There are several ways of defining a stable random variable. First, the initial definition of a stable random variable is based on the sums of random variables. A random variable $X$ is said to have a stable distribution if for any $n\ge 2$, a positive number $C_n$ and real number $D_n$ exist such that:

$X_1+X_2+\cdots +X_n\stackrel{\text{def}}{=}{C}_{n}X+D_n,$ (1)

where $X_1,X_2,\cdots ,X_n$ are independent copies of $X$.

Second, stable variables can be defined via their Lévy measure. For any $\alpha <2$, the Lévy measure of a stable process is given by:

$\nu \left(dx\right)=\left(\frac{C_{+}}{x^{1+\alpha }}{1}_{x>0}+\frac{C_{-}}{x^{1+\alpha }}{1}_{x<0}\right)dx,$ (2)

where $1_{x>0}$ is an indicator function. The calculation of the characteristic function of the stable distribution is based on the Lévy-Khintchine formula. The characteristic function of $S$ given by:

$\Phi \left(t;S\right)=\{\begin{array}{ll}\exp \left(i\mu t-{\left|\sigma t\right|}^{\alpha }\left(1-i\beta \left(signt\right)\tan \left(\frac{\pi \alpha }{2}\right)\right)\right),\hfill & \alpha \ne 1\hfill \\ \exp \left(i\mu t-\sigma \left|t\right|\left(1+i\beta \frac{2}{\pi }\left(signt\right)\ln \left|t\right|\right)\right),\hfill & \alpha =1\hfill \end{array}$ (3)

where

$signt=\{\begin{array}{ll}1,\hfill & t>0\hfill \\ 0,\hfill & t=0\hfill \\ -1,\hfill & t<0\hfill \end{array}$ (4)

$0<\alpha \le 2$, $\sigma \ge 0$, $-1\ge \beta \ge 1$, and $\mu \in ℝ$. The parameter $\alpha$ is the index of stability and it controls the behaviour of the left and right tails. When $\alpha$ is close to 2, the tail becomes thin. $\beta$, $\sigma$, and $\mu$ are the skewness, scale, and location parameters, respectively. When a random variable $S$ follows a stable distribution, we denote it by $S\left(\alpha ,\beta ,\sigma ,\mu \right)$.

In Bayesian Synthetic Likelihood (MCMC-BSL), the objective is to simulate from the summary statistic posterior given by

$p\left(\theta |s_y\right)\propto p\left(\theta \right)p\left(s_y|\theta \right)$ (5)

where $\theta \in \Theta \subseteq {ℝ}^p$ is the parameter that requires estimation with corresponding prior distribution $p\left(\theta \right)$. Here, $y$ is the observed data that is subsequently reduced to a summary statistic $s_y=S\left(y\right)$ where $S(\cdot )$ is the summary statistic function. The dimension of the statistic $d$ must be at least the same size as the parameter dimension, i.e. $d\ge p$.

The BSL involves approximating $p\left(s_y|\theta \right)$ with

$p\left(s_y|\theta \right)\approx p_A\left(s_y|\theta \right)=N\left(s_y|\mu \left(\theta \right),\Sigma \left(\theta \right)\right)$ (6)

The mean $\mu \left(\theta \right)$ and covariance $\Sigma \left(\theta \right)$ are not available in closed form but can be estimated via independent model simulations at $\theta$. The procedure involves drawing $x_{1:n}=\left(x_1,\cdots ,x_n\right)$, where $x_i~p\left(\cdot |\theta \right)$ for $i=1,\cdots ,n$, and calculating the summary statistic for each dataset, $s_{1:n}=\left(s_1,\cdots ,s_n\right)$, where $s_i$ is the summary statistic for $x_i$, $i=1,\cdots ,n$. These simulations can be used to estimate $\mu$ and $\Sigma$ unbiasedly

${\mu }_n\left(\theta \right)=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{s}_i,{\Sigma }_n\left(\theta \right)=\frac{1}{n-1}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left(s_i-{\mu }_n\left(\theta \right)\right){\left(s_i-{\mu }_n\left(\theta \right)\right)}^{\text{T}}$ (7)

We can sample from the approximate posterior using MCMC, see Algorithm 1.

The difference between MCMC-semiBSL and MCMC-BSL lies in the treatment of the likelihood function. MCMC-BSL directly assumes the likelihood function to be a Gaussian distribution with unknown parameters, while MCMC-semiBSL employs a Copula model to estimate the likelihood function. The specific steps are as follows: First, kernel density estimation (KDE) is used to estimate each marginal density. Then, a Gaussian Copula function is applied to estimate the joint density. Finally, the resulting joint density is taken as the estimated likelihood function.

Based on kernel density estimation, the marginal density of the j-th component of the summary statistic $s_y$ can be estimated as:

$\hat{f}\left(s_y^j\right)=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{K}_h\left(s_y^j-x_i^j\right),j=1,\cdots ,d$ (8)

where $s_y^j$ represents the j-th component of the summary statistic $s_y$, and $x_i^j$ denotes the j-th component of the i-th simulated data. The kernel density estimator $K_h\left(u\right)$ is given by $K_h\left(u\right)=h^{-1}K\left(u/h\right)$, where $h$ is the bandwidth, and $K(\cdot )$ is the kernel function, satisfying $K(\cdot )\ge 0$ and ${\displaystyle \int K(\cdot )\text{d}x}=1$. There are many kernel functions that satisfy these conditions, and in MCMC-semiBSL, the Gaussian kernel function $K\left(u\right)=1/\sqrt{2\pi }\exp \left(-u^2/2\right)$ is used. The bandwidth is chosen according to Silverman [12] , with $h=0.9n^{-0.2}\min \left(\delta ,\text{IQR}/1.34\right)$, where IQR denotes the interquartile range. The marginal distribution function $\hat{F}\left(s_y^j\right)$ can be directly estimated from the marginal density function.

After obtaining the marginal distribution, MCMC-semiBSL uses a Gaussian Copula function to model the dependence structure between the components of the summary statistics. The Gaussian Copula density function is given by:

$c\left(u\right)=\frac{1}{\sqrt{\det \left(R\right)}}\exp \left\{-\frac{1}{2}{\eta }^{\text{T}}\left(R^{-1}-I_d\right)\eta \right\}$ (9)

where $I_d$ is the d-dimensional identity matrix, $\eta ={\left({\Phi }^{-1}\left(u_1\right),\cdots ,{\Phi }^{-1}\left(u_d\right)\right)}^{\text{T}}$, ${\Phi }^{-1}(\cdot )$ is the inverse of the standard normal distribution, and $u_j=\hat{F}\left(s_y^j\right)$, for $j=1,\cdots ,d$. If the correlation matrix $R$ in the above expression can be estimated, the likelihood function can be estimated as:

$\begin{array}{c}{p}_{semiBSL}\left(s_y|\theta \right)={\displaystyle \int \frac{1}{\sqrt{\det \left(\hat{R}\right)}}\exp \left\{-\frac{1}{2}{\hat{\eta }}_{s_y}^{\text{T}}\left({\hat{R}}^{-1}-I_d\right){\eta }_{s_y}\right\}}\\ \times \underset{j=1}{\overset{d}{{\displaystyle \prod }}}\hat{f}\left(s_y^j\right)\underset{i=1}{\overset{n}{{\displaystyle \prod }}}p\left(S\left(x_i\right)|\theta \right)\text{d}S\left(x_i\right)\cdots S\left(x_n\right)\end{array}$(10)

MCMC-semiBSL estimates the correlation matrix using the Gaussian rank correlation (GRC) method proposed by Boudt et al. in 2012 [13] . According to the GRC method, the $\left(i,j\right)$-th component $R_{i,j}$ of the correlation matrix $R$ can be estimated as:

${\hat{R}}_{i,j}=\frac{{{\displaystyle \sum }}_{k=1}^n{\Phi }^{-1}\left(\frac{r\left(s_k^i\right)}{n+1}\right){\Phi }^{-1}\left(\frac{r\left(s_k^j\right)}{n+1}\right)}{{{\displaystyle \sum }}_{k=1}^n{\Phi }^{-1}{\left(\frac{k}{n+1}\right)}^2}$ (11)

where $r(\cdot ):ℝ\to \mathcal{A},\mathcal{A}\equiv \left\{1,\cdots ,n\right\}$ is the rank function, and ${\Phi }^{-1}(\cdot )$ is the inverse of the standard normal distribution. After estimating the likelihood function, similar to MCMC-BSL, the MCMC algorithm with iterations T is used to sample from $p\left(s_y|\theta \right)$. The specific algorithm procedure is as follows:

Markov Chain Monte Carlo (MCMC) is a stochastic simulation method based on Markov chains, used to estimate the numerical characteristics of complex probability distributions. The foundational algorithm of MCMC was proposed by W. K. Hastings in 1970, known as the Metropolis-Hastings (M-H) algorithm [14] . In the MCMC-semiBSL method, the M-H algorithm is also used. The basic idea is to generate samples from the target distribution by constructing a Markov sampling chain. The algorithm starts from an initial state and iteratively performs two steps: sampling from the proposal distribution and determining whether to accept the sample, until convergence is reached.

As can be seen from the above algorithm, the M-H algorithm requires the specification of both the proposal distribution and the initial state before execution. The convergence rate of the M-H algorithm depends on the choice of these two factors. When the proposal distribution is poorly chosen or the initial state deviates from the true distribution, the convergence rate of the M-H algorithm can be slow, and it may even become trapped in local optima. To address this limitation of the M-H algorithm, Haario et al. introduced the Adaptive Monte Carlo (AM) algorithm in 2001 [15] . This algorithm continuously adjusts the proposal distribution based on the sampled values during execution, thus accelerating the convergence rate of the algorithm. In the classic M-H algorithm, a Gaussian proposal distribution is typically used as follows:

$q\left(\cdot |X^{\left(t-1\right)}\right)=N\left(\cdot |X^{\left(t-1\right)},C\right)$ (12)

where $C$ is an appropriate covariance matrix. The AM algorithm adjusts the covariance matrix $C$ continuously during the execution of the algorithm based on the sampling results. The specific adjustment method is as follows:

Assume that at time $t-1$, the state samples drawn by the algorithm are $X^{\left(0\right)},\cdots ,X^{\left(t-1\right)}$, where $X^{\left(0\right)}$ is the set initial state. The candidate state at the next time step is then drawn from the proposal distribution $q_t\left(\cdot |X^{\left(0\right)},\cdots ,X^{\left(t-1\right)}\right)$:

$q_t\left(\cdot |X^{\left(0\right)},\cdots ,X^{\left(t-1\right)}\right)=N\left(\cdot |X^{\left(t-1\right)},C_t\right)$ (13)

$C_t=s_{d}cov\left(X^{\left(0\right)},\cdots ,X^{\left(t-1\right)}\right)+s_d\epsilon I_d$ (14)

$cov\left(X^{\left(0\right)},\cdots ,X^{\left(k\right)}\right)=\frac{1}{k}\left(\underset{i=1}{\overset{k}{{\displaystyle \sum }}}{X}^{\left(i\right)}{X}^{\left(i\right)\text{T}}-\left(k+1\right){\bar{X}}^{\left(k\right)}{\bar{X}}^{\left(k\right)\text{T}}\right)$ (15)

${\bar{X}}^{\left(k\right)}=\left(\frac{1}{k+1}\right)\underset{i=1}{\overset{k}{{\displaystyle \sum }}}{X}^{\left(i\right)}$ (16)

where $I_d$ is the d-dimensional identity matrix, $\epsilon >0$ is a small constant chosen by the user, and $s_d$ is a scaling parameter that depends only on the dimensionality $d$. In this paper, $s_d$ is set to $s_d={2.4}^2/d$ as suggested by Gelman et al. [16] .

Theorem 1. Let $\pi$ be the density of a target distribution supported on a bounded measurable subset $S\in {ℝ}^d$, and assume that $\pi$ is bounded from above. Let $\epsilon >0$ and let ${\mu }_0$ be any initial distribution on $S$. Then the AM chain $\left(X_n\right)$ simulates properly the target distribution $\pi$: for any bounded and measurable function $f:S\to ℝ$, the equality

$\underset{n\to \infty }{\lim }\frac{1}{n+1}\left(f\left(X_0\right)+f\left(X_1\right)+\cdots +f\left(X_n\right)\right)={\displaystyle {\int }_{S}f\left(x\right)\pi \text{d}x}$ (17)

holds almost surely.

Theorem 2. Assume that the finite-dimensional distributions of the stochastic process ${\left(X_n\right)}_{n=0}^{\infty }$ on the state space $S$, where the sequence of generalized transition probabilities $\left(K_n\right)$ is assumed to satisfy the following three conditions:

(i) There are a fixed integer $k_0$ and a constant $\lambda \in \left(0,1\right)$ such that

$\delta \left({\left(K_{n,{\tilde{y}}_{n-2}}\right)}^{k_0}\right)\le \lambda <1,\text{for}\text{all}{\tilde{y}}_{n-2}\in S^{n-1}\text{and}n\ge 2$ (18)

(ii) There are a fixed probability measure $\pi$ on $S$ and a constant $c_0>0$ such that

$\Vert \pi K_{n,{\tilde{y}}_{n-2}}-\pi \Vert \le \frac{c_0}{n},\text{for}\text{all}{\tilde{y}}_{n-2}\in S^{n-1}\text{and}n\ge 2$ (19)

(iii) We have the following estimate for the operator norm

$\Vert K_{n,{\tilde{y}}_{n-2}}-K_{n+k,{\tilde{y}}_{n+k-2}}\Vert \le c_1\frac{k}{n}$ (20)

where $c_1$ is a fixed positive constant, $n,k\ge 1$ and one assumes that ${\tilde{y}}_{n+k-2}$ is a direct continuation of ${\tilde{y}}_{n-2}$.

Then, if $f:S\to ℝ$ is bounded and measurable, then the equality (17) holds almost surely.

In what follows, the auxiliary constants $c_i,i=2,3,\cdots$ depend on $S,\epsilon$ or $C_o$, and their actual value is irrelevant to our purposes here.

Proof of Theorem 1. According to Theorem 2, it suffices to prove that the AM chain satisfies conditions (i)-(iii). In order to check condition (i), we observe that, directly from definition and by the fact that $S$ is bounded, all the covariances $C=C_n\left(y_0,\cdots ,y_{n-1}\right)$ satisfy the matrix inequality

$0<c_2{I}_d\le C\le c_3{I}_d$ (21)

Hence the corresponding normal densities $N_c\left(\cdot -x\right)$ are uniformly bounded from below on $S$ for all $x\in S$.

We next verify condition (iii). To that end, we assume that $n\ge 2$ and observe that, for given ${\tilde{y}}_{n+k-2}\in S^{n+k-1}$, one has

$\Vert K_{n,{\tilde{y}}_{n-2}}-K_{n+k,{\tilde{y}}_{n+k-2}}\Vert \le 2{\sup }_{y\in S}\left|K_{n,{\tilde{y}}_{n-2}}\left(y;A\right)-K_{n+k,{\tilde{y}}_{n+k-2}}\left(y;A\right)\right|$ (22)

Fix $y\in S$ and introduce $R_1=C_n\left(y_0,\cdots ,y_{n-2},y\right)$ together with $R_2=C_{n+k}\left(y_0,\cdots ,y_{n+k-2},y\right)$. According to the definition,

$\begin{array}{l}\left|K_{n,{\tilde{y}}_{n-2}}\left(y;A\right)-K_{n+k,{\tilde{y}}_{n+k-2}}\left(y;A\right)\right|=\left|M_{R_1}\left(y;A\right)-M_{R_2}\left(y;A\right)\right|\\ \le |{\displaystyle {\int }_{x\in A}\left(N_{R_1}-N_{R_2}\right)\left(x-y\right)\min \left(1,\frac{\pi \left(x\right)}{\pi \left(y\right)}\right)m\text{d}x}\\ +{\chi }_A\left(x\right){\displaystyle {\int }_{x\in {ℝ}^d}\left(N_{R_1}-N_{R_2}\right)\left(x-y\right)\times \left[1-\min \left(1,\frac{\pi \left(x\right)}{\pi \left(y\right)}\right)\right]m\text{d}x}|\\ \le 2{\displaystyle {\int }_{x\in {ℝ}^d}\left|N_{R_1}\left(z\right)-N_{R_2}\left(z\right)\right|\text{d}z}\le 2{\displaystyle {\int }_{x\in {ℝ}^d}\text{d}z}{\displaystyle {\int }_0^1\text{d}s}\left|\frac{\text{d}}{\text{d}s}{N}_{R_1+s\left(R_2-R_1\right)}\left(z\right)\right|\\ \le c_5\Vert R_1-R_2\Vert \end{array}$ (23)

In general, $\Vert C_t-C_{t-1}\Vert \le \frac{c_6}{t}$, for $t>1$. We easily see that $\Vert R_1-R_2\Vert \le c_7\left(S,C_0,\epsilon \right)\frac{k}{n}$, and hence the previous estimates yield (iii).

In order to check condition (ii), fix ${\tilde{y}}_{n-2}\in S^{n-1}$ and denote $C^{*}=C_{n-1}\left(y_0,\cdots ,y_{n-2}\right)$. It follows that $\Vert C^{*}-C_n\left(y_0,\cdots ,y_{n-2},y\right)\Vert \le c_8/n$, where $c_8$ does not depend on $y\in S$. We may therefore proceed exactly as in (22) and (23) to deduce that

$\Vert K_{n,{\tilde{y}}_{n-2}}-M_{C^{*}}\Vert \le \frac{c_9}{n}$ (24)

Since $\pi M_{C^{*}}=\pi$, we obtain

$\Vert \pi -\pi K_{n,{\tilde{y}}_{n-2}}\Vert =\Vert \pi \left(M_{C^{*}}-K_{n,{\tilde{y}}_{n-2}}\right)\Vert \le \frac{c_9}{n}$ (25)

which completes the proof of Theorem 1.

Traditional kernel density estimation does not perform well for heavy-tail distributions. To address this issue, Wand et al. proposed a kernel density estimation method based on the Box-Cox transformation in 1991 [17] . The basic idea of the method is as follows: First, transform the original data so that the transformed data becomes more uniform. Then, apply traditional kernel density estimation to the transformed data. Finally, the resulting density function is inverted and transformed back to the original data’s kernel density estimate. Based on this concept, this paper extends the transformed kernel density estimation to the framework of BSL.

In 1992, Park et al. [18] demonstrated the effectiveness of this method through numerical experiments. Since then, many studies have proposed different transformation methods, such as the power transformation method proposed by Bolance et al. [19] , the Mobius-like mapping proposed by Clements et al. [20] , the Champernowne transformation proposed by Buch et al. [21] , and the inverse Beta(3,3) transformation proposed by Bolance et al. [22] .

Among these, Bolance et al. introduced a quadratic transformation method that combines the Champernowne distribution function and the inverse Beta(3,3) function. This method first approximates the sample data distribution to a uniform distribution using the Champernowne distribution function, and then transforms the data, which has already been transformed once, to approximately follow a Beta(3,3) distribution using the inverse Beta(3,3) distribution function. This method offers a significant advantage over other methods because of the theory proven by Terrell and Scott in 1985 [23] , which states that the Beta(3,3) density function minimizes the integrated mean squared error of kernel density estimation within the function space $\mathcal{A}=\left\{g:g\text{is}\text{a}\text{density}\text{function}\text{and}g\left(x\right)=0,\left|x\right|\ge 1\right\}$. The kernel density estimation method adopted in this paper is an improved method of this quadratic transformation—the Beta kernel density estimation under the generalized Logistic transformation. This method requires fewer parameters and improves upon the original method by addressing the “boundary bias” issue. The specific method is as follows:

First, the original data is approximately transformed into a uniform distribution on the interval [0, 1]. To achieve this, the transformation function is set to be the cumulative distribution function (CDF) of the original data. However, the true distribution function is unknown, so a generalized Logistic function is used to estimate the empirical distribution function. The expression of the generalized Logistic function is as follows:

$Y=\frac{1}{{\left(1+\lambda \gamma {\text{e}}^{-\alpha X}\right)}^{\frac{1}{\lambda }}}\left(\alpha ,\gamma >0;\lambda \ge 0\right)$ (26)

where $\pi =\left(\alpha ,\gamma ,\lambda \right)$ represents the parameter that needs to be estimated.

Next, the sample data $X_1,\cdots ,X_n$ are arranged in ascending order as order statistics $X_{\left(1\right)},\cdots ,X_{\left(n\right)}$, and the values of the empirical distribution function at $X_{\left(1\right)},\cdots ,X_{\left(n\right)}$ are calculated as $Y_{\left(1\right)},\cdots ,Y_{\left(n\right)}$. A regression is performed on the pairs $\left(X_{\left(1\right)},Y_{\left(1\right)}\right),\cdots ,\left(X_{\left(n\right)},Y_{\left(n\right)}\right)$, and the parameter estimates $\pi =\left(\alpha ,\gamma ,\lambda \right)$ are obtained by least squares. Using the generalized Logistic function as an estimate for the distribution function ${\hat{F}}_{\hat{\pi }}\left(x\right)=\frac{1}{{\left(1+\hat{\lambda }\hat{\gamma }{\text{e}}^{-\hat{\alpha }x}\right)}^{1/\hat{\lambda }}}$, the original data is transformed as $Y_i={\hat{F}}_{\hat{\pi }}\left(X_i\right),i=1,\cdots ,n$.

Then, a Beta kernel density estimate is applied to the transformed sample:

${\hat{f}}_Y\left(y\right)=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{K}_{\frac{y}{h}+1,\frac{1-y}{h}+1}\left(Y_i\right)$ (27)

where $K_{p,q}(\cdot )$ is the probability density function of the Beta(p, q) distribution, and $h$ is the bandwidth.

Finally, the kernel density estimate of the transformed data is converted into the kernel density estimate of the original data as follows:

${\hat{f}}_X\left(x\right)=\left|{{\hat{F}}^{\prime }}_{\hat{\pi }}\right|{\hat{f}}_Y\left({\hat{F}}_{\hat{\pi }}\left(x\right)\right)$ (28)

The AM algorithm and transformed kernel density estimation are incorporated into semiBSL, and the new method is named AM-semiTBSL. The detailed algorithmic process is shown in Algorithm 3.