}/\mathit{y}& =\left(x_1/y_1,\dots ,x_n/y_n\right),\hfill \\ \hfill log\left(\mathit{x}\right)& =\left(log\left(x_1\right),\dots ,log\left(x_n\right)\right),\hfill \\ \hfill \sqrt{\mathit{x}}& =\left(\sqrt{x_1},\dots ,\sqrt{x_n}\right),\hfill \\ \hfill 1/\mathit{x}& =\left(1/x_1,\dots ,1/x_n\right),\hfill \\ \hfill \mathit{x}\pm c& =\left(x_1\pm c,\dots ,x_n\pm c\right).\hfill \end{array}$$

Because $x>0$ in the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$, our ideas for all the transformations that will be used here are to ensure that the transformed data ${\mathit{y}}_i>0$. A modified transformation is

$${\mathit{y}}_2=\mathit{x}-min\left(\mathit{x}\right)+1\times {10}^{-50},$$

where $min\left(\mathit{x}\right)=1.21$ is the minimum of $\mathit{x}$, $\mathit{x}-min\left(\mathit{x}\right)\ge 0$, adding 1 $\times {10}^{-50}$ ensures that ${\mathit{y}}_2>0$, and the range of ${\mathit{y}}_2$ is $\left[1\times {10}^{-50},1,791.79\right]$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=3.388$, ${\beta }_1\left(n\right)=0.009$, and ${\nu }_1\left(n\right)=0.415$. Moreover, the one-sample KS test shows that the D-value is $0.2772$ and the p-value (<2.2 $\times {10}^{-16}$) is less than $0.05$, still indicating a bad fit of the transformed data ${\mathit{y}}_2$ by the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$.

Next, we proceed with a reciprocal transformation:

$${\mathit{y}}_3=\frac{1}{\mathit{x}},$$

where the range of ${\mathit{y}}_3$ is $\left[0.001,0.826\right]$ and ${\mathit{y}}_3>0$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=-93.694$, ${\beta }_1\left(n\right)=-0.123$, and ${\nu }_1\left(n\right)=1.407$. Because the moment estimators of $\alpha$ and $\beta$ are negative, the one-sample KS test cannot be performed. Hence, the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$ cannot be used to fit the transformed data ${\mathit{y}}_3$.

Similarly, we use a modified reciprocal transformation:

$${\mathit{y}}_4=\frac{1}{\mathit{x}}-min\left(\frac{1}{\mathit{x}}\right)+1\times {10}^{-50},$$

so that ${\mathit{y}}_4>0$, where $min\left(1/\mathit{x}\right)=0.0005577245$, and the range of ${\mathit{y}}_4$ is $\left[1\times {10}^{-50},0.826\right]$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=-78.616$, ${\beta }_1\left(n\right)=-0.145$, and ${\nu }_1\left(n\right)=1.388$. Because the moment estimators of $\alpha$ and $\beta$ are negative, the one-sample KS test cannot be performed. Hence, the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$ cannot be used to fit the transformed data ${\mathit{y}}_4$.

After that, we continue with a log transformation:

$${\mathit{y}}_5=log\left(\mathit{x}\right),$$

where the range of ${\mathit{y}}_5$ is $\left[0.191,7.492\right]$ and ${\mathit{y}}_5>0$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=-56.947$, ${\beta }_1\left(n\right)=-0.042$, and ${\nu }_1\left(n\right)=6.081$. Because the moment estimators of $\alpha$ and $\beta$ are negative, the one-sample KS test cannot be performed. Hence, the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$ cannot be used to fit the transformed data ${\mathit{y}}_5$.

Similarly, we carry on with a modified log transformation:

$${\mathit{y}}_6=log\left(\mathit{x}\right)-min\left(log\left(\mathit{x}\right)\right)+1\times {10}^{-50},$$

so that ${\mathit{y}}_6>0$, where $min\left(log\left(\mathit{x}\right)\right)=0.1906204$, and the range of ${\mathit{y}}_6$ is $\left[1\times {10}^{-50},7.301\right]$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=-31.706$, ${\beta }_1\left(n\right)=-0.065$, and ${\nu }_1\left(n\right)=4.866$. Because the moment estimators of $\alpha$ and $\beta$ are negative, the one-sample KS test cannot be performed. Hence, the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$ cannot be used to fit the transformed data ${\mathit{y}}_6$.

Then, we attempt a square root transformation:

$${\mathit{y}}_7=\sqrt{\mathit{x}},$$

where the range of ${\mathit{y}}_7$ is $\left[1.100,42.344\right]$ and ${\mathit{y}}_7>0$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=5.072$, ${\beta }_1\left(n\right)=2.109$, and ${\nu }_1\left(n\right)=33.695$. Moreover, the one-sample KS test shows that the D-value is $0.032274$ and the p-value (=0.0001452) is less than $0.05$, still indicating a bad fit of the transformed data ${\mathit{y}}_7$ by the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$.

Finally, we seek out a modified square root transformation:

$${\mathit{y}}_8=\sqrt{\mathit{x}}-min\left(\sqrt{\mathit{x}}\right)+1\times {10}^{-50},$$

so that ${\mathit{y}}_8>0$, where $min\left(\sqrt{\mathit{x}}\right)=1.100$, and the range of ${\mathit{y}}_8$ is $\left[1\times {10}^{-50},41.244\right]$. The moment estimators of the hyperparameters are ${\alpha }_1\left(n\right)=5.375$, ${\beta }_1\left(n\right)=0.257$, and ${\nu }_1\left(n\right)=3.175$. Moreover, the one-sample KS test shows that the D-value is $0.018468$ and the p-value (=0.08825) is larger than $0.05$, indicating a good fit of the transformed data ${\mathit{y}}_8$ by the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$.

The transformations, the moment estimators of the hyperparameters of the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$, and the D-value and p-value of the KS test are summarized in

Table 5. The transformations, the moment estimators of the hyperparameters of the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$, and the D-value and p-value of the KS test.

Transformations	Moment Estimators			KS Test
	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\nu }$	D-Value	p-Value
${\mathit{y}}_1=\mathit{x}$	$3.367$	$0.010$	$0.482$	$0.3020$	<2.2 $\times {10}^{-16}$
${\mathit{y}}_2=\mathit{x}-min\left(\mathit{x}\right)+1\times {10}^{-50}$	$3.388$	$0.009$	$0.415$	$0.2772$	<2.2 $\times {10}^{-6}$
${\mathit{y}}_3=1/\mathit{x}$	$-93.694$	$-0.123$	$1.407$	NA	NA
${\mathit{y}}_4=1/\mathit{x}-min\left(1/\mathit{x}\right)+1\times {10}^{-50}$	$-78.616$	$-0.145$	$1.388$	NA	NA
${\mathit{y}}_5=log\left(\mathit{x}\right)$	$-56.947$	$-0.042$	$6.081$	NA	NA
${\mathit{y}}_6=log\left(\mathit{x}\right)-min\left(log\left(\mathit{x}\right)\right)+1\times {10}^{-50}$	$-31.706$	$-0.065$	$4.866$	NA	NA
${\mathit{y}}_7=\sqrt{\mathit{x}}$	$5.072$	$2.109$	$33.695$	$0.0323$	$0.0001$
${\mathit{y}}_8=\sqrt{\mathit{x}}-min\left(\sqrt{\mathit{x}}\right)+1\times {10}^{-50}$	$5.375$	$0.257$	$3.175$	$0.0185$	$0.0883$

. In the table, the NA value indicates that the one-sample KS test cannot be performed due to some errors. From the table, we see that only the modified square root transformation can be statistically and significantly used to fit the transformed data ${\mathit{y}}_8$ by the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$. The transformation is

$${\mathit{y}}_8=\sqrt{\mathit{x}}-1.1$$

or

$$\mathit{x}={\left({\mathit{y}}_8+1.1\right)}^2.$$

The histogram of the transformed data ${\mathit{y}}_8$ with a fit marginal density $m_1\left(x|\alpha =5.375,\beta =0.257,\nu =3.175\right)$ is plotted in Figure 12. From the figure, we see that the marginal density $m_1\left(x|\alpha =5.375,\beta =0.257,\nu =3.175\right)$ fits the transformed data ${\mathit{y}}_8$ very well.

5. Conclusions and Discussion

For the 16 hierarchical models of the gamma and inverse gamma distributions, there are only 8 of them that have conjugate priors, that is $G-IG$, $G-\widetilde{IG}$, $\tilde{G}-G$, $\tilde{G}-\tilde{G}$, $IG-IG$, $IG-\widetilde{IG}$, $\widetilde{IG}-G$, and $\widetilde{IG}-\tilde{G}$. We first discussed some common typical problems for the eight hierarchical models that do not have conjugate priors. Then, we calculated the Bayesian posterior densities and marginal densities of the eight hierarchical models that have conjugate priors. After that, we discussed the relations among the eight analytical marginal densities of the hierarchical models that have conjugate priors and found that there is only one family of marginal densities. Furthermore, we found some relations (6) among the random variables of the marginal densities and the beta densities. Moreover, we discussed the random variable generations for the gamma and inverse gamma distributions $G\left(\alpha ,\beta \right)$, $IG\left(\alpha ,\beta \right)$, $\tilde{G}\left(\alpha ,\beta \right)$, and $\widetilde{IG}\left(\alpha ,\beta \right)$ by using the R software.

In the numerical simulations, we plotted the marginal densities $m_1\left(x|\alpha ,\beta ,\nu \right)$ given by (2) for various hyperparameters $\alpha$, $\beta$, and $\nu$. Moreover, we illustrated the theoretical result that the random variables can be generated from $m_i\left(x|\alpha ,\beta ,\nu \right)\left(i=1,2,3,4\right)$ by three methods from the almost identical histograms of the samples. Furthermore, we illustrated that the transformations of the moment estimators of the hyperparameters of one of the 8 hierarchical models that have conjugate priors can be used to obtain the moment estimators of the hyperparameters of another one of the 8 hierarchical models that have conjugate priors. In addition, we obtained some conclusions about the properties of the eight marginal densities that do not have a closed form by simulation techniques and numerical integration, taking the hierarchical $G-G$ model as an example.

We illustrated our method by a real data example, in which the original and transformed data, the close prices of the SSB A shares, were fit by the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$ with different hyperparameters. Because $x>0$ in the marginal density $m_1\left(x|\alpha ,\beta ,\nu \right)$, our ideas for all the transformations that were used were to ensure that the transformed data ${\mathit{y}}_i>0$, $i=1,\dots ,8$.

We emphasize that the 8 hierarchical models that have conjugate priors are suitable for the study of positive, continuous, and right-skewed data, as the marginal densities of the 8 hierarchical models are positive, continuous, and right-skewed distributions. Moreover, the KS test can be used to judge whether the positive, continuous, and right-skewed data have a good fit by the marginal densities.

The eight hierarchical models that have conjugate priors are of particular interest in empirical Bayes analysis, which relies on conjugate prior modeling, where the hyperparameters are estimated from the observations by the marginal densities and the estimated prior is then used as a regular prior in the subsequent inference. See [40,48,49,50] and the references therein.