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СпутникиРНSpaceXРоскосмосNASAEstimating the parameters of the generalized logistic distribution has posed a challenge for many inference approaches, particularly in the frequentist approach. Surprisingly, there have been very few attempts to study the objective Bayesian analysis approach for this problem. In our study, we investigate objective Bayesian analysis using Jeffreys prior, reference priors, matching priors, and the maximal data information prior. We demonstrate that using these non-informative priors does not result in proper posterior distributions. Additionally, we develop a Bayesian analysis based on reference priors with partial information, which yields proper posterior distributions. To evaluate the performance of these priors, we conduct a small Markov Chain Monte Carlo (MCMC) study examining three of these prior distributions. The results demonstrate strong performance in terms of mean squared error and coverage probability. Finally, we utilize these priors to obtain estimations and credible sets for the distribution parameters in a specific example.
Estimating the parameters of the generalized logistic distribution has posed a challenge for many inference approaches, particularly in the frequentist approach. Surprisingly, there have been very few attempts to study the objective Bayesian analysis approach for this problem. In our study, we investigate objective Bayesian analysis using Jeffreys prior, reference priors, matching priors, and the maximal data information prior. We demonstrate that using these non-informative priors does not result in proper posterior distributions. Additionally, we develop a Bayesian analysis based on reference priors with partial information, which yields proper posterior distributions. To evaluate the performance of these priors, we conduct a small Markov Chain Monte Carlo (MCMC) study examining three of these prior distributions. The results demonstrate strong performance in terms of mean squared error and coverage probability. Finally, we utilize these priors to obtain estimations and credible sets for the distribution parameters in a specific example.
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He’s work is supported by the National Social Science Foundation of China (grant No. 21BTJ034) and the Natural Science Foundation of Anhui Province (grant No. 2408085MA005).
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Mohammed K. Shakhatreh
School of Mathematics and Statistics, Anhui Normal University, Wuhu, China
Daojiang He
Authors
Correspondence to Daojiang He.
The authors declare that there is no conflict of interest.
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Here, we introduce the proofs for Theorems 2.4, 2.6 and 3.1.
Lemma 6.1The Fisher information matrix for \(\varvec{\vartheta }= (\theta , \sigma , \alpha )'\) in the GLO distribution Eq. 1, which can be found in Nadarajah (2004) but with some minor corrections is given by \(\mathbf {I(\varvec{\vartheta })}=(I_{ij})_{3 \times 3},\) where
$$\begin{aligned} I_{11}&= \alpha \sigma ^{-2}(\alpha +2)^{-1}\\ I_{22}&= \sigma ^{-2}+\alpha \sigma ^{-2}(\alpha +2)^{-1}\left\{ \psi '(\alpha +1)+\psi '(2)+[\psi (\alpha +1)-\psi (2)]^2\right\} \\ I_{33}&= \alpha ^{-2},\,\,I_{12} = \alpha \sigma ^{-2}(\alpha +2)^{-1}[\psi (\alpha +1)-\psi (2)]\\ I_{13}&= \sigma ^{-1}(\alpha +1)^{-1},\,\,I_{23} = \sigma ^{-1}(\alpha +1)^{-1}[\psi (\alpha )-\psi (2)] \end{aligned}$$
and \(I_{21} = I_{12}, I_{31} = I_{13}, I_{32} = I_{23}.\)
Proof of Theorem 2.4It suffices to prove the reference prior for the ordering group \(\{\theta ,\sigma ,\alpha \}\), as all the other cases can be obtained in a similar way. Let \(\textbf{S}\) be the inverse of the Fisher information matrix \(\textbf{I}(\varvec{\vartheta })\), that is, \(\textbf{S}=\textbf{I}^{-1}= (s_{ij})_{3 \times 3},\) then it can be shown that
$$\begin{aligned} s_{11}= & \frac{\sigma ^2(\alpha +2)}{\alpha ^3[\psi '(\alpha +1)+\psi '(2)]} \Big \{ \alpha ^2(\alpha +1)^2[\psi '(\alpha +1)+\psi '(2)] \\ & ~~~~ + \alpha ^2 [\psi (\alpha ) - \psi (2)]^2 + 2\alpha (\alpha +1)^2[\psi (\alpha ) - \psi (2)] + (\alpha +1)^4 \Big \}, \\ s_{12}= & - \frac{\sigma ^2(\alpha +2)}{\alpha ^2[\psi '(\alpha +1)+\psi '(2)]} \Big \{ \alpha [\psi (\alpha ) - \psi (2)] + (\alpha +1)^2 \Big \}, \\ s_{13}= & - \frac{\sigma (\alpha +1)(\alpha +2)}{\alpha [\psi '(\alpha +1)+\psi '(2)]} \Big \{ \alpha [\psi (\alpha ) - \psi (2)] + (\alpha +1)^2 + \alpha ^2 [\psi '(\alpha +1)+\psi '(2)] \Big \}, \\ s_{22}= & \frac{\sigma ^2(\alpha +2)}{\alpha [\psi '(\alpha +1)+\psi '(2)]}, \\ s_{23}= & \frac{\sigma (\alpha +1)(\alpha +2)}{\psi '(\alpha +1)+\psi '(2)}, \\ s_{33}= & \frac{\alpha (\alpha +1)^2}{\psi '(\alpha +1)+\psi '(2)} \Big \{ (\alpha +2) + \alpha [\psi '(\alpha +1)+\psi '(2)] \Big \}. \end{aligned}$$
Following the notations in Bernardo (n.d.), the h functions here are:\(h_1 = s^{-1}_{11}; h_2 = \sigma ^{-2}\cdot h_{2}^*(\alpha ); h_3 = \alpha ^{-2}.\) where
$$\begin{aligned} h_{2}^*(\alpha )= & \frac{1}{\alpha (\alpha +1)^2(\alpha +2)} \Big \{ \alpha ^2(\alpha +1)^2[\psi '(\alpha +1)+\psi '(2)] \\ \nonumber & ~~~~ + \alpha ^2 [\psi (\alpha ) - \psi (2)]^2 + 2\alpha (\alpha +1)^2[\psi (\alpha ) - \psi (2)] + (\alpha +1)^4 \Big \}. \end{aligned}$$
For \((\theta ,\sigma ,\alpha )\), we choose compact set series \(\Omega _l = [c_{1l}, d_{1l}] \times [c_{2l}, d_{2l}] \times [c_{3l}, d_{3l}]\), \(l = 1,2,\ldots \), such that \(c_{1l}\rightarrow -\infty \), \(c_{2l}, c_{3l} \rightarrow 0\) and \(d_{1l}, d_{2l}, d_{3l} \rightarrow +\infty \) as \(l\rightarrow +\infty \). Then we have
$$\begin{aligned} \pi _3^l(\alpha |\theta ,\sigma )=\frac{|h_3|^{1/2} \textbf{1}_{[c_{3l}, d_{3l}]}(\alpha )}{\int _{c_{3l}}^{d_{3l}}|h_3|^{1/2} \text {d}\alpha }:= \frac{1}{k_1 \alpha }\textbf{1}_{[c_{3l}, d_{3l}]}(\alpha ), \end{aligned}$$
where \(\textbf{1}_{[a, b]}(\cdot )\) stands for the indicator function on the interval [a, b], and \(k_1=\log (d_{3l})-\log (c_{3l})\) is a constant. It follows that
$$\begin{aligned} E_2^{l}\left( \log |h_2| | \theta , \sigma \right) =\int _{c_{3l}}^{d_{3l}}\log |h_2|\cdot \frac{1}{k_1 \alpha } \text {d}\alpha := k_2 - \log \sigma ^2, \end{aligned}$$
where \(k_2 = \int _{c_{3l}}^{d_{3l}} \log h_{2}^*(\alpha ) \text {d}\alpha \) is constant. Subsequently, we have
$$\begin{aligned} \pi _2^l(\alpha , \sigma | \theta )= & \frac{\pi _3^{l}(\alpha |\theta , \sigma )\cdot \exp \left\{ \frac{1}{2} E_2^{l}\left( \log |h_2| | \theta , \sigma \right) \right\} \textbf{1}_{[c_{2l}, d_{2l}]}(\sigma )}{\int _{c_{2l}}^{d_{2l}} \exp \left\{ \frac{1}{2} E_2^{l}\left( \log |h_2| | \theta , \sigma \right) \right\} \text {d}\sigma } \nonumber \\= & \frac{\pi _3^{l}(\alpha |\theta , \sigma ) \sigma ^{-1} \textbf{1}_{[c_{1l}, d_{1l}]}(\alpha )}{\int _{c_{2l}}^{d_{2l}} \sigma ^{-1} \text {d}\alpha } \nonumber \\:= & \frac{1}{k_3 \alpha \sigma }\textbf{1}_{[c_{1l}, d_{1l}]}(\alpha )\textbf{1}_{[c_{2l}, d_{2l}]}(\sigma ),\nonumber \end{aligned}$$
where \(k_3=k_1\{\log (d_{2l})-\log (c_{2l})\}\). Observe that
$$\begin{aligned} E_1^{l}\left( \log |h_1| \vert \theta \right) =\int _{c_{3l}}^{d_{3l}} \int _{c_{2l}}^{d_{2l}} \log |h_1|\cdot \frac{1}{k_3 \alpha \sigma } \text {d}\sigma \text {d}\alpha := k_4 \end{aligned}$$
is constant, it follows that
$$\begin{aligned} \pi _1^l(\theta , \sigma , \alpha )= & \frac{\pi _2^{l}(\alpha , \sigma | \theta ) \cdot \exp \left\{ \frac{1}{2} E_1^{l}\left( \log |h_1| | \theta \right) \right\} \textbf{1}_{[c_{1l}, d_{1l}]}(\theta )}{\int _{c_{1l}}^{d_{1l}}\exp \left\{ \frac{1}{2} E_1^{l}\left( \log |h_1| | \theta \right) \right\} \text {d}\theta } \nonumber \\:= & \frac{1}{k_5 \alpha \sigma }\textbf{1}_{[c_{1l}, d_{1l}]}(\theta )\textbf{1}_{[c_{2l}, d_{2l}]}(\sigma )\textbf{1}_{[c_{3l}, d_{3l}]}(\alpha ),\nonumber \end{aligned}$$
where \(k_5 = k_3 (d_{1l}- c_{1l})\). Finally, let \((\theta ^*,\sigma ^*,\alpha ^*)\) be an inner point of \(\Omega _l\), then the reference prior under the ordering group \(\{\theta ,\sigma ,\alpha \}\) is given by
$$\begin{aligned} \pi _{R_1} = \lim _{l\rightarrow \infty } \frac{\pi _1^l(\theta ,\sigma ,\alpha )}{\pi _1^l(\theta ^*,\sigma ^*,\alpha ^*)} \propto \frac{1}{\alpha \sigma }. \end{aligned}$$
Thus, the proof of Theorem 2.4 is completed. \(\square \)
Proof of Theorem 2.6Here we prove (i) and the remaining cases can be treated similarly. Suppose that the parameter of interest is \(\theta .\) So \(\tau (\varvec{\vartheta })=\tau (\theta ,\sigma , \alpha )=\theta .\) The gradient vector is, \(\nabla _{\tau }(\varvec{\vartheta })=(1,0,0)'\) and after some algebra, we have that
$$\begin{aligned} \zeta (\varvec{\vartheta })&=\left( \frac{\sigma (\alpha +2)^{1/2}g_{11}(\alpha )}{\alpha ^{3/2}}, -\frac{\sigma (2+\alpha )^{1/2}g_{22}(\alpha )}{\alpha ^{1/2}} ,-(1+\alpha )[\alpha (2+\alpha )]^{1/2}g_{33}(\alpha )\right) ', \end{aligned}$$
where \(g_{11}(\alpha )=g_{2}^{-1/2}(\alpha )[g_{1}^{2}(\alpha )+\alpha ^{2}(1+\alpha )^{2}g_{2}(\alpha )]^{1/2},\) \(g_{22}(\alpha )=g_{1}(\alpha )g_{2}(\alpha )^{-1/2}[g_{1}^{2}(\alpha )+\alpha ^{2}(1+\alpha )^{2}g_{2}(\alpha )]^{-1/2}\) and \(g_{33}(\alpha )=[g_{1}(\alpha )+\alpha ^{2}g_{2}(\alpha )]g_{2}^{-1/2}(\alpha )[g_{1}^{2}(\alpha )+\alpha ^{2}(1+\alpha )^{2}g_{2}(\alpha )]^{-1/2}\). On using Eq. 9, then the required matching prior must solve the following partial differential equation
$$\begin{aligned} \frac{\partial }{\partial \theta }\left\{ \frac{\sigma (\alpha +2)^{1/2}g_{11}(\alpha )}{\alpha ^{3/2}}\,\pi \right\}&-\frac{\partial }{\partial \sigma }\left\{ \frac{\sigma (2+\alpha )^{1/2}g_{22}(\alpha )}{\alpha ^{1/2}}\,\pi \right\} \\ &-\frac{\partial }{\partial \alpha }\left\{ (1+\alpha )[\alpha (2+\alpha )]^{1/2}g_{33}(\alpha )\,\pi \right\} =0. \end{aligned}$$
A particular solution of the above partial differential equation is given in Eq. 10. \(\square \)
Proof of Theorem 3.1We show only (i) and the remaining parts can be proved similarly. By the assumption that there exists a subjective prior distribution for \(\alpha \) with density function \(\pi (\theta ,\sigma )\), then we seek for non-informative conditional prior of \((\theta ,\sigma )'\) given \(\alpha \) denoted by \(\pi ^{r}(\theta , \sigma |\alpha ).\) According to Sun and Berger (1998), the form of this prior distribution should take the following \(\pi (\alpha |\theta , \sigma )\propto |\textbf{I}(\theta ,\sigma )|^{1/2},\) where the sub-matrix; \(\textbf{I}(\theta ,\sigma )\) is the per unit Fisher information matrix of \(\theta \) and \(\sigma \) keeping \(\alpha \) is fixed. We have that
$$\textbf{I}(\theta ,\sigma )=\left( \begin{array}{cc} I_{11} & I_{12} \\ I_{21} & I_{22}\\ \end{array} \right) ,$$
where the elements of \(\textbf{I}(\theta ,\sigma )\) is given in Theorem 6.1. The determinant of \(\textbf{I}(\theta ,\sigma )\) is
$$|\textbf{I}(\theta ,\sigma )|=\frac{1}{\sigma ^{4}}G_{2}(\alpha ) \left[ 1+G_{2}^{2}(\alpha )g_{2}(\alpha )\right] =\Delta _{1}(\sigma )\Delta _{2}(\alpha ),$$
where \(\Delta _{1}(\sigma )=1/\sigma ^{4}\) and \(\Delta _{2}(\alpha )=G_{2}(\alpha ) \left[ 1+G_{2}^{2}(\alpha )g_{2}(\alpha )\right] .\) Notice that the determinant of the matrix, \(\textbf{I}(\theta ,\sigma )\) factorize into two independent functions of \(\Delta _{1}(\sigma )\) and \(\Delta _{2}(\alpha ).\) Therefore, on using Theorem 2 of Sun and Berger (1998), it follows that \(\pi ^{r}(\theta ,\sigma |\alpha )\propto \Delta _{2}^{1/2}(\sigma )=1/\sigma ^{2}.\) Thus, the joint density of \(\theta ,\) \(\sigma \) and \(\alpha \) is
$$\begin{aligned} \pi _{R_{1}}^{r}(\varvec{\vartheta })=\pi ^{r}(\alpha |\theta ,\sigma )\pi (\theta , \sigma )=\sigma ^{-2}~\pi (\alpha ), \end{aligned}$$
as required. \(\square \)
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Shakhatreh, M.K., He, D. Objective Bayesian Analysis for the Generalized Logistic Distribution. Sankhya A (2026). https://doi.org/10.1007/s13171-026-00448-7
Received: 14 May 2025
Accepted: 13 July 2026
Published: 22 July 2026
Version of record: 22 July 2026
DOI: https://doi.org/10.1007/s13171-026-00448-7