Bayesian inference for Laplace distribution based on complete and censored samples with illustrations

Abstract

In this paper, Bayesian estimates are derived for the location and scale parameters of the Laplace distribution based on complete, Type-I, and Type-II censored samples under different prior settings. Subsequently, Bayesian point and interval estimates, as well as the associated statistical inference, are discussed in detail. The developed methods are then applied to two real data sets for illustrative purposes. Moreover, a detailed Monte Carlo simulation study is carried out for evaluating the performance of the inferential methods developed here. Finally, we provide a brief discussion of the established results to demonstrate their practical utility and present some associated problems of further interest. Overall, this study fills an existing gap in the development of Bayesian inferential techniques for the parameters of the two-parameter Laplace distribution, making this research innovative and offering more investigative implications. It showcases the potential for broader methodological applications of Bayesian inference for complex real-world data sets, especially in scenarios involving different forms of censoring. This research provides a critical tool for statistical analysis in different fields such as engineering and finance, where the Laplace distribution is frequently adopted as a fundamental model.

1. Introduction

In 1774, Pierre-Simon Laplace derived the Laplace distribution, also known as the double exponential distribution [24]. The Laplace distribution arises from the difference of two independent and identically distributed exponential random variables [20]. Compared to the normal distribution, this distribution possesses a steeper peak and heavier tail, and is therefore used for analyzing heavier (than Gaussian) tailed noise data or in robustness studies; it has also found some important applications in deep neural networks [26], analysis of filter-bank multi-carrier with offset quadrature amplitude modulation (a promising waveform for 5G) [23], and incipient fault detection [30]. Interested readers may also refer to [21] for a detailed review and various applications of Laplace distribution.

Several inferential results for the location and scale parameters of Laplace distribution have been developed in the literature; see, for example, [17,20]. Based on ordered observations, linear estimation methods have been discussed in [12,13]. Best linear unbiased estimators of Laplace parameters based on complete and right censored samples have been discussed by Balakrishnan and Chandramouleeswaran [3]. Balakrishnan et al. [6] further discussed confidence intervals (CIs) and hypothesis tests under Type-II censored samples. Bain and Engelhardt [2] developed approximate CIs and Kappenman [18] discussed conditional CI for the location and scale parameters of Laplace distribution. Subsequently, Kappenman [19] derived exact tolerance interval for a Laplace proportion. This work was followed up by Childs and Balakrishnan [8,10], who presented a generalized procedure for obtaining conditional exact CI as well as tolerance interval. Furthermore, maximum likelihood estimates (MLEs) of the two parameters have been derived in [4,9]. Using those estimates, MLE-based inference under different censoring schemes have been discussed extensively in [1,5,15,16,27,31–33].

But, limited research seems to exist for inference under the Bayesian framework for the Laplace distribution. So, in this paper, we develop Bayesian methods for the two parameters of the Laplace distribution under different priors.

Previous studies on estimating the Laplace distribution have primarily focused on frequentist approach. However, there appears to be limited work on the Bayesian framework for the Laplace distribution. Implementing frequentist approach for parameter distribution estimation is challenging when data are sparse or samples are censored, where precise measurements are often hard to obtain. In contrast to frequentist statistics, which typically fit the fixed model parameter with repeated data observations, Bayesian statistics can provide a robust alternative by quantitatively integrating prior knowledge with experimental data to derive posterior distributions [25]. This adaptability is crucial for making reliable inference given suitable prior settings from incomplete data sets, a common issue in fields such as engineering and finance, where the Laplace distribution is used extensively due to its heavy tails [28]. In addition, use of the Bayesian approach helps to alleviate the frequently criticized opaque subjectivity inherent in frequentist interpretations of P-values and confidence intervals, providing a natural way of quantifying uncertainty [25,28].

Therefore, given the strengths of Bayesian methods and the significant applications of Laplace distribution in different fields, this paper extends the Bayesian framework to the two parameters of the Laplace distribution. In practice, Bayesian techniques handle the subjectivity of statistical analysis by employing a range of prior distributions [28]. The choice of prior often depends on available prior knowledge independent of the experimental data. For instance, a uniform distribution might be used as a noninformative prior when specific prior information is lacking. The noninformative prior simply incorporates vague or general information about the parameters with the most limited impact on the posterior distribution. While noninformative prior is an objective choice that allows the data itself to dominate the posterior, informative priors would contain more meaningful knowledge about the parameters, enhancing the probability of parameters falling within specific intervals and simultaneously reducing outlier probabilities. This flexibility is particularly advantageous when dealing with incomplete data. For a more comprehensive analysis, this study derives posterior distributions for the two-parameter Laplace distribution under various priors, including uniform, normal, and Laplace distributions for the location parameter, and inverse-gamma for the scale parameter. This not only addresses a significant gap that is present in the literature, but also equips practitioners with more robust and flexible inferential tools, enhancing the reliability and applicability of statistical analysis in scenarios comprising incomplete or censored data.

The rest of this paper is organized as follows. In Section 2, we present detailed procedures on Bayesian inference assuming non-informative and inverse gamma priors for location and scale parameters, respectively. In Section 3, adopting a similar approach, we develop results when the prior for the location parameter is taken to be normal and Laplace distributions. We further extend these results to Type-I and Type-II censoring cases in Section 4. In Section 5, the developed inferential results are applied to two real data sets and the performance of the developed results are evaluated by means of Monte Carlo simulations. Finally, some discussion of the developed results and some possible future works are provided in Section 6.

2. Bayesian inference

In this section, we discuss in detail the Bayesian estimation of the two parameters of Laplace distribution. The priors for the location parameter μ and the scale parameter σ are assumed to be independent, following uniform $U(a,b)$ and inverse gamma $\mathrm{IG}(\alpha ,\beta )$ distributions, respectively. The prior for μ used here is a non-informative prior, and we will later consider normal and Laplace priors in the following section. The corresponding marginal densities of the considered priors are

$$\begin{array}{rl}f(\mu )& =\frac{1}{b-a},a\le \mu \le b,\\ f(\sigma )& =\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\sigma }^{-\alpha -1}\exp (-\frac{\beta }{\sigma }),\sigma ,\alpha ,\beta >0,\end{array}$$

where a, b, α and β are hyper-parameters assumed to be fixed.

Based on a random sample of size n from the Laplace $(\mu ,\sigma )$ distribution with density

$$f(x;\mu ,\sigma )=\frac{1}{2\sigma }{e}^{-\frac{|x-\mu |}{\sigma }},-\mathrm{\infty }<x<\mathrm{\infty },$$

let $x_{1:n}<x_{2:n}<\cdots <x_{n:n}$ be the corresponding ordered observations. Suppose $x_{j:n}<\mu \le x_{j+1:n}$ and $\mathbf{x}\mathit{=}\{{\mathit{x}}_{\mathit{1}\mathit{:}\mathit{n}}\mathit{,}{\mathit{x}}_{\mathit{2}\mathit{:}\mathit{n}}\mathit{,}\dots \mathit{,}{\mathit{x}}_{\mathit{n}\mathit{:}\mathit{n}}{\}}^{\mathrm{T}}$. Then, the joint density of $(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}$ is

$$\begin{array}{rl}f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}& \propto \frac{{\beta }^{\alpha }}{2^n(b-a){\sigma }^{n+\alpha +1}\mathrm{\Gamma }(\alpha )}\exp (-\frac{\sum _{i=1}^j(\mu -x_{i:n})}{\sigma }-\frac{\sum _{i=j+1}^n(x_{i:n}-\mu )}{\sigma }-\frac{\beta }{\sigma })\\ & =C{\sigma }^{-(n+\alpha +1)}\exp (-\frac{1}{\sigma }g(\beta ,\mathit{x},j,\mu )),x_{j:n}<\mu \le x_{j+1:n},\end{array}$$ (1)

where C is a constant that depends on a, b, α, β, and sample size n, and $g(\beta ,\mathit{x},j,d)=\beta +\sum _{i=j+1}^n(x_{j:n}-d)+\sum _{i=1}^j(d-x_{j:n})$.

2.1. Marginal distribution of $\mathit{x}$

We will first derive the joint probability density function (PDF) of x and then consider the conditional PDF of the Laplace parameters. By solving the double integral of the joint PDF across the support of μ and σ, the PDF of sample x can be attained. In particular, two assumptions are made to restrict the upper and lower bounds of the location parameter, as a and b. Without loss of any generality, let us assume that $x_{n_1:n}<a\le x_{n_1+1:n}$ and $x_{n_2:n}<b\le x_{n_2+1:n}$ with $0\le n_1\le n_2\le n$ and $x_{0:n}=-\mathrm{\infty }$ and $x_{n+1:n}=\mathrm{\infty }$ throughout the paper. Then, the required integration is divided into two cases, depending on the parity of the sample size n.

Case I: n = 2k + 1, $k\in {\mathbb{Z}}^{+}$ In this case, we have

$$\begin{array}{rl}f(\mathbf{x}\mathit{)}& ={\int }_0^{\mathrm{\infty }}[{\int }_a^{x_{n_1+1:n}}+\cdots +{\int }_{x_{j:n}}^{x_{j+1:n}}+\cdots +{\int }_{x_{n_2:n}}^b]f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}\mathrm{d}\mu \mathrm{d}\sigma \\ & =C\sum _{j=n_1}^{n_2}{\int }_0^{\mathrm{\infty }}\frac{1}{n-2j}{\sigma }^{-(\alpha +n)}[\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}}{\sigma })-\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}}{\sigma })]\mathrm{d}\sigma \\ & =C\mathrm{\Gamma }(\alpha +n-1)\sum _{j=n_1}^{n_2}{h}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{,}\end{array}$$

where

$$h^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{=}[{\mathit{h}}^{\mathit{-}\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}\mathit{(}\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}\mathit{-}{\mathit{h}}^{\mathit{-}\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}\mathit{(}\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}]\mathit{/}\mathit{(}\mathit{n}\mathit{-}\mathit{2}\mathit{j}\mathit{)}$$ (2)

and

$$h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{l}\mathit{)}\mathit{=}\{\begin{array}{ll}\beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-x_{l:n})+{\displaystyle \sum _{i=1}^j(x_{l:n}-x_{i:n})}}& \mathrm{if}{n}_1<l\le n_2,\\ \beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-a)+{\displaystyle \sum _{i=1}^j(a-x_{i:n})}}& \mathrm{if}l=n_1,\\ \beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-b)+{\displaystyle \sum _{i=1}^j(b-x_{i:n})}}& \mathrm{if}l=n_2+1.\end{array}$$

Case II: n = 2k, $k\in {\mathbb{Z}}^{+}$ In this case, we have

$$\begin{array}{rl}f(\mathbf{x}\mathit{)}& =C\sum _{\begin{array}{c}j=n_1\\ j\ne k\end{array}}^{n_2}\frac{1}{n-2j}{\int }_0^{\mathrm{\infty }}{\sigma }^{-(\alpha +n)}[\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}}{\sigma })-\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}}{\sigma })]\mathrm{d}\sigma \\ & +\mathbb{I}(n_1\le k\le n_2)C{\int }_0^{\mathrm{\infty }}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})}{\sigma }){\sigma }^{-(\alpha +n+1)}(x_{k+1:n}-x_{k:n})\mathrm{d}\sigma \\ & =C\mathrm{\Gamma }(\alpha +n-1)[\sum _{\begin{array}{c}j=n_1\\ j\ne k\end{array}}^{n_2}{h}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{+}\mathbb{I}\mathit{(}{\mathit{n}}_{\mathit{1}}\le \mathit{k}\le {\mathit{n}}_{\mathit{2}}\mathit{)}\frac{({\mathit{x}}_{\mathit{k}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}\mathit{-}{\mathit{x}}_{\mathit{k}\mathit{:}\mathit{n}})\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}{{\left(\mathit{g}(\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}^{\alpha \mathit{+}\mathit{n}}}],\end{array}$$

where $\mathbb{I}(\cdot )$ denotes an indictor function.

For simplicity and convenience, let us define

$$w=\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}{h}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{+}\mathbb{I}\mathit{(}\mathit{n}\mathit{=}\mathit{2}\mathit{k}\mathit{,}{\mathit{n}}_{\mathit{1}}\le \mathit{k}\le {\mathit{n}}_{\mathit{2}}\mathit{)}\left[\frac{({\mathit{x}}_{\mathit{k}\mathit{:}\mathit{n}}\mathit{-}{\mathit{x}}_{\mathit{k}\mathit{:}\mathit{n}})\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}{{\left(\mathit{g}(\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}^{\alpha \mathit{+}\mathit{n}}}\right]\mathit{.}$$

Note here that we use the notation $\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}$ instead of $\sum _{\begin{array}{c}j=n_1\\ j\ne k\end{array}}^{n_2}$ so that the results for cases n = 2k and n = 2k + 1 can be combined together. For an odd sample size, $2j$ will never equal n. Hence, the result coincides with the result in Case I. But, for even sample size, $2j\ne n$ is equivalent to $j\ne k$, and thus the result, coincides with the result in Case II.

Now, the marginal PDF for sample x is given as

$$f(\mathbf{x}\mathit{)}\mathit{=}\mathit{C\Gamma }\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}\mathit{w}\mathit{.}$$

In particular, this PDF for sample x also holds true when $n_1=n_2$.

2.2. Posterior distributions

From the above result, we can readily obtain the joint posterior PDF for $(\mu ,\sigma )|\mathit{x}$ as

$$f(\mu ,\sigma ;\mathbf{x}\mathit{)}\mathit{=}\frac{\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}\mathit{IG}\mathit{(}\sigma \mathit{;}\alpha \mathit{+}\mathit{n}\mathit{,}\sum _{\mathit{i}\mathit{=}\mathit{1}}^{\mathit{n}}|{\mathit{x}}_{\mathit{i}\mathit{:}\mathit{n}}\mathit{-}\mu |\mathit{+}\beta \mathit{)}}{{(\sum _{\mathit{i}\mathit{=}\mathit{1}}^{\mathit{n}}|{\mathit{x}}_{\mathit{i}\mathit{:}\mathit{n}}\mathit{-}\mu |\mathit{+}\beta )}^{\alpha \mathit{+}\mathit{n}}\mathit{w}}\mathit{,}\mathit{a}\le \mu \le \mathit{b}\mathit{,}\sigma \mathit{>}\mathit{0}\mathit{,}$$

where $\mathrm{IG}(x;\alpha ,\beta )$ denote the PDF of an inverse gamma distribution with shape parameter α and scale parameter β.

Then, by integrating with respect to σ, we obtain the posterior PDF of $\mu |\mathit{x}$ to be

$$\begin{array}{rl}f(\mu ;\mathbf{x}\mathit{)}& ={\int }_{{\mathrm{\Theta }}_{\sigma }}f(\mu ,\sigma ;\mathbf{x}\mathit{)}\mathrm{d}\sigma \\ & ={\int }_0^{\mathrm{\infty }}\frac{(\alpha +n-1)\mathrm{IG}(\sigma ;\alpha +n,\sum _{i=1}^n|x_{i:n}-\mu |+\beta )}{{(\sum _{i=1}^n|x_{i:n}-\mu |+\beta )}^{\alpha +n}w}\mathrm{d}\sigma \\ & =\frac{(\alpha +n-1)}{{(\sum _{i=1}^n|x_{i:n}-\mu |+\beta )}^{\alpha +n}w}.\end{array}$$

Likewise, we can also derive the posterior PDF of $\sigma |\mathit{x}$ to be

$$\begin{array}{rl}f(\sigma ;\mathbf{x}\mathit{)}& ={\int }_a^{x_{n_1+1:n}}f(\mu ,\sigma ;\mathbf{x}\mathit{)}\mathrm{d}\mu \mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{\mathit{j}\mathit{:}\mathit{n}}}^{{\mathit{x}}_{\mathit{j}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}}\mathit{f}\mathit{(}\mu \mathit{,}\sigma \mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mu \mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{{\mathit{n}}_{\mathit{2}}\mathit{:}\mathit{n}}}^{\mathit{b}}\mathit{f}\mathit{(}\mu \mathit{,}\sigma \mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mu \\ & =w^{-1}\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}\{\frac{\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}}{\sigma }){\sigma }^{-\alpha -n}}{\mathrm{\Gamma }(\alpha +n-1)(n-2j)}-\frac{\exp (-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}}{\sigma }){\sigma }^{-\alpha -n}}{\mathrm{\Gamma }(\alpha +n-1)(n-2j)}\}\\ & +\mathbb{I}(n_1\le k\le n_2,n=2k)w^{-1}\frac{(\alpha +n-1)(x_{k+1:n}-x_{k:n})}{{\left[g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right]}^{(\alpha +n)}}\\ & \times \mathrm{IG}(\sigma ;\alpha +n,g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}))\\ & =\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}[c_{1j}\mathrm{IG}(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)})-c_{2j}\mathrm{IG}(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)})]\\ & +c_3\mathrm{IG}(\sigma ;\alpha +n,g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})),\end{array}$$

where $c_{1j}=\frac{1}{w(n-2j)[h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}{\mathit{]}}^{\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}}$, $c_{2j}=\frac{1}{w(n-2j)[h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}{\mathit{]}}^{\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}}$, and $c_3=\mathbb{I}(n_1\le k\le n_2,n=2k)\frac{(\alpha +n-1)(x_{k+1:n}-x_{k:n})}{w[g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{)}{\mathit{]}}^{\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{)}}}$.

As $\alpha +n-1$, $h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}$, $h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}$ and $g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{)}$ are all positive for $n_1\le j\le n_2$, the posterior density of $\sigma |\mathit{x}$ is indeed a generalized mixture of inverse-gamma distributions with components $\mathrm{IG}(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}\mathit{)}$, $\mathrm{IG}(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}\mathit{)}$ and $\mathrm{IG}(\sigma ;\alpha +n,g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{)}\mathit{)}$ with corresponding mixing probabilities $c_{1j}$, $-c_{2j}$ and $c_3$, for j from $n_1$ to $n_2$.

2.3. Bayesian posterior point estimation

In this subsection, we will develop three commonly used Bayesian point estimates using posterior mode, posterior mean and posterior median.

2.3.1. Posterior mode

The parameter value that maximizes the posterior PDF for $\mu |\mathit{x}$ is

$$\begin{array}{rl}{\hat{\mu }}_{\mathrm{MD}}& =\underset{\mu }{\mathrm{Argmax}}f(\mu ;\mathbf{x}\mathit{)}\\ & =\underset{\mu }{\mathrm{Argmax}}\left(\frac{(\alpha +n-1)}{(\sum _{i=1}^n|x_{i:n}-\mu |+\beta {)}^{\alpha +n}}{w}^{-1}\right)\\ & =\underset{\mu }{\mathrm{Argmin}}(\sum _{i=1}^n|x_{i:n}-\mu |)\\ & =\{\begin{array}{ll}{x}_{k+1:n}& \mathrm{if}n=2k+1,\\ {\displaystyle \frac{1}{2}(x_{k:n}+x_{k+1:n})}& \mathrm{if}n=2k.\end{array}\end{array}$$

For the case when n = 2k, actually the posterior mode estimate could be any value in the interval $[x_{k:n},x_{k+1:n}]$. Here, we use $\frac{1}{2}(x_{k:n}+x_{k+1:n})$ for convenience. In particular, the numerical value of the posterior mode estimate will be equal to the MLE.

In a similar vein, upon solving the equation of the first derivative of the posterior PDF being zero, the posterior mode estimate for the scale parameter σ can be obtained. In this regard, the derivative of the posterior PDF of $\sigma |\mathit{x}$ is given by

$$\begin{array}{rl}{f}^{\prime }(\sigma ;\mathbf{x}\mathit{)}& =\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}[c_{1j}I{G}^{\prime }(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)})-c_{2j}I{G}^{\prime }(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)})]\\ & +c_{3}I{G}^{\prime }(\sigma ;\alpha +n,g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})),\end{array}$$

where $I{G}^{\prime }(\sigma ;\alpha ,\beta )$ denotes the first derivative of the PDF with respective to σ, of the form

$$I{G}^{\prime }(\sigma ;\alpha ,\beta )=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}[\beta -(\alpha +1)\sigma ]{\sigma }^{-(\alpha +3)}\exp (-\frac{\beta }{\sigma }).$$

2.3.2. Posterior mean

The posterior mean estimator for μ is

$$\begin{array}{rl}{\hat{\mu }}_E& =E(\mu ;\mathbf{x}\mathit{)}\\ & ={\int }_a^{x_{n_1+1:n}}\mathrm{\mu f}(\mu ;\mathbf{x}\mathit{)}\mathrm{d}\mu \mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{\mathit{j}\mathit{:}\mathit{n}}}^{{\mathit{x}}_{\mathit{j}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}}\mu \mathit{f}\mathit{(}\mu \mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mu \mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{{\mathit{n}}_{\mathit{2}}\mathit{:}\mathit{n}}}^{\mathit{b}}\mu \mathit{f}\mathit{(}\mu \mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mu \\ & =[{\int }_a^{x_{n_1+1:n}}+\cdots +{\int }_{x_{j:n}}^{x_{j+1:n}}+\cdots +{\int }_{x_{n_2:n}}^b]\mu \frac{(\alpha +n-1)}{{(\sum _{i=1}^n|x_{i:n}-\mu |+\beta )}^{\alpha +n}w}\mathrm{d}\mu \\ & =w^{-1}\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}\frac{1}{n-2j}{[h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mu \mathit{)}]}^{-\alpha -n+1}\{\mu -\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mu \mathit{)}}{(\alpha +n-2)(n-2j)}\}{|}_{\mu =\mu (j)}^{\mu =\mu (j+1)}\\ & +\mathbb{I}(n=2k,n_1\le k\le n_2)w^{-1}\frac{\alpha +n-1}{2{\left(g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}^{\alpha +n}}(x_{k+1:n}^2-x_{k:n}^2),\end{array}$$

where

$$\mu (j)=\{\begin{array}{ll}{x}_{j:n}& \mathrm{if}{n}_1<j\le n_2,\\ a& \mathrm{if}j=n_1,\\ b& \mathrm{if}j=n_2+1.\end{array}$$

Using the fact that the marginal distribution of the scale parameter is a generalized mixture of inverse-gamma distributions, the posterior mean estimate of σ is determined readily as the weighted average of the means of the corresponding inverse-gamma distributions. We thus have

$${\hat{\sigma }}_E=\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}\{c_{1j}\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}}{(\alpha +n-2)}-c_{2j}\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}}{(\alpha +n-2)}\}+c_3\frac{\left(g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}{(\alpha +n-1)}$$

for $\alpha +n-1>1$.

2.3.3. Posterior median

For finding the posterior median estimate, we first obtain the posterior CDF for $\mu |\mathbf{x}$ as follows:

$$\begin{array}{rl}F(\mu ;\mathbf{x}\mathit{)}& ={\int }_a^{x_{n_1+1:n}}f(s;\mathbf{x}\mathit{)}\mathrm{d}\mathit{s}\mathit{+}{\int }_{{\mathit{x}}_{{\mathit{n}}_{\mathit{1}}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}}^{{\mathit{x}}_{{\mathit{n}}_{\mathit{1}}\mathit{+}\mathit{2}\mathit{:}\mathit{n}}}\mathit{f}\mathit{(}\mathit{s}\mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mathit{s}\mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{\mathit{m}\mathit{:}\mathit{n}}}^{\mu }\mathit{f}\mathit{(}\mathit{s}\mathit{;}\mathbf{x}\mathit{)}\mathrm{d}\mathit{s}\\ & =[{\int }_a^{x_{n_1+1:n}}+{\int }_{x_{n_1+1:n}}^{x_{n_1+2:n}}+\cdots +{\int }_{x_{m:n}}^{\mu }]\frac{(\alpha +n-1)}{w(\sum _{i=1}^n|x_{i:n}-s|+\beta {)}^{\alpha +n}}\mathrm{d}s\\ & =-\left\{w^{-1}\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^m\frac{1}{(2j-n)}{[g(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{s}\mathit{)}]}^{-(\alpha +n-1)}\right\}{|}_{s={\mu }^{\ast }(j)}^{s={\mu }^{\ast }(j+1)}\\ & =-w^{-1}\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^m{h}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{-}\mathbb{I}\mathit{(}\mathit{2}\mathit{m}\ne \mathit{n}\mathit{)}{\mathit{w}}^{\mathit{-}\mathit{1}}\frac{\mathit{1}}{\mathit{2}\mathit{m}\mathit{-}\mathit{n}}\mathit{h}\mathit{(}\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{s}\mathit{)}{|}_{\mathit{s}\mathit{=}{\mathit{x}}_{\mathit{m}\mathit{:}\mathit{n}}}^{\mathit{s}\mathit{=}\mu }\\ & -\mathbb{I}(n=2k,n_1\le k<m)w^{-1}\frac{[x_{k+1:n}-x_{k:n}](\alpha +n-1)}{{\left(g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}^{\alpha +n}}\\ & -\mathbb{I}(n=2k,k=m)w^{-1}\frac{[\mu -x_{k:n}](\alpha +n-1)}{{\left(g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})\right)}^{\alpha +n}},x_{m:n}<\mu \le x_{m+1:n},\end{array}$$

where $h^{\ast }(\cdot ,\cdot ,\cdot )$ is as defined in (2) and

$${\mu }^{\ast }(j)=\{\begin{array}{ll}{x}_{j:n}& \mathrm{if}{n}_1<j\le m,\\ a& \mathrm{if}j=n_1,\\ b& \mathrm{if}j=m+1.\end{array}$$

In practice, considering $a\le \mu \le b$ and $F({\hat{\mu }}_{\mathrm{ME}})=\frac{1}{2}$, we can derive the exact value of the posterior median estimate leveraging user-defined functions detailing the CDF as well as the built-in function in a software.

On the other hand, due to the fact that the marginal distribution of the scale parameter $\sigma |\mathit{x}$ is a generalized mixture of inverse gamma distributions, the CDF of $\sigma |\mathbf{x}$ will also be a generalized mixture of inverse-gamma CDFs of the form

$$\begin{array}{rl}F(\sigma ;\mathbf{x}\mathit{)}& =\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}[c_{1j}I{G}^{\ast }(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)})-c_{2j}I{G}^{\ast }(\sigma ;\alpha +n-1,h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)})]\\ & +c_{3}I{G}^{\ast }(\sigma ;\alpha +n,g(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0})),\end{array}$$

where $I{G}^{\ast }(\sigma ;\alpha ,\beta )$ denotes the CDF of an inverse gamma distribution, evaluated at σ, with shape parameter α and scale parameter β.

2.4. Bayesian posterior interval estimation

We can readily develop credible intervals with $100(1-\alpha )\mathrm{\%}$ confidence level as an interval with upper bound being the value of the $100(1-\frac{\alpha }{2})$th percentile and the lower bound being the value of the $100\frac{\alpha }{2}$th percentile of the derived posterior distributions. For instance, we can get a 95 $\mathrm{\%}$ credible interval by finding the 2.5th percentile and the 97.5th percentile of the posterior distributions.

3. Normal and Laplace priors for μ

The last section developed Bayesian inference when the prior for the location parameter μ was taken to be an uniform distribution. In this section, we present the results for two other prior settings.

3.1. $\mu \sim N(a,b)$ $\mathrm{\&}$ $\sigma \sim \mathrm{IG}(\alpha ,\beta )$

In this case, the prior densities for the two Laplace parameters are considered as

$$\begin{array}{rl}f(\mu )& =\frac{1}{\sqrt{2\pi }b}\exp [-\frac{(\mu -a{)}^2}{2b^2}],-\mathrm{\infty }<\mu <\mathrm{\infty },-\mathrm{\infty }<a<\mathrm{\infty },b>0,\\ f(\sigma )& =\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\sigma }^{-\alpha -1}\exp (-\frac{\beta }{\sigma }),\sigma >0,\alpha >0,\beta >0,\end{array}$$

where a, b, α and β are fixed constants.

Assuming that μ and σ are independent, the joint PDF of $(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}$ is given by

$$\begin{array}{rl}f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}& =\frac{1}{\sqrt{2\pi }b}{\left(\frac{1}{2}\right)}^n\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\sigma }^{-\alpha -n-1}\exp [-\frac{(\mu -a{)}^2}{2b^2}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\\ & =C_n{\sigma }^{-\alpha -n-1}\exp [-\frac{(\mu -a{)}^2}{2b^2}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }],\end{array}$$

where $C_n=\frac{1}{\sqrt{2\pi }b}(\frac{1}{2}{)}^n\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}$.

Then, the PDF of $\mathbf{x}$ can be obtained as

$$\begin{array}{rl}f(\mathbf{x}\mathit{)}& ={\int }_0^{\mathrm{\infty }}\{{\int }_{x_{0:n}}^{x_{1:n}}f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}\mathrm{d}\mu \mathit{+}\cdots \mathit{+}{\int }_{{\mathit{x}}_{\mathit{n}\mathit{:}\mathit{n}}}^{{\mathit{x}}_{\mathit{n}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}}\mathit{f}\mathit{(}\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}\mathrm{d}\mu \}\mathrm{d}\sigma \\ & =\sum _{j=0}^n{\int }_0^{\mathrm{\infty }}{\int }_{x_{j:n}}^{x_{j+1:n}}{C}_n{\sigma }^{-\alpha -n-1}\exp [-\frac{(\mu -a{)}^2}{2b^2}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\mathrm{d}\mu \mathrm{d}\sigma \\ & =\sum _{j=0}^n{\int }_0^{\mathrm{\infty }}{C}_n{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathit{x},j,0)}{\sigma }+\frac{{[a+\frac{b^2(n-2j)}{\sigma }]}^2-a^2}{2b^2})\\ & \times {\int }_{x_{j:n}}^{x_{j+1:n}}\exp [-\frac{{(\mu -[a+\frac{b^2(n-2j)}{\sigma }])}^2}{2b^2}]\mathrm{d}\mu \mathrm{d}\sigma \\ & =\sqrt{2\pi }b{C}_n\sum _{j=0}^n{\int }_0^{\mathrm{\infty }}{\sigma }^{-\alpha -n-1}k(\sigma ,a,b,\mathit{x},n,j)\mathrm{d}\sigma ,\end{array}$$

where

$$\begin{array}{rl}k(\sigma ,a,b,\mathit{x},n,j)& =\exp (-\frac{g(\beta ,\mathit{x},j,0)}{\sigma }+\frac{{[a+\frac{b^2(n-2j)}{\sigma }]}^2-a^2}{2b^2})\\ & \times \{\mathrm{\Phi }\left(\frac{x_{j+1:n}-[a+\frac{b^2(n-2j)}{\sigma }]}{b}\right)-\mathrm{\Phi }\left(\frac{x_{j:n}-[a+\frac{b^2(n-2j)}{\sigma }]}{b}\right)\}.\end{array}$$

Subsequently, the posterior densities of $\mu |\mathit{x}$ and $\sigma |\mathit{x}$ can be obtained as follows:

$$\begin{array}{rl}f(\mu ;\mathbf{x}\mathit{)}& =\frac{1}{f(\mathit{x})}{\int }_0^{\mathrm{\infty }}{C}_n{\sigma }^{-\alpha -n-1}\exp [-\frac{(\mu -a{)}^2}{2b^2}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\mathrm{d}\sigma \\ & =\frac{C_n}{f(\mathbf{x}\mathit{)}}\mathrm{\Gamma }(\alpha +n){[\sum _{i=1}^n|x_{i:n}-\mu |+\beta ]}^{-(\alpha +n)}\exp [-\frac{(\mu -a{)}^2}{2b^2}]\end{array}$$

and

$$\begin{array}{rl}f(\sigma ;\mathbf{x}\mathit{)}& =\frac{C_n}{f(\mathit{x})}\sum _{j=0}^n{\int }_{x_{j:n}}^{x_{j+1:n}}{\sigma }^{-\alpha -n-1}\exp [-\frac{h(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mu \mathit{)}}{\sigma }-\frac{(\mu -a{)}^2}{2b^2}]\mathrm{d}\mu \\ & =\frac{\sqrt{2\pi }b{C}_n}{f(\mathit{x})}\sum _{j=0}^n{\sigma }^{-\alpha -n-1}k(\sigma ,a,b,\mathit{x},n,j).\end{array}$$

3.2. $\mu \sim L(a,b)$ $\mathrm{\&}$ $\sigma \sim \mathrm{IG}(\alpha ,\beta )$

In this case, the prior densities for the two Laplace parameters are considered as

$$\begin{array}{rl}& f(\mu )=\frac{1}{2b}\exp (-\frac{|\mu -a|}{b}),-\mathrm{\infty }<\mu <\mathrm{\infty },-\mathrm{\infty }<a<\mathrm{\infty },b>0,\\ & f(\sigma )=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\sigma }^{-\alpha -1}\exp (-\frac{\beta }{\sigma }),\sigma >0,\alpha >0,\beta >0,\end{array}$$

where a, b, α and β are fixed constants.

Also, the joint PDF of $(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}$ is given by

$$\begin{array}{rl}f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}& =\left(\frac{1}{2b}\right){\left(\frac{1}{2}\right)}^n\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\sigma }^{-\alpha -n-1}\exp [-\frac{|\mu -a|}{b}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\\ & =C_L{\sigma }^{-\alpha -n-1}\exp [-\frac{|\mu -a|}{b}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }],\end{array}$$

where $C_L=(\frac{1}{2b})(\frac{1}{2}{)}^n\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}$.

Without loss of any generality, let m be such that $x_{m:n}<a\le x_{m+1:n}$. Then, the PDF of $\mathbf{x}$ can be given as

$$\begin{array}{rl}f(\mathbf{x}\mathit{)}& =\sum _{j=0}^{m-1}{\int }_0^{\mathrm{\infty }}{\int }_{x_{j:n}}^{x_{j+1:n}}{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{a-\mu }{b}-\frac{g(\beta ,\mathit{x},j,\mu )}{\sigma }]\mathrm{d}\mu \mathrm{d}\sigma \\ & +{\int }_0^{\mathrm{\infty }}{\int }_{x_{m:n}}^a{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{a-\mu }{b}-\frac{g(\beta ,\mathit{x},m,\mu )}{\sigma }]\mathrm{d}\mu \mathrm{d}\sigma \\ & +{\int }_0^{\mathrm{\infty }}{\int }_a^{x_{m+1:n}}{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{\mu -a}{b}-\frac{g(\beta ,\mathit{x},m,\mu )}{\sigma }]\mathrm{d}\mu \mathrm{d}\sigma \\ & +\sum _{j=m+1}^n{\int }_0^{\mathrm{\infty }}{\int }_{x_{j:n}}^{x_{j+1:n}}{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{\mu -a}{b}-\frac{g(\beta ,\mathit{x},j,\mu )}{\sigma }]\mathrm{d}\mu \mathrm{d}\sigma \\ & =\sum _{j=0}^{m-1}{f}_1(\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{+}{\mathit{f}}_{\mathit{2}}\mathit{(}\mathbf{x}\mathit{,}\mathit{m}\mathit{)}\mathit{+}{\mathit{f}}_{\mathit{3}}\mathit{(}\mathbf{x}\mathit{,}\mathit{m}\mathit{)}\mathit{+}\sum _{\mathit{j}\mathit{=}\mathit{m}\mathit{+}\mathit{1}}^{\mathit{n}}{\mathit{f}}_{\mathit{4}}\mathit{(}\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{,}\end{array}$$

where $f_1(\cdot ,\cdot )$, $f_2(\cdot ,\cdot )$, $f_3(\cdot ,\cdot )$, and $f_4(\cdot ,\cdot )$ are as presented in the Appendix.

From it, we can obtain the posterior densities of $\mu |\mathit{x}$ and $\sigma |\mathit{x}$ to be

$$\begin{array}{rl}f(\mu ;\mathbf{x}\mathit{)}& =\frac{1}{f(\mathbf{x}\mathit{)}}{\int }_0^{\mathrm{\infty }}{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{|\mu -a|}{b}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\mathrm{d}\sigma \\ & =\frac{C_L}{f(\mathbf{x}\mathit{)}}\mathrm{\Gamma }(\alpha +n){[\sum _{i=1}^n|x_{i:n}-\mu |+\beta ]}^{-(\alpha +n)}\exp [-\frac{|\mu -a|}{b}]\end{array}$$

and

$$\begin{array}{rl}f(\sigma ;\mathbf{x}\mathit{)}& =\frac{1}{f(\mathbf{x}\mathit{)}}\sum _{\begin{array}{c}j=0\\ j\ne m\end{array}}^n{\int }_{x_{j:n}}^{x_{j+1:n}}{C}_L{\sigma }^{-\alpha -n-1}\exp [-\frac{|\mu -a|}{b}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\mathrm{d}\mu \\ & +\frac{1}{f(\mathbf{x}\mathit{)}}[{\int }_{x_{m:n}}^a+{\int }_a^{x_{m+1:n}}]C_L{\sigma }^{-\alpha -n-1}\exp [-\frac{|\mu -a|}{b}-\frac{\sum _{i=1}^n|x_{i:n}-\mu |+\beta }{\sigma }]\mathrm{d}\mu \\ & =\sum _{j=0}^{m-1}{f}_1(\sigma ;\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{+}{\mathit{f}}_{\mathit{2}}\mathit{(}\sigma \mathit{;}\mathbf{x}\mathit{,}\mathit{m}\mathit{)}\mathit{+}{\mathit{f}}_{\mathit{3}}\mathit{(}\sigma \mathit{;}\mathbf{x}\mathit{,}\mathit{m}\mathit{)}\mathit{+}\sum _{\mathit{j}\mathit{=}\mathit{m}\mathit{+}\mathit{1}}^{\mathit{n}}{\mathit{f}}_{\mathit{4}}\mathit{(}\sigma \mathit{;}\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{,}\end{array}$$

where $f_1(\sigma ;\mathbf{x}\mathit{,}\mathit{j}\mathit{)}$, $f_2(\sigma ;\mathbf{x}\mathit{,}\mathit{m}\mathit{)}$, $f_3(\sigma ;\mathbf{x}\mathit{,}\mathit{m}\mathit{)}$ and $f_4(\sigma ;\mathbf{x}\mathit{,}\mathit{j}\mathit{)}$ are as presented in the Appendix.

4. Bayesian inference for Type-I and Type-II censored data

In this section, we will extend the results derived in the preceding sections to Type-I and Type-II censored data. For illustrative purpose, we use uniform prior $U(a,b)$ and inverse gamma prior $\mathrm{IG}(\alpha ,\beta )$ for the location and scale parameters, while similar results can also be developed for other priors as well.

Suppose the life-testing experiment gets terminated at time U. For a Type-II censored sample, the experiment gets terminated at the time of the rth failure, where r is a pre-fixed number. For a Type-I censored sample, the experiment gets terminated at a pre-fixed time T, and let r denote the observed number of failures before time T. Here, r has been used for both Type-I and Type-II censored samples only for notational ease. Actually, r is a constant in the case of Type-II censoring, but a random variable in the case of Type-I censoring. Further, let U denote the termination time; i.e. $U=X_{r:n}$ and U = T for a Type-II censored sample and Type-I censored sample, respectively.

Let us assume that the observed data is $X_{1:n}<X_{2:n}<\cdots <X_{r:n}\le U$. Further, let us replace all unobservable censored data, $X_{r+1:n},\dots ,X_{n:n}$, by U, and define ${\mathit{x}}^{\prime }=(x_{1:n},\dots ,x_{r:n},U\cdots ,U)$. We shall use this new $\mathit{x}$ from hereon.

Then, the joint distribution of $(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}$ is

$$f(\mathbf{x}\mathit{,}\mu \mathit{,}\sigma \mathit{)}\propto \{\begin{array}{ll}C{\sigma }^{-(n-r+\alpha +1)}\exp (-{\displaystyle \frac{g_c(\beta ,\mathit{x},j,d,n-r)}{\sigma }})& \mathrm{if}{X}_{j:n}<\mu \le X_{j+1:n},\\ & a\le \mu \le b,\\ C{\displaystyle \sum _{l=0}^{n-r}(\genfrac{}{}{0ex}{}{n-r}{l})(-1{)}^l{2}^{n-r-l}{\sigma }^{-(n-r+\alpha +1)}\exp (-{\displaystyle \frac{g_c(\beta ,\mathit{x},r,d,l)}{\sigma }})}& \mathrm{if}\mu >U,a\le \mu \le b\\ 0& \mathrm{otherwise},\end{array}$$

where C is as defined in (1), and

$$\begin{array}{rl}{g}_c(\beta ,\mathit{x},j,d,k)& =\beta +\sum _{i=j+1}^r(x_{i:n}-d)+k(U-d)\\ & +\sum _{i=1}^j(d-x_{i:n}).\end{array}$$

In the second case when $\mu >U$, we have used the binomial expansion, $(1-F(U;\mu ,\sigma ){)}^{n-r}=\sum _{l=0}^{n-r}(\genfrac{}{}{0ex}{}{n-r}{l})(-1{)}^l{F}^l(U;\mu ,\sigma )$, where $F(\cdot ;\cdot ,\cdot )$ is the CDF of the Laplace distribution. Clearly, $g_c(\beta ,\mathit{x},j,d,n-r)=g(\beta ,\mathit{x},j,d)$, according to the definition of $\mathit{x}$.

As the joint distribution here takes a form similar to the one for the complete sample case, we can readily obtain the corresponding inferential results by carefully modifying the results in Section 2. We will now present the results based on cases $b\le U$ and b>U. We choose $n_1$ and $n_2$ such that $x_{n_1:n}<a\le x_{n_1+1:n}$ and $x_{n_2:n}<b\le x_{n_2+1:n}$. Note that $x_{i:n}=U$ if i>r, based on the definition of $\mathit{x}$.

We then have

$$f(\mathbf{x}\mathit{)}\mathit{=}\mathit{C\Gamma }\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{1}\mathit{)}{\mathit{w}}_{\mathit{c}}\mathit{,}$$

where

$$\begin{array}{rl}{w}_c& =\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}{h}_c^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{+}\mathbb{I}\mathit{(}\mathit{n}\mathit{=}\mathit{2}\mathit{k}\mathit{,}{\mathit{n}}_{\mathit{1}}\le \mathit{k}\le {\mathit{n}}_{\mathit{2}}\mathit{)}\left[\frac{({\mathit{x}}_{\mathit{k}\mathit{+}\mathit{1}\mathit{:}\mathit{n}}\mathit{-}{\mathit{x}}_{\mathit{k}\mathit{:}\mathit{n}})\mathit{(}\alpha \mathit{-}\mathit{r}\mathit{+}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}{{({\mathit{g}}_{\mathit{c}}\mathit{(}\beta \mathit{,}\mathit{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{,}\mathit{n}\mathit{-}\mathit{r}\mathit{)})}^{\alpha \mathit{-}\mathit{r}\mathit{+}\mathit{n}}}\right]\\ & +\mathbb{I}(b>U)\sum _{l=0}^{n-r}{h}_{\mathrm{cl}}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{l}\mathit{)}\mathit{,}\end{array}$$

$h_c^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{)}\mathit{=}\mathit{[}{\mathit{h}}_{\mathit{c}}^{\mathit{-}\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{1}\mathit{)}}\mathit{(}\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}\mathit{-}{\mathit{h}}_{\mathit{c}}^{\mathit{-}\mathit{(}\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{1}\mathit{)}}\mathit{(}\beta \mathit{,}\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}\mathit{]}\mathit{/}\mathit{(}\mathit{n}\mathit{-}\mathit{2}\mathit{j}\mathit{)}\mathit{,}$ $h_{\mathrm{cl}}^{\ast }(\beta ,\mathbf{x}\mathit{,}\mathit{l}\mathit{)}\mathit{=}(\genfrac{}{}{0ex}{}{\mathit{n}\mathit{-}\mathit{r}}{\mathit{l}})\mathit{(}\mathit{-}\mathit{1}{\mathit{)}}^{\mathit{l}}{\mathit{2}}^{\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{l}}\mathit{(}\mathit{r}\mathit{+}\mathit{l}{\mathit{)}}^{\mathit{-}\mathit{1}}\mathit{[}\frac{\mathit{1}}{\mathit{[}\beta \mathit{+}\sum _{\mathit{i}\mathit{=}\mathit{1}}^{\mathit{r}}\mathit{(}\mathit{U}\mathit{-}{\mathit{x}}_{\mathit{i}\mathit{:}\mathit{n}}\mathit{)}{\mathit{]}}^{\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{1}}}\mathit{-}\frac{\mathit{1}}{\mathit{[}\beta \mathit{+}\mathit{l}\mathit{(}\mathit{b}\mathit{-}\mathit{U}\mathit{)}\mathit{+}\sum _{\mathit{i}\mathit{=}\mathit{1}}^{\mathit{r}}\mathit{(}\mathit{b}\mathit{-}{\mathit{x}}_{\mathit{i}\mathit{:}\mathit{n}}\mathit{)}{\mathit{]}}^{\alpha \mathit{+}\mathit{n}\mathit{-}\mathit{r}\mathit{-}\mathit{1}}}\mathit{]}$, and

$$h_c(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{l}\mathit{)}\mathit{=}\{\begin{array}{ll}\beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-x_{l:n})+{\displaystyle \sum _{i=1}^j(x_{l:n}-x_{i:n})}}& \mathrm{if}{n}_1<l\le n_2,\\ \beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-a)+{\displaystyle \sum _{i=1}^j(a-x_{i:n})}}& \mathrm{if}l=n_1,\\ \beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-b)+{\displaystyle \sum _{i=1}^j(b-x_{i:n})}}& \mathrm{if}l=n_2+1,b\le U,\\ \beta +{\displaystyle \sum _{i=j+1}^n(x_{i:n}-U)+{\displaystyle \sum _{i=1}^j(U-x_{i:n})}}& \mathrm{if}l=n_2+1,b>U.\end{array}$$

The corresponding posterior PDFs of $\mu |\mathit{x}$ and $\sigma |\mathit{x}$ are

$$f(\mu ;\mathbf{x}\mathit{)}\mathit{=}\{\begin{array}{ll}{\displaystyle \frac{(\alpha +n-r-1)}{{\left({\displaystyle \sum _{i=1}^r|x_{i:n}-\mu |+(n-r)(U-\mu )+\beta }\right)}^{\alpha +n-r}{w}_c}}& \mathrm{if}\mu \le U,\\ {\displaystyle \frac{1}{w_c}{\displaystyle \sum _{l=0}^{n-r}(\genfrac{}{}{0ex}{}{n-r}{l})(-1{)}^l{2}^{n-r-l}{[\sum _{i=1}^r(\mu -x_{i:n})+l(\mu -U)+\beta ]}^{-(\alpha +n-r)}}}& \mathrm{if}\mu >U,\end{array}$$

and

$$\begin{array}{rl}f(\sigma ;\mathbf{x}\mathit{)}& =\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}{c}_{c1j}\mathrm{IG}(\sigma ;\alpha +n-r-1,h_c(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)})\\ & +\sum _{\begin{array}{c}j=n_1\\ 2j\ne n\end{array}}^{n_2}-c_{c2j}\mathrm{IG}(\sigma ;\alpha +n-r-1,h_c(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)})\\ & +c_{c3}\mathrm{IG}(\sigma ;\alpha +n-r,g_c(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{,}\mathit{n}\mathit{-}\mathit{r}))\\ & +\mathbb{I}(b>\mu )\sum _{l=0}^{n-r}{c}_{c4l}\mathrm{IG}(\sigma ;\alpha +n-r-1,g_c(\beta ,\mathbf{x}\mathit{,}\mathit{r}\mathit{,}\mathit{b}\mathit{,}\mathit{l}))\\ & +\mathbb{I}(b>\mu )\sum _{l=0}^{n-r}-c_{c5l}\mathrm{IG}(\sigma ;\alpha +n-r-1,g_c(\beta ,\mathbf{x}\mathit{,}\mathit{r}\mathit{,}\mathit{U}\mathit{,}\mathit{l})),\end{array}$$

where

$$\begin{array}{rl}{c}_{c1j}& =\frac{1}{w_c(n-2j){[h_c(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{+}\mathit{1}\mathit{)}]}^{(\alpha +n-r-1)}},\\ c_{c2j}& =\frac{1}{w_c(n-2j){[h_c(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{j}\mathit{)}]}^{(\alpha +n-r-1)}},\\ c_{c3}& =\mathbb{I}(n_1\le k\le n_2,n=2k)\frac{(\alpha +n-r-1)(x_{k+1:n}-x_{k:n})}{w_c{\left[g_c(\beta ,\mathbf{x}\mathit{,}\mathit{k}\mathit{,}\mathit{0}\mathit{,}\mathit{n}\mathit{-}\mathit{r})\right]}^{(\alpha +n-r)}},\\ c_{c4l}& =(\genfrac{}{}{0ex}{}{n-r}{l})(-1{)}^l(r+l{)}^{-1}{w}_c^{-1}{2}^{n-r-l}[g_c(\beta ,\mathbf{x}\mathit{,}\mathit{r}\mathit{,}\mathit{U}\mathit{,}\mathit{l}){]}^{-(\alpha +n-r-1)},\\ c_{c5l}& =(\genfrac{}{}{0ex}{}{n-r}{l})(-1{)}^l(r+l{)}^{-1}{w}_c^{-1}{2}^{n-r-l}[g_c(\beta ,\mathbf{x}\mathit{,}\mathit{r}\mathit{,}\mathit{b}\mathit{,}\mathit{l}){]}^{-(\alpha +n-r-1)}.\end{array}$$

Here again, $\sigma |\mathit{x}$ is a generalized mixture of inverse-gamma distributions.

5. Illustrative examples

In this section, we apply the developed Bayesian inferential methods to some known data sets from the literature, and then use a simulation study to evaluate the performance of the methods.

Example 5.1

The data, presented in Table 1, are from [22]. These real data present the failure time of bearings when using an aviation gas turbine lubricant O-64-2 with two specific testers. The original experiment employed 10 different testers and here we choose two of them, for illustrative purpose.

Let us now assume that both data sets follow Laplace distributions with same location parameter μ and scale parameter σ. Further, the information obtained from tester 4 will be used as prior information while analyzing data from tester 2. Applying the results in [7] to tester 4 data, we first obtain the MLEs, $\hat{\mu }$ and $\hat{\sigma }$, as well as $E(\hat{\mu })$, $E(\hat{\sigma })$, $\mathrm{Var}(\hat{\mu })$, $\mathrm{Var}(\hat{\sigma })$ and CIs for μ and σ. These are subsequently used to build prior distributions for μ and σ before analyzing data from tester 2. The prior distribution for μ is assumed to be uniform $U(a,b)$, where a and b are two end points of the $99.9\mathrm{\%}$ CI of μ obtained from data from tester 4. The prior distribution for σ is assumed to be $\mathrm{IG}(\alpha ,\beta )$, where α and β are chosen so that the mean and variance of $\mathrm{IG}(\alpha ,\beta )$ coincide with $E(\hat{\sigma })$ and $\mathrm{Var}(\hat{\sigma })$. The priors so obtained are $U(169.57,262.03)$ and $\mathrm{IG}(2.5,73.10)$, respectively.

We then apply the developed Bayesian inference along with the two priors for data from tester 2. The results so obtained are presented in Table 2. For visualization purpose, the posterior PDF and CDF plots for μ and σ are presented in Figures 1–4. From the plots of the posterior CDFs of the two Laplace parameters, we can readily give 95% credible intervals for μ and σ. More importantly, according to the posterior PDF plots of $\mu |\mathit{x}$ and $\sigma |\mathit{x}$, we can observe the unimodality of the posterior distribution under the two selected samples, which ensures the unique existence of the posterior mode estimates for the two Laplace parameters.

Figure 2.

Posterior CDF of $\sigma |\mathit{x}$.

Figure 3.

Posterior PDF of $\mu |\mathit{x}$.

Table 1.

Failure times of bearings, taken from [22].

Tester
2	243.6	242.1	239.0	202.1	190.5	159.8	275.5	192.4	183.8	203.7
4	183.4	276.9	210.3	262.8	115.3	242.2	293.5	221.3	108.9	191.5

Table 2.

Bayesian inference for tester 2 data using tester 4 data for priors.

Estimates	σ	μ
Posterior Mode Estimate	27.2453	(202.1, 203.7)
Posterior Mean Estimate	32.1851	207.1467
Posterior Median Estimate	30.3501	205.0151
95 $\mathrm{\%}$ credible interval	$[17.9640,57.0657]$	$[185.1328,238.6305]$

Figure 1.

Posterior PDF of $\sigma |\mathit{x}$.

Figure 4.

Posterior CDF of $\mu |\mathit{x}$.

Example 5.2

The second data, presented in Table 3, is from [14]. This data set, describing the lifetimes of lamps, involves 31 observations.

We now assume these data follow Laplace distribution $L(\mu ,\sigma )$. In this case, the prior mean for the location parameter μ, which is the expected lifetime of lamps, is set to be 1500  h as the manufacturer's initial technical specification claim. In addition, the prior standard deviation for μ is set to be 2000  h representing a relatively large uncertainty. With these, the prior distribution for μ is assumed to be $U(0,1500+2000\sqrt{3})$ to coincide with the above mentioned values of prior mean and standard deviation. The left end is set to be 0, due to the non-negativity of lifetimes. Furthermore, to match the moments of prior distribution of σ, the prior for σ is chosen to be $\mathrm{IG}(3,2)$.

Then, by applying the inferential results developed in the preceding section, we obtained the results presented in Table 4. Once again, for visualization purpose, the posterior PDF and CDF plots of $\mu |\mathit{x}$ and $\sigma |\mathit{x}$ are presented in Figures 5–8.

Figure 6.

Posterior CDF of $\sigma |\mathit{x}$.

Figure 7.

Posterior PDF of $\mu |\mathit{x}$.

Table 3.

Projector lamp failure times, taken from [14].

Sample 2	387	182	244	600	627	332	418	300	798	584	660
	39	274	174	50	34	1895	158	974	345	1755	1752
	473	81	954	1407	230	464	380	131	1205

Table 4.

Bayesian inference for projector lamp failure times.

Estimates	σ	μ
Posterior Mode Estimate	337.3038	387
Posterior Mean Estimate	358.0949	403.553
Posterior Median Estimate	350.8794	400.0494
95 $\mathrm{\%}$ credible interval	$[254.2519,503.2918]$	$[268.5265,558.9243]$

Figure 5.

Posterior PDF of $\sigma |\mathit{x}$.

Figure 8.

Posterior CDF of $\sigma |\mathit{x}$.

Example 5.3

In this example, we carry out a Monte Carlo simulation study for evaluating the performance of the Bayesian inference developed in the preceding sections. We perform two simulations, one with n = 10 and one with n = 31, i.e. covering both even and odd sample size cases. When n = 10, the prior distributions were assumed to be $\mu \sim U(169.57,262.03)$ and $\sigma \sim \mathrm{IG}(2.5,73.1)$, coinciding with the priors in Example 5.1. When n = 31, the prior distributions were assumed to be $\mu \sim U(0,1500+2000\sqrt{3})$ and $\sigma \sim \mathrm{IG}(3,2)$, coinciding with the priors in Example 5.2. Based on 10,000 simulations, we first generated μ and σ from the specified prior distributions and then generated data from the associated Laplace distribution. Next, we obtained $95\mathrm{\%}$ credible intervals based on the posterior distribution and evaluated whether the true values μ and σ were contained in the corresponding intervals or not. The obtained results are presented in Table 5, revealing the accuracy of the derived inference.

Table 5.

Coverage probabilities of the $95\mathrm{\%}$ credible intervals.

Simulations	Even sample size	Odd sample size
Success Ratio(σ)	0.9495	0.9509
Success Ratio(μ)	0.9484	0.9488
Sample size	40	31
Simulation size	10,000	10,000

6. Concluding remarks

In this paper, we have developed Bayesian inference for the two parameters of the Laplace distribution based on three widely used priors. Upon using the non-informative and inverse gamma priors for location and scale parameters, respectively. We have presented the detailed procedure for obtaining the posterior distributions and then made use of them to develop point and interval estimation. The simulation study carried out shows the accuracy of the developed inferential methods. The Bayesian inference developed in this study demonstrates its suitability in handling real-world data and its potential to enhance analytical precision in fields that rely heavily on robust statistical modeling, such as finance and engineering, opening new avenues for applying advanced Bayesian techniques to other complex parametric distributions. Upon integrating prior knowledge with observed data, Bayesian methods offer a dynamic framework that is particularly effective in scenarios comprising incomplete or uncertain data.

There are still some limitations to the current research, which can be identified as the necessity for further research. Firstly, In this work, the unimodality of the posterior distributions has been graphically evaluated. It will be of interest to establish this property formally. Secondly, extensions of the results to multivariate Laplace distribution will be of great interest as this model is used extensively in practice; see, for example, [11,29]. Finally, it will also be of great interest to extend the results to some general censoring schemes such as progressive censoring, hybrid censoring and so on. We are currently working on these problems and hope to explore these directions in our future work, aiming to extend our findings to more broader applications.

Appendix.

We have

$$\begin{array}{rl}{f}_1(\mathbf{x}\mathit{,}\mathit{j}\mathit{)}& =C_L{\int }_0^{\mathrm{\infty }}\frac{\mathrm{\sigma b}}{b(n-2j)+\sigma }{\sigma }^{-(\alpha +n+1)}{\exp [-\frac{a-\mu }{b}-\frac{h(\beta ,\mathit{x},j,\mu )}{\sigma }]|}_{\mu =x_{j:n}}^{\mu =x_{j+1:n}}\mathrm{d}\sigma \\ & =\frac{C_L\mathrm{\Gamma }(n+\alpha -1)}{[h(\beta ,\mathit{x},j,\mu ){]}^{n+\alpha -1}}\exp [-\frac{a-\mu }{b}]{E_{\mathrm{IG}}[\frac{b}{b(n-2j)+\sigma };n+\alpha -1,h(\beta ,\mathit{x},j,\mu )]|}_{\mu =x_{j:n}}^{\mu =x_{j+1:n}},\\ f_2(\mathbf{x}\mathit{,}\mathit{m}\mathit{)}& =\frac{C_L\mathrm{\Gamma }(n+\alpha -1)}{[h(\beta ,\mathit{x},m,\mu ){]}^{n+\alpha -1}}\exp [-\frac{a-\mu }{b}]{E_{\mathrm{IG}}[\frac{b}{b(n-2m)+\sigma };n+\alpha -1,h(\beta ,\mathit{x},m,\mu )]|}_{\mu =x_{m:n}}^{\mu =a},\\ f_3(\mathbf{x}\mathit{,}\mathit{m}\mathit{)}& =\frac{C_L\mathrm{\Gamma }(n+\alpha -1)}{[h(\beta ,\mathit{x},m,\mu ){]}^{n+\alpha -1}}\exp [-\frac{\mu -a}{b}]{E_{\mathrm{IG}}[\frac{b}{b(n-2m)-\sigma };n+\alpha -1,h(\beta ,\mathit{x},m,\mu )]|}_{\mu =a}^{\mu =x_{m+1:n}},\\ f_4(\mathbf{x}\mathit{,}\mathit{j}\mathit{)}& =\frac{C_L\mathrm{\Gamma }(n+\alpha -1)}{[h(\beta ,\mathit{x},j,\mu ){]}^{n+\alpha -1}}\exp [-\frac{\mu -a}{b}]{E_{\mathrm{IG}}[\frac{b}{b(n-2j)-\sigma };n+\alpha -1,h(\beta ,\mathit{x},j,\mu )]|}_{\mu =x_{j:n}}^{\mu =x_{j+1:n}},\end{array}$$

where $E_{\mathrm{IG}}(g(X);\alpha ,\beta )$ denotes the mean of $g(X)$ with X following an inverse gamma distribution with shape and scale parameters α and β, respectively. Further,

$$\begin{array}{rl}{f}_1(\sigma ;\mathbf{x}\mathit{,}\mathit{j}\mathit{)}& ={\int }_{x_{j:n}}^{x_{j+1:n}}\frac{C_L{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{0}\mathit{)}}{\sigma }-\frac{a}{b})}{f(\mathbf{x}\mathit{)}}\exp \left[(\frac{1}{b}+\frac{n-2j}{\sigma })\mu \right]\mathrm{d}\mu \\ & =\frac{\mathrm{b\sigma }}{\sigma +b(n-2j)}\frac{C_L{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{0}\mathit{)}}{\sigma }-\frac{a}{b})}{f(\mathbf{x}\mathit{)}}\\ & \times \{\exp \left[(\frac{1}{b}+\frac{n-2j}{\sigma })x_{j+1:n}\right]-\exp \left[(\frac{1}{b}+\frac{n-2j}{\sigma })x_{j:n}\right]\},\\ f_2(\sigma ;\mathbf{x}\mathit{,}\mathit{m}\mathit{)}& =\frac{\mathrm{b\sigma }}{\sigma +b(n-2m)}\frac{C_L{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{m}\mathit{,}\mathit{0}\mathit{)}}{\sigma }-\frac{a}{b})}{f(\mathbf{x}\mathit{)}}\\ & \times \{\exp \left[(\frac{1}{b}+\frac{n-2m}{\sigma })a\right]-\exp \left[(\frac{1}{b}+\frac{n-2m}{\sigma })x_{m:n}\right]\},\\ f_3(\sigma ;\mathbf{x}\mathit{,}\mathit{m}\mathit{)}& =\frac{\mathrm{b\sigma }}{b(n-2m)-\sigma }\frac{C_L{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{m}\mathit{,}\mathit{0}\mathit{)}}{\sigma }+\frac{a}{b})}{f(\mathbf{x}\mathit{)}}\\ & \times \{\exp \left[(-\frac{1}{b}+\frac{n-2m}{\sigma })x_{m+1:n}\right]-\exp \left[(-\frac{1}{b}+\frac{n-2m}{\sigma })a\right]\},\\ f_4(\sigma ;\mathbf{x}\mathit{,}\mathit{j}\mathit{)}& =\frac{\mathrm{b\sigma }}{b(n-2j)-\sigma }\frac{C_L{\sigma }^{-\alpha -n-1}\exp (-\frac{g(\beta ,\mathbf{x}\mathit{,}\mathit{j}\mathit{,}\mathit{0}\mathit{)}}{\sigma }+\frac{a}{b})}{f(\mathbf{x}\mathit{)}}\\ & \times \{\exp \left[(-\frac{1}{b}+\frac{n-2j}{\sigma })x_{j+1:n}\right]-\exp \left[(-\frac{1}{b}+\frac{n-2j}{\sigma })x_{j:n}\right]\}.\end{array}$$

Funding Statement

The second author was supported by Natural Science Foundation of Jiangsu Province, China – The Excellent Young Scholar Programme [No. BK20220098], National Natural Science Foundation of China – Young Scientists Fund [No. 11801459], and Research Enhancement Fund of Xi'an-Jiaotong Liverpool University [No. REF-22-01-012]. The third author was supported by Natural Science Foundation of the Jiangsu Higher Education Institutions of China Programme – General Programme [No. 24KJB110027], and Research Development Fund of Xi'an-Jiaotong Liverpool University [No. RDF- 19-02-19].

Disclosure statement

No potential conflict of interest was reported by the author(s).