Parameter Estimation for Two Logistic Populations with Fuzzy Data: A Comparative Study of MLE and Bayesian Methods

Abstract

This study addresses the challenge of estimating parameters for two logistic populations that share a common scale parameter but have different location parameters in the presence of fuzzy data. To handle these complexities, both Maximum Likelihood Estimation (MLE) and Bayesian methods are employed. Asymptotic confidence intervals are constructed using ML estimates. For Bayesian estimation, a conjugate prior is utilized, and Bayes estimators are approximated using Lindley’s method due to the lack of closed-form solutions. Furthermore, Approximate Bayesian Computation (ABC) and Markov Chain Monte Carlo (MCMC) techniques, including Hamiltonian Monte Carlo (HMC) and the Metropolis–Hastings (MH) algorithm, are utilized to sample from the posterior distributions and construct Highest Posterior Density (HPD) intervals. A detailed comparative analysis of MLE, Lindley’s approximation, ABC, HMC, and MH is conducted to assess their performance. The effectiveness of the proposed methodology is demonstrated using a real-world dataset under fuzzy conditions.

Keywords

logistic distribution fuzzy data Bayesian estimation Lindley approximation Approximate Bayesian Computation Metropolis-Hastings Hamiltonian Monte Carlo

1. Introduction

The logistic distribution is widely used in various practical applications due to its flexibility in modeling real-world datasets. For instance, it has been effectively applied in demographic studies to describe growth models (Balakrishnan and Basu, 1995). Additionally, it is frequently employed in agricultural production modeling, categorical data analysis, biochemical data analysis, bioassays, and life testing studies (Kotz et al., 2019; Rashad et al., 2016). While its shape resembles that of the normal distribution, the logistic distribution differs in key aspects, it has heavier tails and a more pronounced peak at the center, making it more robust for analyzing datasets with extreme values.

Beyond its applications, a key statistical challenge is parameter estimation under equality restrictions, which plays a crucial role in comparative analysis and risk assessment across different populations. This problem has been extensively studied in normal distributions and is commonly referred to as the “common mean problem.” One of the earliest works in this area was by Graybill (1959), who proposed a combined estimator that dominates individual sample means in terms of variance under specific sample size conditions. Subsequent advancements extended this concept to two or more normal populations with unknown variances, with notable contributions by Moore (1997), Tripathy and Nagamani (2015), and others.

While researchers have extensively studied parameter estimation in normal populations, significant attention has also been given to non-normal distributions, such as the exponential, gamma, and inverse Gaussian families. Ghosh and Meeden (1984) estimated the common location parameter for two exponential distributions, while Jin (1992) introduced improved estimators that outperformed the Maximum Likelihood Estimator (MLE). Tripathy and Nagamani (2014) investigated the estimation of a common location parameter for two exponential populations under order-restricted failure rates. More recently, Azhad and Tripathy (2021) explored parameter estimation for heterogeneous exponential distributions using MLE, modified MLE, and the uniformly minimum variance unbiased estimator (UMVUE).

Yang (2007) explored the estimation of a common shape parameter for two Weibull populations. Later, Nagamani and Tripathy (2017) studied the estimation of a common scale parameter for two gamma populations, while in their other work, Tripathy and Nagamani (2017) focused on estimating the common shape parameter for the same gamma populations. Nagamani and Tripathy (2018) extended this work by examining the common dispersion parameter across multiple Gaussian populations. These studies underscore the growing significance of parameter estimation in non-normal statistical models, particularly in practical applications such as survival analysis, reliability studies, and risk assessment.

Despite these advancements, research on parameter estimation for two logistic populations remains limited. Recently, Nagamani and Tripathy (2020) explored Bayesian estimation of the common scale parameter for two logistic populations with unknown and possibly unequal location parameters. In another study, Lingutla and Nagamani (2024) examined two logistic populations with a common location parameter and possibly unequal scale parameters. However, these studies do not fully address the complexities introduced by fuzzy data in logistic distribution modeling.

While significant research has focused on classical estimation methods, handling fuzzy data in multi-population contexts remains an open problem. Many real-world datasets contain uncertainty, making traditional estimation techniques inadequate for precise inference. Several researchers have investigated parameter estimation under fuzzy conditions for single-population models. Pak and Parham (2013) analyzed the Weibull distribution using Maximum Likelihood Estimation (MLE), the method of moments, and Bayesian estimation under fuzzy conditions. Khoolenjani and Jafari (2016) estimated parameters of the weighted exponential distribution using Bayesian techniques with both symmetric and asymmetric loss functions, employing Lindley’s approximation. Pak (2020) analyzed the generalized exponential model with Bayesian methods under fuzzy data, illustrating its application in reliability and risk assessment scenarios. Roohanizadeh and Pak (2022) extended Bayesian inference techniques to fuzzy datasets in the context of generalized exponential and Weibull distributions. More recently, Ashok and Nagamani (2025) explored parameter estimation for the gamma distribution under fuzzy conditions. These studies highlight the increasing importance of incorporating fuzzy data into statistical models to enhance the robustness and applicability of estimation methods in uncertain environments.

Despite these contributions, a critical gap remains in developing methodologies that effectively handle fuzzy data in multi-population statistical models. Addressing this challenge is essential for improving decision-making processes in real-world scenarios characterized by imprecise or uncertain information.

This study focuses on parameter estimation for two logistic populations that share a common scale parameter but have different location parameters, integrating fuzzy data into the estimation process.

1.1. Key Contributions

The key contributions of this study are summarized as follows:

We develop a unified estimation framework for two logistic populations sharing a common scale parameter in the presence of fuzzy data, extending classical logistic modeling to accommodate data imprecision.

We provide a comprehensive comparative analysis of classical and Bayesian estimation methods, including EM-based MLE, Lindley’s approximation, Approximate Bayesian Computation (ABC), Hamiltonian Monte Carlo (HMC), and the Metropolis–Hastings (MH) algorithm, within a fuzzy data setting.

The performance of the proposed estimators is systematically evaluated through extensive Monte Carlo simulation studies across different sample sizes, using bias and mean square error as performance criteria.

The practical applicability of the methodology is demonstrated using a real engineering dataset, illustrating the impact of fuzzy data modeling on parameter estimation and uncertainty quantification.

The remainder of the paper is structured as follows: Section 2 introduces foundational concepts related to fuzzy data and triangular fuzzy numbers. Section 3 presents the MLE approach for logistic parameters using the Expectation-Maximization (EM) algorithm. Section 4 discusses Bayesian estimation techniques including Lindley’s approximation, ABC, and MCMC methods (HMC and MH). Section 5 contains simulation studies evaluating estimator performance. Section 6 applies the methods to real-world data. Finally, Section 7 concludes with a summary of findings and suggestions for future research.

2. Theoretical Background

2.1. Logistic Distribution

The logistic distribution is widely used in various fields, including survival analysis, economics, and machine learning, due to its similarity to the normal distribution but with heavier tails. The probability density function (PDF) of the logistic distribution for a random variable $X$ is given by:

\begin{aligned} f (x; μ, s) = \frac{e^{- (x - μ) / s}}{s {(1 + e^{- (x - μ) / s})}^{2}}, - \infty < x < \infty . \end{aligned}

where:

–
$μ$ is the location parameter, which represents the median of the distribution.
–
$s > 0$ is the scale parameter, which controls the spread of the distribution.

2.2. Fuzzy Set Theory

Fuzzy set theory, introduced by Zadeh (1965), provides a mathematical framework to handle uncertainty and imprecision by allowing membership in sets to be gradual rather than binary. In classical set theory, an element either belongs to a set or does not; in fuzzy sets, an element can have a degree of membership, defined by a membership function $μ_{A} (x)$ , which maps each element $x$ to a value in the interval $[0, 1]$ .

For a fuzzy set $A$ , the membership function $μ_{A} (x)$ satisfies the following:

\begin{aligned} μ_{A} (x) \in [0, 1], for all x . \end{aligned}

To address the imprecision in fuzzy data, Zadeh (1968) introduced the concept of fuzzy probability. The probability of a fuzzy event

\tilde{A}

is given by:

\begin{aligned} P (\tilde{A}) = \int μ_{\tilde{A}} (x) d P (x), \end{aligned}

(1)

The membership function reflects the degree of truth of an element’s belonging to a set. For example, in medical data, this could represent the degree of certainty about a patient’s survival time or diagnosis based on imprecise or incomplete information.

In the context of survival analysis, fuzzy methods are particularly useful in accommodating incomplete or uncertain data, such as imprecise survival times, due to factors like missing information, recall bias, or measurement errors. Fuzzy methods allow us to represent data that is not entirely precise, which is often the case in medical and clinical studies.

2.3. Triangular Membership Function

A Triangular membership function is commonly used to model fuzzy numbers, particularly when expert knowledge or subjective judgment is used to quantify uncertain data. A triangular fuzzy number is characterized by three parameters: the lower bound $a$ , the peak $b$ , and the upper bound $c$ , which represent the best estimate, the lower bound, and the upper bound of a fuzzy quantity, respectively.

The membership function $μ_{A} (x)$ for a triangular fuzzy number is given by:

\begin{aligned} μ_{A} (x) = {\begin{cases} \frac{x - a}{b - a} & for a \leq x \leq b, \\ \frac{c - x}{c - b} & for b < x \leq c, \\ 0 & otherwise. \end{cases} \end{aligned}

This function is piecewise linear and reflects the degree of membership of an element

x

in the fuzzy set

A

. Triangular fuzzy numbers are often used in situations where precise data is unavailable, but expert opinions or estimates are available, making them suitable for handling uncertainty in medical datasets, including survival analysis.

In medical data, triangular fuzzy numbers are frequently used to represent uncertain survival times or other clinical parameters. For example, if an expert estimates that a patient’s survival time is likely to be around 12 months, but could range between 10 and 15 months, the fuzzy number $(10, 12, 15)$ can be used to represent this uncertainty.

In the subsequent sections we estimate the parameters, including Maximum Likelihood Estimation (MLE) and Bayesian estimation techniques. Specifically, we employ Lindley’s approximation, Approximate Bayesian Computation (ABC), Hamiltonian Monte Carlo (HMC), and the Metropolis-Hastings (MH) algorithm.

3. Maximum Likelihood Estimation

Consider two independent random samples, $X = (x_{1}, x_{2}, \dots, x_{m})$ and $Y = (y_{1}, y_{2}, \dots, y_{n})$ , drawn from logistic distributions representing two populations with a common scale parameter $s$ but distinct location parameters $μ_{1}$ and $μ_{2}$ , respectively. The probability density functions (PDFs) for each population are given by:

\begin{aligned} f_{X} (x; μ_{1}, s) = \frac{e^{- (x - μ_{1}) / s}}{s {(1 + e^{- (x - μ_{1}) / s})}^{2}}, - \infty < x < \infty . \\ f_{Y} (y; μ_{2}, s) = \frac{e^{- (y - μ_{2}) / s}}{s {(1 + e^{- (y - μ_{2}) / s})}^{2}}, - \infty < y < \infty . \end{aligned}

The fuzzy likelihood function is given by:

\begin{aligned} L (μ_{1}, μ_{2}, s ∣ {\tilde{X}}_{i}, {\tilde{Y}}_{j}) = \prod_{i = 1}^{m} P ({\tilde{X}}_{i}) \prod_{j = 1}^{n} P ({\tilde{Y}}_{j}), \end{aligned}

(2)

where

P ({\tilde{X}}_{i})

and

P ({\tilde{Y}}_{j})

are fuzzy probabilities associated with each observed value. Expanding these probabilities:

\begin{aligned} L (μ_{1}, μ_{2}, s ∣ {\tilde{X}}_{i}, {\tilde{Y}}_{j}) = \prod_{i = 1}^{m} \int_{a}^{c} μ_{{\tilde{X}}_{i}} (x) f_{X} (x; μ_{1}, s) d x \prod_{j = 1}^{n} \int_{a}^{c} μ_{{\tilde{Y}}_{j}} (y) f_{X} (y; μ_{2}, s) d y, \\ L = \frac{1}{s^{m + n}} \prod_{i = 1}^{m} \int \frac{\exp (- \frac{x - μ_{1}}{s})}{{(1 + \exp (- \frac{x - μ_{1}}{s}))}^{2}} μ_{{\tilde{X}}_{i}} (x) d x \prod_{j = 1}^{n} \int \frac{\exp (- \frac{y - μ_{2}}{s})}{{(1 + \exp (- \frac{y - μ_{2}}{s}))}^{2}} μ_{{\tilde{Y}}_{j}} (y) d y . \end{aligned}

Taking the natural logarithm, we obtain:

\begin{aligned} ℓ & = \sum_{i = 1}^{m} \log \int \frac{\exp (- \frac{x - μ_{1}}{s})}{{(1 + \exp (- \frac{x - μ_{1}}{s}))}^{2}} μ_{{\tilde{X}}_{i}} (x) d x + \sum_{j = 1}^{n} \log \int \frac{\exp (- \frac{y - μ_{2}}{s})}{{(1 + \exp (- \frac{y - μ_{2}}{s}))}^{2}} μ_{{\tilde{Y}}_{j}} (y) d y \\ - (m + n) \log s . \end{aligned}

(3)

To estimate $μ_{1}, μ_{2},$ and $s$ , we solve the system of nonlinear equations obtained by differentiating the log-likelihood function:

\begin{aligned} \sum_{i = 1}^{m} \frac{\int (\frac{x - μ_{1}}{s^{2}}) \exp (- \frac{x - μ_{1}}{s}) μ_{{\tilde{X}}_{i}} (x) d x}{\int f_{1} (x; μ_{1}, s) μ_{{\tilde{X}}_{i}} (x) d x} = 0, \\ \sum_{j = 1}^{n} \frac{\int (\frac{y - μ_{2}}{s^{2}}) \exp (- \frac{y - μ_{2}}{s}) μ_{{\tilde{Y}}_{j}} (y) d y}{\int f_{2} (y; μ_{2}, s) μ_{{\tilde{Y}}_{j}} (y) d y} = 0, \\ \sum_{i = 1}^{m} \frac{\int \frac{(x - μ_{1})^{2}}{s^{3}} \exp (- \frac{x - μ_{1}}{s}) μ_{{\tilde{X}}_{i}} (x) d x}{\int f_{1} (x; μ_{1}, s) μ_{{\tilde{X}}_{i}} (x) d x} + \sum_{j = 1}^{n} \frac{\int \frac{(y - μ_{2})^{2}}{s^{3}} \exp (- \frac{y - μ_{2}}{s}) μ_{{\tilde{Y}}_{j}} (y) d y}{\int f_{2} (y; μ_{2}, s) μ_{{\tilde{Y}}_{j}} (y) d y} - \frac{m + n}{s} = 0. \end{aligned}

3.1. Expectation-Maximization Algorithm for Logistic Distribution

The Expectation-Maximization (EM) (Dempster et al., 1977) algorithm is a widely used iterative method for estimating parameters in models with latent variables, particularly when dealing with incomplete, missing, or fuzzy data. For two logistic populations with a common scale parameter $s$ and different location parameters $μ_{1}$ and $μ_{2}$ , the complete-data log-likelihood function is given by:

\begin{aligned} ℓ = + \sum_{i = 1}^{m} \log \int \frac{\exp (- \frac{x - μ_{1}}{s})}{{(1 + \exp (- \frac{x - μ_{1}}{s}))}^{2}} μ_{{\tilde{X}}_{i}} (x) d x + \sum_{j = 1}^{n} \log \int \frac{\exp (- \frac{y - μ_{2}}{s})}{{(1 + \exp (- \frac{y - μ_{2}}{s}))}^{2}} μ_{{\tilde{Y}}_{j}} (y) d y . \end{aligned}

(4)

The iterative EM procedure for estimating the parameters is as follows:

Assume initial values $μ_{1}^{(0)}$ , $μ_{2}^{(0)}$ , and $s^{(0)}$ , and set iteration index $h = 0$ .

E-step: Compute the conditional expectations under fuzzy observations:

\begin{aligned} E (x_{i} ∣ {\tilde{x}}_{i}) & = \frac{\int x_{i} e^{- \frac{x_{i} - μ_{1}}{s}} μ_{{\tilde{A}}_{i}} (x) d x}{\int e^{- \frac{x_{i} - μ_{1}}{s}} μ_{{\tilde{A}}_{i}} (x) d x}, \\ E (y_{j} ∣ {\tilde{y}}_{j}) & = \frac{\int y_{j} e^{- \frac{y_{j} - μ_{2}}{s}} μ_{{\tilde{B}}_{j}} (y) d y}{\int e^{- \frac{y_{j} - μ_{2}}{s}} μ_{{\tilde{B}}_{j}} (y) d y} . \end{aligned}

M-step: Update the parameter estimates by solving the following equations:

\begin{aligned} μ_{1}^{(h + 1)} & = \frac{\sum E (x_{i} ∣ {\tilde{x}}_{i})}{m}, \\ μ_{2}^{(h + 1)} & = \frac{\sum E (y_{j} ∣ {\tilde{y}}_{j})}{n}, \\ s^{(h + 1)} & = \frac{\sum | E (x_{i} ∣ {\tilde{x}}_{i}) - μ_{1}^{(h + 1)} | + \sum | E (y_{j} ∣ {\tilde{y}}_{j}) - μ_{2}^{(h + 1)} |}{m + n} . \end{aligned}

If $| μ_{1}^{(h + 1)} - μ_{1}^{(h)} | < ϵ, | μ_{2}^{(h + 1)} - μ_{2}^{(h)} | < ϵ, and | s^{(h + 1)} - s^{(h)} | < ϵ,$ then the algorithm has converged and the resulting values are taken as the required posterior samples. Otherwise, set $h = h + 1$ and repeat from the M-step.

Once the MLEs of the parameters are obtained, the asymptotic confidence intervals for $μ_{1}$ , $μ_{2}$ , and $s$ are given by:

\begin{aligned} {\hat{μ}}_{1} \pm 1.96 \sqrt{I_{11}^{- 1}}, {\hat{μ}}_{2} \pm 1.96 \sqrt{I_{22}^{- 1}}, \hat{s} \pm 1.96 \sqrt{I_{33}^{- 1}}, \end{aligned}

Where

I^{- 1}

is the inverse of the Fisher information matrix. Utilizing the idea of Nadarajah (2004) we can derive the Fisher information matrix.

4. Bayesian Estimation

Bayesian estimation provides a flexible framework for parameter inference by incorporating prior information together with the observed data. Let $θ = (μ_{1}, μ_{2}, s)$ denote the vector of unknown parameters. Given the fuzzy observations $\tilde{X}$ and $\tilde{Y}$ , the posterior density of $θ$ is

\begin{aligned} p (θ ∣ \tilde{X}, \tilde{Y}) = \frac{p (\tilde{X}, \tilde{Y} ∣ θ) p (θ)}{\int p (\tilde{X}, \tilde{Y} ∣ θ) p (θ) d θ}, \end{aligned}

(5)

where

p (\tilde{X}, \tilde{Y} ∣ θ)

is the fuzzy likelihood function and

p (θ)

denotes the prior distribution.

The posterior expectation of any function $π (θ)$ is

\begin{aligned} E [π (θ) ∣ \tilde{X}, \tilde{Y}] = \frac{\int π (θ) p (\tilde{X}, \tilde{Y} ∣ θ) p (θ) d θ}{\int p (\tilde{X}, \tilde{Y} ∣ θ) p (θ) d θ} . \end{aligned}

(6)

Here, $π (θ)$ represents function of the parameters. Because the fuzzy likelihood involves nonlinear integrals that are difficult to evaluate analytically, closed-form expressions for the posterior distribution are not available. Consequently, numerical and simulation-based methods are employed to approximate the Bayesian estimators. In this study, we consider the following approaches:

Lindley’s Approximation: A second-order Taylor expansion of the log-posterior about the posterior mode (typically the maximum likelihood estimate) provides an analytical approximation to posterior expectations. This method is computationally efficient and particularly useful for low-dimensional parameter spaces.

Approximate Bayesian Computation (ABC): A likelihood-free technique that approximates the posterior distribution by matching summary statistics from simulated data with those of the observed fuzzy data. ABC is applicable when the likelihood is either unavailable or too complex.

Hamiltonian Monte Carlo (HMC): An advanced Markov chain Monte Carlo (MCMC) method that uses Hamiltonian dynamics to explore the posterior distribution efficiently. HMC greatly improves mixing in high-dimensional or highly correlated parameter spaces.

Metropolis–Hastings Algorithm: A widely used MCMC procedure based on the acceptance rejection principle, which constructs a Markov chain whose stationary distribution is the desired posterior. This method is useful when the posterior distribution has a complicated form but is computationally manageable.

These methods allow us to obtain approximate Bayesian estimators and posterior summaries for the parameters $μ_{1}$ , $μ_{2}$ , and $s$ under uncertainty.

4.1. Lindley’s Approximation

The approximation proposed by Lindley (1980) is a numerical technique for computing Bayes estimators when the posterior distribution does not possess a closed form. Under the squared error loss function (SELF), the Bayes estimator of a function $u (θ)$ is its posterior expectation,

\begin{aligned} E [u (θ) ∣ x] = \frac{\int u (θ) v (θ) \exp {L (θ)} d θ}{\int v (θ) \exp {L (θ)} d θ}, \end{aligned}

(7)

where

L (θ)

is the log–likelihood of the data

X = (x_{1}, \dots, x_{n})

v (θ)

is the prior density and

θ = (θ_{1}, \dots, θ_{m})

Lindley’s approximation to equation (7) is

\begin{aligned} E [u (θ) ∣ x] = {[u + \frac{1}{2} \sum_{i} \sum_{j} (u_{i j} + 2 u_{i} ρ_{j}) σ_{i j} + \frac{1}{2} \sum_{i} \sum_{j} \sum_{k} \sum_{r} L_{i j k} σ_{i j} σ_{k r} u_{r}]}_{{\hat{θ}}_{M L}} + O (\frac{1}{n^{2}}) . \end{aligned}

(8)

Ignoring terms of order $(m + n)^{- 2}$ , the Bayes estimator of $u (θ)$ under SELF is

\begin{aligned} E [u (θ) ∣ \tilde{x}, \tilde{y}] & = u + u_{1} a_{1} + u_{2} a_{2} + u_{3} a_{3} + \frac{1}{2} [A (u_{1} σ_{11} + u_{2} σ_{12} + u_{3} σ_{13}) \\ + B (u_{1} σ_{21} + u_{2} σ_{22} + u_{3} σ_{23}) + D (u_{1} σ_{31} + u_{2} σ_{32} + u_{3} σ_{33})], \end{aligned}

(9)

where

θ = (μ_{1}, μ_{2}, s)

We assume independent conjugate priors,

\begin{aligned} μ_{1} \sim N (α_{1}, τ_{1}^{2}), μ_{2} \sim N (α_{2}, τ_{2}^{2}), s \sim Gamma (b, c) . \end{aligned}

The corresponding log–prior derivatives are

\begin{aligned} ρ_{1} = - \frac{μ_{1} - α_{1}}{τ_{1}^{2}}, ρ_{2} = - \frac{μ_{2} - α_{2}}{τ_{2}^{2}}, ρ_{3} = \frac{b - 1}{s} - c . \end{aligned}

The constants

a_{1}, a_{2}, a_{3}

are

\begin{aligned} a_{i} = ρ_{1} σ_{i 1} + ρ_{2} σ_{i 2} + ρ_{3} σ_{i 3}, i = 1, 2, 3. \end{aligned}

The elements

σ_{i j}

are obtained from the observed information matrix

\begin{aligned} I (\hat{θ}) = - {[\frac{\partial^{2} ℓ}{\partial θ_{i} \partial θ_{j}}]}_{θ = \hat{θ}}, σ_{i j} = (I^{- 1})_{i j} . \end{aligned}

Using Lindley’s approximation evaluated at the MLEs, the approximate Bayes estimators of the parameters are

\begin{aligned} E (μ_{1} ∣ \tilde{x}, \tilde{y}) = {\hat{μ}}_{1 M L} + a_{1} + \frac{1}{2} (A σ_{11} + B σ_{21} + D σ_{31}), \\ E (μ_{2} ∣ \tilde{x}, \tilde{y}) = {\hat{μ}}_{2 M L} + a_{2} + \frac{1}{2} (A σ_{12} + B σ_{22} + D σ_{32}), \\ E (s ∣ \tilde{x}, \tilde{y}) = {\hat{s}}_{M L} + a_{3} + \frac{1}{2} (A σ_{13} + B σ_{23} + D σ_{33}) . \end{aligned}

The quantities

σ_{i j}

together with the third–order derivatives

A

B

, and

D

appearing in the above expressions are obtained from the second- and third–order derivatives of the log-likelihood. Since closed-form expressions are not available for the fuzzy logistic model, all these derivatives were computed numerically in R.

To address potential concerns regarding numerical stability, central difference schemes with carefully chosen step sizes were employed for numerical differentiation. These step sizes were selected to ensure stable numerical evaluation of the derivatives, and the resulting Bayes estimates were found to be numerically well behaved across all simulation settings.

4.2. Approximate Bayesian Computation (ABC)

Approximate Bayesian Computation (ABC) is particularly useful when the exact likelihood function is analytically intractable, as is the case for fuzzy and imprecise observations arising from the logistic distribution. ABC avoids explicit likelihood evaluations by comparing observed data with synthetic data generated from the model, thereby approximating the posterior distribution through simulation. This makes ABC especially suitable for Bayesian inference in reliability and survival models involving uncertainty, fuzziness, or non-standard likelihoods.

4.2.1. ABC Procedure

Two populations are modeled using logistic distributions with a common scale parameter:

\begin{aligned} X \sim Logistic (μ_{1}, s), Y \sim Logistic (μ_{2}, s), \end{aligned}

where

μ_{1}

and

μ_{2}

are location parameters and

s

is the common scale parameter.

Independent priors are assigned as

\begin{aligned} μ_{1} \sim N (α_{1}, τ_{1}^{2}), μ_{2} \sim N (α_{2}, τ_{2}^{2}), s \sim Gamma (b, c) . \end{aligned}

For each iteration, parameter values $θ^{(k)} = (μ_{1}^{(k)}, μ_{2}^{(k)}, s^{(k)})$ are drawn from the prior and synthetic fuzzy data $({\tilde{X}}^{*}, {\tilde{Y}}^{*})$ are generated from the logistic model using these sampled parameters.

Suitable summary statistics are computed for the observed fuzzy data ( $S_{obs}$ ) and for the simulated data ( $S_{sim}^{(k)}$ ). Typical choices include the sample mean, variance, median, or quantile-based statistics adapted to fuzzy observations.

A discrepancy measure is calculated, commonly the Euclidean distance:

\begin{aligned} d^{(k)} = ‖ S_{obs} - S_{sim}^{(k)} ‖ . \end{aligned}

A tolerance level $ϵ$ is specified, and the sampled parameters $θ^{(k)}$ are accepted whenever

\begin{aligned} d^{(k)} \leq ϵ . \end{aligned}

Repeating the simulation until a sufficient number of accepted samples is obtained yields an approximate posterior sample, from which Bayesian point estimates (posterior mean, median) and credible intervals are computed.

In summary, ABC provides a flexible and likelihood-free Bayesian inference framework capable of handling fuzzy observations and complex data structures. By relying on simulations rather than analytic likelihoods, ABC effectively accommodates uncertainty and supports robust parameter estimation for logistic reliability models under fuzzy data.

In the present study, the summary statistics were selected as the fuzzy sample mean, fuzzy median, and interquartile range, as these statistics jointly capture both location and dispersion while remaining robust under data imprecision. The tolerance level $ϵ$ was chosen to retain a small proportion of the simulated samples (approximately 1–2%), following common practice in ABC implementations, so as to balance posterior approximation accuracy and computational feasibility. The resulting acceptance rate was sufficient to obtain stable posterior summaries for the parameters of interest.

4.3. Hamiltonian Monte Carlo (HMC)

Hamiltonian Monte Carlo (HMC) is an efficient Markov chain Monte Carlo (MCMC) method that uses Hamiltonian dynamics to generate proposals for sampling from complex posterior distributions. Unlike traditional random-walk MCMC methods, HMC uses gradient information from the log-posterior to guide the sampling trajectory, enabling rapid exploration of the parameter space and reducing autocorrelation among successive samples. This makes HMC particularly suitable for Bayesian inference involving fuzzy observations and nonstandard likelihood functions such as those arising from the logistic model considered here.

4.3.1. Working Rule for HMC

The two populations are modeled as

\begin{aligned} X \sim Logistic (μ_{1}, s), Y \sim Logistic (μ_{2}, s), \end{aligned}

where

μ_{1}

and

μ_{2}

are location parameters and

s

is the common scale parameter. The observed data consist of triangular fuzzy numbers

(L_{i}, P_{i}, U_{i})

representing lower, modal, and upper bounds.

Independent prior distributions are assigned as

\begin{aligned} μ_{1} \sim N (α_{1}, τ_{1}^{2}), μ_{2} \sim N (α_{2}, τ_{2}^{2}), s \sim Gamma (b, c) . \end{aligned}

The posterior density is proportional to the product of the fuzzy-data likelihood and the priors:

\begin{aligned} π (μ_{1}, μ_{2}, s ∣ X, Y) \propto L (μ_{1}, μ_{2}, s ∣ X, Y) π (μ_{1}) π (μ_{2}) π (s) . \end{aligned}

The gradient of the log-posterior is required for Hamiltonian dynamics:

\begin{aligned} \nabla_{θ} \log π (θ ∣ X, Y), θ = (μ_{1}, μ_{2}, s) . \end{aligned}

For fuzzy observations, this gradient is computed numerically because the log-likelihood involves fuzzy integrals.

HMC augments the parameter vector $θ$ with momentum variables $p = (p_{μ_{1}}, p_{μ_{2}}, p_{s})$ drawn from a standard normal distribution. The Hamiltonian is defined as

\begin{aligned} H (θ, p) = - \log π (θ ∣ X, Y) + \frac{1}{2} p^{⊤} p . \end{aligned}

Each HMC iteration consists of:

1.
Initialization: Start with current parameter vector $θ^{(t)}$ .
2.
Momentum Sampling: Draw $p^{(t)} \sim N (0, I)$ .
3.
Leapfrog Integration: Evolve the system using $L$ leapfrog steps of size $ϵ$ :
$\begin{aligned} p^{(t + \frac{1}{2})} = p^{(t)} + \frac{ϵ}{2} \nabla_{θ} \log π (θ^{(t)}), \\ θ^{(t + 1)} = θ^{(t)} + ϵ p^{(t + \frac{1}{2})}, \\ p^{(t + 1)} = p^{(t + \frac{1}{2})} + \frac{ϵ}{2} \nabla_{θ} \log π (θ^{(t + 1)}) . \end{aligned}$

4.
Metropolis–Hastings Correction: Accept the proposal $(θ^{(t + 1)}, p^{(t + 1)})$ with probability
$\begin{aligned} α = min {1, \exp (H (θ^{(t)}, p^{(t)}) - H (θ^{(t + 1)}, p^{(t + 1)}))} . \end{aligned}$
Otherwise, retain $θ^{(t)}$ .

After discarding burn-in iterations, the retained samples form an approximate posterior distribution. From these samples we compute: –
posterior means of $μ_{1}$ , $μ_{2}$ , and $s$ ,
–
Highest Posterior Density (HPD) credible intervals.

HMC thus provides an efficient and scalable sampling framework for Bayesian inference in the presence of fuzzy data. Its ability to explore the posterior distribution effectively makes it particularly well suited for the logistic model with fuzzy observations considered in this study.
4.4. Metropolis–Hastings (MH) Algorithm

The Metropolis–Hastings (MH) algorithm is a widely used Markov chain Monte Carlo (MCMC) technique for generating samples from a posterior distribution that cannot be sampled from directly. MH constructs a Markov chain whose stationary distribution is the target posterior, using a proposal mechanism followed by an acceptance–rejection step.

4.4.1. Working Rule for MH

The two populations under study follow logistic distributions with a common scale parameter:

\begin{aligned} X \sim Logistic (μ_{1}, s), Y \sim Logistic (μ_{2}, s), \end{aligned}

where

μ_{1}

and

μ_{2}

are location parameters and

s

is the shared scale parameter. The observed data

\tilde{X}

and

\tilde{Y}

consist of triangular fuzzy numbers.

The posterior density is proportional to the fuzzy likelihood multiplied by the prior:

\begin{aligned} π (θ ∣ \tilde{X}, \tilde{Y}) \propto L (\tilde{X}, \tilde{Y} ∣ θ) π (θ), θ = (μ_{1}, μ_{2}, s) . \end{aligned}

A proposal distribution $q (θ * ∣ θ)$ is selected to generate candidate values. A common choice is a multivariate normal random walk:

\begin{aligned} θ * \sim q (θ * ∣ θ_{t}) = N (θ_{t}, Σ), \end{aligned}

where

Σ

is a covariance matrix controlling the step size.

Given the current state $θ_{t}$ , generate a proposal $θ *$ from $q (θ * ∣ θ_{t})$ .

Compute the acceptance probability

\begin{aligned} α = min {1, \frac{L (\tilde{X}, \tilde{Y} ∣ θ *) π (θ *)}{L (\tilde{X}, \tilde{Y} ∣ θ_{t}) π (θ_{t})}}, \end{aligned}

which simplifies because the proposal is symmetric.

Accept or reject the proposal according to:

\begin{aligned} θ_{t + 1} = {\begin{cases} θ *, & if u \leq α, \\ θ_{t}, & otherwise, \end{cases} u \sim Uniform (0, 1) . \end{aligned}

Repeat this process for $N$ iterations to obtain the Markov chain

\begin{aligned} {θ_{1}, θ_{2}, \dots, θ_{N}}, \end{aligned}

which, after a burn-in period, provides dependent samples from the posterior distribution

π (θ ∣ \tilde{X}, \tilde{Y})

Posterior summaries are obtained using the retained samples, including:

$\cdot$ posterior means:

\begin{aligned} {\hat{θ}}_{Bayes} = E [θ ∣ \tilde{X}, \tilde{Y}], \end{aligned}

$\cdot$ credible intervals based on posterior quantiles:

\begin{aligned} [q_{2.5 %}, q_{97.5 %}] . \end{aligned}

The MH algorithm thus provides a flexible and likelihood-free tool for Bayesian inference under fuzzy observations, enabling sampling from complex posterior structures when analytical forms are not available.

5. Data Analysis

In this section, we present a comparative evaluation of different estimation methods under varying sample sizes. Specifically, we estimate the common scale parameter $s$ and the distinct scale parameters $μ_{1}$ and $μ_{2}$ for two fuzzy logistic populations using Maximum Likelihood Estimation (MLE), Lindley’s approximation, Approximate Bayesian Computation (ABC), Hamiltonian Monte Carlo (HMC), and the Metropolis-Hastings (MH) algorithm.

We consider the following parameter settings:

Set 1: $(μ_{1}, μ_{2}, s) = (1, 2, 2.5)$ (see Table 1)

Set 2: $(μ_{1}, μ_{2}, s) = (1.5, 2.5, 3)$ (see Table 2)

Table 1.
For Various Sample Sizes, We Compare estimated Value and Mean Square Errors of Multiple Estimators under Squared Error Loss for $θ = (μ_{1}, μ_{2}, s) = (1, 1.5, 2)$ .

EM(Fuzzy) Bayes(Lindley) Bayes(ABC) Bayes(HMC)) Bayes(MH))

(m,n) Estimate Error Estimate Error Estimate Error Estimate Error Estimate Error

(20,30) 1.0072 0.5984 1.9989 0.1876 0.8249 0.6545 1.2861 0.3766 1.0013 0.3465

1.5314 0.4129 2.2268 0.2147 1.0628 0.3994 1.2726 0.4667 1.6015 0.3878

1.9454 0.0597 2.8664 0.1305 1.5221 0.6771 2.0777 0.0684 2.0613 0.064

(50,40) 1.0097 0.2425 2.0526 0.1516 1.0451 0.5973 1.1567 0.2551 1.006 0.1802

1.5263 0.3153 2.3441 0.2271 1.3757 0.3285 1.3949 0.2079 1.4821 0.3081

1.9827 0.0296 2.9876 0.1129 1.9719 0.5918 2.0232 0.0326 2.0152 0.0335

(70,70) 0.9740 0.1713 1.5899 0.0720 0.8685 0.5763 1.3476 0.2609 0.9815 0.1383

1.5027 0.1814 1.6908 0.1395 1.6442 0.2293 1.4101 0.1495 1.7477 0.2104

1.9824 0.0195 2.3549 0.0782 2.0862 0.5859 1.9613 0.0212 1.9567 0.0217

(80,90) 1.0085 0.1533 1.5832 0.0629 0.9523 0.4781 0.9613 0.1042 0.8861 0.1318

1.4980 0.1264 1.7031 0.1005 1.6583 0.2256 1.3346 0.1449 1.3945 0.1093

1.9867 0.0170 2.3434 0.0598 2.0876 0.5707 1.876 0.0305 1.871 0.0311

(100,100) 0.9961 0.1116 1.5631 0.0467 1.0020 0.4252 0.8873 0.1057 0.8709 0.1109

1.4872 0.1242 1.7311 0.0937 1.5153 0.1918 1.6729 0.1178 1.6603 0.1201

1.9824 0.0142 2.3296 0.0563 1.9890 0.4526 1.827 0.0419 1.823 0.0431

	EM(Fuzzy)	Bayes(Lindley)	Bayes(ABC)	Bayes(HMC))	Bayes(MH))
(20,30)	1.0072	0.5984	1.9989	0.1876	0.8249	0.6545	1.2861	0.3766	1.0013	0.3465
	1.5314	0.4129	2.2268	0.2147	1.0628	0.3994	1.2726	0.4667	1.6015	0.3878
	1.9454	0.0597	2.8664	0.1305	1.5221	0.6771	2.0777	0.0684	2.0613	0.064
(50,40)	1.0097	0.2425	2.0526	0.1516	1.0451	0.5973	1.1567	0.2551	1.006	0.1802
	1.5263	0.3153	2.3441	0.2271	1.3757	0.3285	1.3949	0.2079	1.4821	0.3081
	1.9827	0.0296	2.9876	0.1129	1.9719	0.5918	2.0232	0.0326	2.0152	0.0335
(70,70)	0.9740	0.1713	1.5899	0.0720	0.8685	0.5763	1.3476	0.2609	0.9815	0.1383
	1.5027	0.1814	1.6908	0.1395	1.6442	0.2293	1.4101	0.1495	1.7477	0.2104
	1.9824	0.0195	2.3549	0.0782	2.0862	0.5859	1.9613	0.0212	1.9567	0.0217
(80,90)	1.0085	0.1533	1.5832	0.0629	0.9523	0.4781	0.9613	0.1042	0.8861	0.1318
	1.4980	0.1264	1.7031	0.1005	1.6583	0.2256	1.3346	0.1449	1.3945	0.1093
	1.9867	0.0170	2.3434	0.0598	2.0876	0.5707	1.876	0.0305	1.871	0.0311
(100,100)	0.9961	0.1116	1.5631	0.0467	1.0020	0.4252	0.8873	0.1057	0.8709	0.1109
	1.4872	0.1242	1.7311	0.0937	1.5153	0.1918	1.6729	0.1178	1.6603	0.1201
	1.9824	0.0142	2.3296	0.0563	1.9890	0.4526	1.827	0.0419	1.823	0.0431

Table 2.

For Various Sample Sizes, We Compare estimated Value and Mean Square Errors of Multiple Estimators under Squared Error Loss for $θ = (μ_{1}, μ_{2}, s) = (1.5, 2, 2.5)$ .

	EM(Fuzzy)		Bayes(Lindley)		Bayes(ABC)		Bayes(HMC)		Bayes(MH)
(m,n)	Estimate	Error	Estimate	Error	Estimate	Error	Estimate	Error	Estimate	Error
(20,30)	$\begin{array}{l} 1.0811 \\ 1.6154 \\ 1.9244 \end{array}$	$\begin{array}{l} 0.0645 \\ 0.1221 \\ 0.1767 \end{array}$	$\begin{array}{l} 1.9448 \\ 2.3534 \\ 3.1726 \end{array}$	$\begin{array}{l} 0.7771 \\ 0.6046 \\ 0.0940 \end{array}$	$\begin{array}{l} 0.8249 \\ 1.0628 \\ 1.5221 \end{array}$	$\begin{array}{l} 0.6545 \\ 0.3994 \\ 0.6771 \end{array}$	$\begin{array}{l} 1.1237 \\ 1.9887 \\ 2.5668 \end{array}$	$\begin{array}{l} 0.6672 \\ 0.3864 \\ 0.0966 \end{array}$	$\begin{array}{l} 1.1271 \\ 1.927 \\ 2.5475 \end{array}$	$\begin{array}{l} 0.5769 \\ 0.5006 \\ 0.0949 \end{array}$
(50,40)	$\begin{array}{l} 1.5122 \\ 2.0328 \\ 2.4784 \end{array}$	$\begin{array}{l} 0.3789 \\ 0.4927 \\ 0.0463 \end{array}$	$\begin{array}{l} 1.2836 \\ 2.0041 \\ 3.4070 \end{array}$	$\begin{array}{l} 0.3792 \\ 0.4737 \\ 0.0493 \end{array}$	$\begin{array}{l} 1.0451 \\ 1.3757 \\ 1.9719 \end{array}$	$\begin{array}{l} 0.5973 \\ 0.3285 \\ 0.5918 \end{array}$	$\begin{array}{l} 1.3001 \\ 1.9164 \\ 2.5157 \end{array}$	$\begin{array}{l} 0.3158 \\ 0.3239 \\ 0.0495 \end{array}$	$\begin{array}{l} 1.3112 \\ 1.8432 \\ 2.5059 \end{array}$	$\begin{array}{l} 0.2982 \\ 0.4692 \\ 0.0501 \end{array}$
(70,70)	$\begin{array}{l} 1.4675 \\ 2.0033 \\ 2.4780 \end{array}$	$\begin{array}{l} 0.2677 \\ 0.2834 \\ 0.0305 \end{array}$	$\begin{array}{l} 1.2585 \\ 1.7044 \\ 2.8371 \end{array}$	$\begin{array}{l} 0.2987 \\ 0.3056 \\ 0.0388 \end{array}$	$\begin{array}{l} 0.8685 \\ 1.6442 \\ 2.0862 \end{array}$	$\begin{array}{l} 0.5763 \\ 0.2293 \\ 0.5859 \end{array}$	$\begin{array}{l} 1.3343 \\ 2.2383 \\ 2.443 \end{array}$	$\begin{array}{l} 0.231 \\ 0.2583 \\ 0.033 \end{array}$	$\begin{array}{l} 1.3423 \\ 2.1855 \\ 2.4389 \end{array}$	$\begin{array}{l} 0.2117 \\ 0.2757 \\ 0.0359 \end{array}$
(80,90)	$\begin{array}{l} 1.5107 \\ 1.9974 \\ 2.4834 \end{array}$	$\begin{array}{l} 0.2395 \\ 0.1974 \\ 0.0266 \end{array}$	$\begin{array}{l} 1.3143 \\ 2.0649 \\ 2.8170 \end{array}$	$\begin{array}{l} 0.3073 \\ 0.2357 \\ 0.0275 \end{array}$	$\begin{array}{l} 0.9523 \\ 1.6583 \\ 2.0876 \end{array}$	$\begin{array}{l} 0.4781 \\ 0.2256 \\ 0.5707 \end{array}$	$\begin{array}{l} 1.255 \\ 1.8316 \\ 2.3368 \end{array}$	$\begin{array}{l} 0.2311 \\ 0.1785 \\ 0.0489 \end{array}$	$\begin{array}{l} 1.2591 \\ 1.8001 \\ 2.3339 \end{array}$	$\begin{array}{l} 0.2204 \\ 0.2161 \\ 0.0503 \end{array}$
(100,100)	$\begin{array}{l} 1.4951 \\ 1.9841 \\ 2.4780 \end{array}$	$\begin{array}{l} 0.1744 \\ 0.1941 \\ 0.0221 \end{array}$	$\begin{array}{l} 1.4221 \\ 2.0841 \\ 2.8688 \end{array}$	$\begin{array}{l} 0.2284 \\ 0.2302 \\ 0.0207 \end{array}$	$\begin{array}{l} 1.0020 \\ 1.5153 \\ 1.9890 \end{array}$	$\begin{array}{l} 0.4252 \\ 0.1918 \\ 0.4526 \end{array}$	$\begin{array}{l} 1.2689 \\ 2.1527 \\ 2.2791 \end{array}$	$\begin{array}{l} 0.1904 \\ 0.1534 \\ 0.0682 \end{array}$	$\begin{array}{l} 1.2484 \\ 2.1293 \\ 2.2768 \end{array}$	$\begin{array}{l} 0.1966 \\ 0.1665 \\ 0.0679 \end{array}$

Fuzzy numbers for $X$ and $Y$ are generated as follows:

Peak values are sampled from the respective logistic distributions.

Lower and upper bounds of the triangular fuzzy numbers are constructed by subtracting and adding a small random value (sampled from a uniform distribution) to the peak.

A triangular membership function is applied to determine the membership degree across the fuzzy intervals.

We assess the performance of each estimator using the following metrics:

Average Value (Estimate)—the mean of the estimated values across replications.

Mean Square Error (MSE)—to quantify estimation accuracy and variability.

To ensure robust performance comparison, we simulate 10,000 datasets from each parameter setting across different sample sizes. The prior hyperparameters used in Bayesian estimation are fixed as follows: $μ_{1} \sim N (α_{1}, τ_{1}^{2})$ , $μ_{2} \sim N (α_{2}, τ_{2}^{2})$ and $s \sim Gamma (b, c)$ , where $α_{1} = α_{2} = 1$ , $τ_{1}^{2} = τ_{2}^{2} = 2$ , $b = 1$ , and $c = 2$ .

Tables 1 and 2 summarize the estimated values and MSEs for each method. The first column denotes the sample size, followed by the true parameter values. The subsequent columns report the estimation results for MLE, Lindley’s method, ABC, HMC, and MH.

Tables 3 and 4 provide interval estimates:

The third column shows the 95% asymptotic confidence intervals derived using the observed Fisher information matrix.

The fourth column lists the 95% Highest Posterior Density (HPD) intervals obtained via HMC and MH sampling.

Table 3.

95 $%$ asymptotic and HPD Intervals using ML Estimates and HMC, MH Estimates for the Parameters $(μ_{1}, μ_{2}, s) = (1, 1.5, 2)$ at Various Sample Sizes.

(m,n)	Observed	Asymptotic	HMC	MH
(20,30)	$\begin{array}{l} μ_{1} = 1 \\ μ_{2} = 1.5 \\ s = 2 \end{array}$	$\begin{array}{l} [- 0.5097, 2.5241] \\ [0.2729, 2.7899] \\ [1.4786, 2.4122] \end{array}$	$\begin{array}{l} [0.0538, 2.5740] \\ [0.2422, 2.3758] \\ [1.6174, 2.5737] \end{array}$	$\begin{array}{l} [- 0.067, 2.1280] \\ [0.37602.7514] \\ [1.6087, 2.5577] \end{array}$
(50,40)	$\begin{array}{l} μ_{1} = 1 \\ μ_{2} = 1.5 \\ s = 2 \end{array}$	$\begin{array}{l} [0.0443, 1.9751] \\ [0.4263, 2.6263] \\ [1.6470, 2.3184] \end{array}$	$\begin{array}{l} [0.2470, 2.1354] \\ [0.5038, 2.2334] \\ [1.6948, 2.3852] \end{array}$	$\begin{array}{l} [0.2421, 1.8756] \\ [0.3991, 2.4925] \\ [1.6784, 2.3855] \end{array}$
(70,70)	$\begin{array}{l} μ_{1} = 1 \\ μ_{2} = 1.5 \\ s = 2 \end{array}$	$\begin{array}{l} [0.1640, 1.7840] \\ [0.6676, 2.3378] \\ [1.7107, 2.2541] \end{array}$	$\begin{array}{l} [0.6185, 2.0783] \\ [0.7059, 2.1800] \\ [1.6905, 2.2350] \end{array}$	$\begin{array}{l} [0.2767, 1.7129] \\ [0.9726, 2.5050] \\ [1.6857, 2.2510] \end{array}$
(80,90)	$\begin{array}{l} μ_{1} = 1 \\ μ_{2} = 1.5 \\ s = 2 \end{array}$	$\begin{array}{l} [0.2410, 1.7760] \\ [0.8009, 2.1951] \\ [1.7320, 2.2414] \end{array}$	$\begin{array}{l} [0.3199, 1.5647] \\ [0.6403, 1.9826] \\ [1.6362, 2.1123] \end{array}$	$\begin{array}{l} [0.2317, 1.5851] \\ [0.7905, 2.0122] \\ [1.6493, 2.1185] \end{array}$
(100,100)	$\begin{array}{l} μ_{1} = 1 \\ μ_{2} = 1.5 \\ s = 2 \end{array}$	$\begin{array}{l} [0.3410, 1.6512] \\ [0.7966, 2.1778] \\ [1.7517, 2.2131] \end{array}$	$\begin{array}{l} [0.3109, 1.5039] \\ [1.0920, 2.2471] \\ [1.6294, 2.0518] \end{array}$	$\begin{array}{l} [0.2700, 1.4642] \\ [1.0873, 2.2744] \\ [1.6113, 2.0376] \end{array}$

Table 4.

95 $%$ asymptotic and HPD Intervals using ML Estimates, HMC and MH Estimates for the Parameters $(μ_{1}, μ_{2}, s) = (1.5, 2, 2.5)$ at Various Sample Sizes.

(m,n)	Observed	Asymptotic	HMC	MH
(20,30)	$\begin{array}{l} μ_{1} = 1.5 \\ μ_{2} = 2 \\ s = 2.5 \end{array}$	$\begin{array}{l} [0.3873, 3.4051] \\ [0.4661, 3.6123] \\ [1.8482, 3.0152] \end{array}$	$\begin{array}{l} [- 0.1926, 2.6342] \\ [0.8152, 3.2256] \\ [1.9993, 3.1610] \end{array}$	$\begin{array}{l} [- 0.0804, 2.3503] \\ [0.5273, 3.2666] \\ [1.9382, 3.1059] \end{array}$
(50,40)	$\begin{array}{l} μ_{1} = 1.5 \\ μ_{2} = 2 \\ s = 2.5 \end{array}$	$\begin{array}{l} [0.3054, 2.7190] \\ [0.6579, 3.4077] \\ [2.0587, 2.8981] \end{array}$	$\begin{array}{l} [0.2503, 2.3118] \\ [0.8597, 3.0649] \\ [2.1105, 2.9706] \end{array}$	$\begin{array}{l} [0.3716, 2.3133] \\ [0.5841, 3.1257] \\ [2.0850, 2.9604] \end{array}$
(70,70)	$\begin{array}{l} μ_{1} = 1.5 \\ μ_{2} = 2 \\ s = 2.5 \end{array}$	$\begin{array}{l} [0.4549, 2.4801] \\ [0.9595, 3.0471] \\ [2.1383, 2.8177] \end{array}$	$\begin{array}{l} [0.4267, 2.1903] \\ [1.3697, 3.1229] \\ [2.1339, 2.8055] \end{array}$	$\begin{array}{l} [0.5557, 2.2184] \\ [1.2812, 3.1793] \\ [2.1064, 2.8111] \end{array}$
(80,90)	$\begin{array}{l} μ_{1} = 1.5 \\ μ_{2} = 2 \\ s = 2.5 \end{array}$	$\begin{array}{l} [0.5513, 2.4701] \\ [1.1261, 2.8687] \\ [2.1651, 2.8017] \end{array}$	$\begin{array}{l} [0.4427, 2.0591] \\ [1.0992, 2.6149] \\ [2.0376, 2.6199] \end{array}$	$\begin{array}{l} [0.4775, 2.0345] \\ [0.9585, 2.5821] \\ [2.0489, 2.6400] \end{array}$
(100,100)	$\begin{array}{l} μ_{1} = 1.5 \\ μ_{2} = 2 \\ s = 2.5 \end{array}$	$\begin{array}{l} [0.6762, 2.3140] \\ [1.1208, 2.8474] \\ [2.1896, 2.7664] \end{array}$	$\begin{array}{l} [0.5363, 1.9854] \\ [1.4404, 2.8447] \\ [2.0194, 2.5593] \end{array}$	$\begin{array}{l} [0.6058, 2.0155] \\ [1.3774, 2.8560] \\ [2.0053, 2.5342] \end{array}$

The simulation study leads to the following observations:

MSE consistently decreases with increasing sample size, confirming the consistency of all estimators.

Bayesian methods (Lindley, ABC, HMC, MH) provide stable and accurate parameter estimates, especially in small-sample scenarios.

HMC performs particularly well for small sample sizes, yielding lower bias and MSE than MLE, especially for the scale parameters $s_{1}$ and $s_{2}$ .

Lindley’s approximation offers computationally efficient and competitive estimates, showing performance comparable to MLE and other Bayesian methods.

ABC is effective in cases with intractable likelihoods and performs well with fuzzy data, though it may require careful tuning of the tolerance parameter.

HPD intervals from HMC and MH are generally narrower and more informative than traditional asymptotic intervals, particularly for the fuzzy data setting.

Overall, the integration of fuzzy data into the Bayesian framework enhances estimation robustness and provides a more realistic reflection of uncertainty in survival data analysis.

5.1. Remarks on Consistency and Robustness under Fuzzy Data

Although explicit theoretical proofs are difficult to obtain for likelihood-based inference under fuzzy observations, the proposed estimators inherit desirable asymptotic properties from their classical counterparts. As the sample size increases and the degree of fuzziness decreases, the fuzzy likelihood converges to the classical likelihood, implying consistency of the EM-based maximum likelihood estimators as well as the approximate Bayesian estimators. This behavior is empirically supported by the simulation results, where the mean square errors consistently decrease with increasing sample size. Moreover, Bayesian estimators obtained via HMC and MH demonstrate robustness to data imprecision by providing stable posterior summaries and wider, yet informative, HPD intervals that appropriately reflect uncertainty under fuzzy data.

6. Real Aata Analysis

To assess the performance of our model, we utilized data from an engineering dataset provided by Pak and Parham (2013), which includes a case study on the light-emitting diodes (LED) manufacturing process, specifically focusing on the luminous intensities of LED sources. The data is shown in Table 5. We validated that the distribution of this data adheres to a logistic distribution using the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests, with the results displayed in Table 6.

Table 5.
The Breakdown Lifetimes at 32 and 34 KV of an Insulating Fluid.

Data Sets

Set 1 (2.163, 2.738, 3.068), (5.972, 6.353, 8.150), (1.032, 1.971, 2.642), (1.336, 2.750, 3.284), (0.628, 0.964, 1.735), (2.995, 3.442, 5.066), (3.766, 5.814, 6.212), (5.434, 7.093, 7.655), (0.974, 1.839, 2.045), (4.352, 5.206, 5.988), (3.920, 4.762, 6.121), (4.065, 5.312, 7.480), (1.375, 2.195, 3.086), (0.618, 0.839, 2.217), (4.575, 6.050, 6.734)

Set 2 (6.032, 7.746, 8.529), (1.027, 1.218, 3.116), (6.279, 8.156, 9.435), (2.821, 3.409, 5.272), (2.247, 2.990, 4.128), (7.125, 8.470, 9.044), (5.443, 6.231, 7.395), (1.766, 2.190, 2.638), (7.261, 8.325, 8.871), (7.155, 8.013, 8.352), (0.830, 1.288, 2.541), (3.590, 4.169, 4.899), (4.634, 5.780, 7.058), (5.965, 7.344, 8.019), (3.177, 3.600, 4.213)

	Data Sets
Set 1	(2.163, 2.738, 3.068), (5.972, 6.353, 8.150), (1.032, 1.971, 2.642), (1.336, 2.750, 3.284), (0.628, 0.964, 1.735), (2.995, 3.442, 5.066), (3.766, 5.814, 6.212), (5.434, 7.093, 7.655), (0.974, 1.839, 2.045), (4.352, 5.206, 5.988), (3.920, 4.762, 6.121), (4.065, 5.312, 7.480), (1.375, 2.195, 3.086), (0.618, 0.839, 2.217), (4.575, 6.050, 6.734)
Set 2	(6.032, 7.746, 8.529), (1.027, 1.218, 3.116), (6.279, 8.156, 9.435), (2.821, 3.409, 5.272), (2.247, 2.990, 4.128), (7.125, 8.470, 9.044), (5.443, 6.231, 7.395), (1.766, 2.190, 2.638), (7.261, 8.325, 8.871), (7.155, 8.013, 8.352), (0.830, 1.288, 2.541), (3.590, 4.169, 4.899), (4.634, 5.780, 7.058), (5.965, 7.344, 8.019), (3.177, 3.600, 4.213)

Table 6.

KS and AD Test Results.

		KS Test		AD Test
Data	Parameter	Statistic	$P$ -value	Statistic	$P$ -value
Set 1	$\begin{array}{l} μ_{1} = 4, \\ s = 1.5 \end{array}$	0.1739	0.692	0.6061	0.0937
Set 2	$\begin{array}{l} μ_{2} = 5, \\ s = 1.5 \end{array}$	0.2092	0.4652	0.6579	0.0686

6.1. Goodness-of-Fit Analysis

To evaluate whether the breakdown lifetimes at 32 and 34 KV follow a Logistic distribution, we applied the Kolmogorov-Smirnov (KS) test and the Anderson-Darling (AD) test. These tests assess whether an empirical distribution significantly deviates from a theoretical distribution.

The KS test is a non-parametric test comparing the empirical cumulative distribution function (ECDF) with the theoretical logistic distribution, while the AD test places more weight on the tails of the distribution, making it more sensitive to deviations in extreme values. These tests were selected to ensure that the fitted distribution is robust across different parts of the dataset.

For Set 1 (32 KV), the KS ( $p = 0.692$ ) and AD ( $p = 0.09376$ ) tests confirm that the data does not significantly deviate from the Logistic distribution. Similarly, for Set 2 (34 KV), KS ( $p = 0.4652$ ) and AD ( $p = 0.06864$ ) tests validate the goodness-of-fit. These results suggest that the logistic model is appropriate for both datasets.

6.2. Wald Test Results

The Wald test was performed to evaluate the null hypothesis that the scale parameters of the two populations are equal ( $H_{0} : s_{1} = s_{2}$ ). Table 7 presents the test results, which include the estimated scale parameters, their variances, and the test statistics. The computed Wald statistic is 0.03844, with a corresponding $p$ -value of 0.8445. Since the $p$ -value is significantly greater than the conventional significance level ( $α = 0.05$ ), we fail to reject the null hypothesis.

Table 7.
Wald Test for Equality of Scale Parameters.

Wald Test

Data $\hat{s}$ V( $s$ ) Statistic $P$ -value

Set 1 1.2324 0.1444 0.3552 0.5511

Set 2 1.5236 0.0944

	Wald Test
Set 1	1.2324	0.1444	0.3552	0.5511
Set 2	1.5236	0.0944

This suggests that there is no statistically significant evidence to conclude that the scale parameters of the two populations differ. Practically, this implies that the two datasets share similar variability, supporting the assumption of a common scale parameter.

In our study, each data point is represented using triangular fuzzy numbers ${\tilde{x}}_{i}$ and ${\tilde{y}}_{j}$ , which capture the possibility distributions associated with the imprecise values $x_{i}$ and $y_{j}$ . These are considered as realizations of the random variables $X_{i}$ and $Y_{j}$ , respectively. To estimate the parameters $(μ_{1}, μ_{2}, s)$ of the logistic distribution, we employed both Maximum Likelihood Estimation (MLE) and Bayesian approaches, explicitly incorporating the fuzzy nature of the data. For Bayesian inference, we utilized Lindley’s approximation, Approximate Bayesian Computation (ABC), Hamiltonian Monte Carlo (HMC), and the Metropolis–Hastings (MH) algorithm. Additionally, we computed asymptotic confidence intervals based on MLE and Highest Posterior Density (HPD) intervals through HMC and MH methods to evaluate the precision and reliability of the parameter estimates.

The outcomes, summarized in Tables 8 and 9, provide comparative insights into the effectiveness of the different estimation strategies under uncertainty. Figure 1 displays the density plots for the parameters $(μ_{1}, μ_{2}, s)$ , which validate the convergence and distributional properties of the Bayesian estimates. Trace plots confirm that the Markov chains have reached stationarity, with mean lines indicating stable behavior. The corresponding density plots reveal the posterior distributions, highlighting concentrated regions around the estimated values, thereby supporting the credibility of the Bayesian approach under fuzzy data.

Figure 1.

Illustration of Density plots of parameters.

Table 8.

Maximum Likelihood and Bayesian Estimators of the Combined Model.

Method	$μ_{1}$	$μ_{2}$	$s$
MLE(EM)	3.8080	5.3669	1.3799
Lindley	3.7994	5.3667	1.3546
ABC	3.5334	4.7358	1.3001
HMC	3.7115	5.2395	1.4767
MH	3.7780	5.2964	1.4734

Table 9.

95% Asymptotic and HPD Intervals using ML Estimates, HMC and MH Estimates for the Parameters $(μ_{1}, μ_{2}, s)$ .

Method	$μ_{1}$	$μ_{2}$	$s$
EM	[2.5923, 5.0238]	[4.0469, 6.6869]	[0.9881, 1.7715]
HMC	[2.4724, 4.9862]	[3.9293, 6.5657]	[1.0797, 1.9541]
MH	[2.4719, 4.9885]	[3.8754, 6.8019]	[1.0707, 1.9974]

Table 8 shows that Bayesian estimates closely align with those obtained via MLE. Lindley’s method gives results almost identical to the EM estimates, indicating its strong performance with imprecise inputs. ABC slightly underestimates the parameters, whereas HMC and MH produce consistent yet slightly conservative estimates, suggesting their better accommodation of data uncertainty.

As observed in Table 9, the 95% HPD intervals from HMC and MH are generally wider than the MLE-based asymptotic intervals, particularly for the scale parameter $s$ . This reflects the Bayesian framework’s ability to incorporate data fuzziness, resulting in more cautious uncertainty quantification. Furthermore, the HPD intervals from both HMC and MH are comparable, indicating robustness and reliability across these sampling techniques.

From a computational perspective, the EM-based maximum likelihood estimation and Lindley’s approximation were found to be the most efficient methods, as they rely on deterministic optimization and low-dimensional numerical calculations. Approximate Bayesian Computation required moderate computational effort due to repeated simulation of synthetic datasets and acceptance–rejection steps. Among the Bayesian sampling methods, Hamiltonian Monte Carlo and the Metropolis–Hastings algorithm were the most computationally intensive, owing to their iterative nature and the need for a large number of iterations to ensure adequate convergence. However, these methods provide richer posterior inference and more comprehensive uncertainty quantification, which may justify the additional computational cost in applied fuzzy data settings.

7. Conclusion and Future Work

This study investigated parameter estimation for two logistic populations sharing a common scale parameter under fuzzy data conditions. Both classical and Bayesian estimation approaches were examined, including EM-based maximum likelihood estimation, Lindley’s approximation, Approximate Bayesian Computation, Hamiltonian Monte Carlo, and the Metropolis–Hastings algorithm. Simulation results demonstrated the consistency of all estimators, with Bayesian methods exhibiting superior performance in small-sample scenarios and providing improved uncertainty quantification under fuzziness. The real-data application further illustrated the practical relevance of the proposed framework and confirmed the suitability of the logistic model for fuzzy observations. Overall, this study highlights the importance of incorporating fuzzy data into multi-population inference and provides practical guidance for selecting appropriate estimation methods in the presence of data imprecision.

Footnotes

ORCID iDs

Lingutla Vijay Kumar

Nadiminti Nagamani

R. K. Davala

Funding Statement

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Ashok

Nagamani

(2025). Adaptive estimation: Fuzzy data-driven gamma distribution via Bayesian and maximum likelihood approaches. AIMS Mathematic, 10(1), 438–451. https://doi.org/10.3934/math.2025021

Azhad

Tripathy

M. R.

(2021). Estimation of parameters of two Weibull populations. Statistics in Transition, 22, 1–22.

Balakrishnan

Basu

A. P.

(1995). The exponential distribution: Theory, methods and applications. Gordon and Breach Science Publishers.

Dempster

A. P.

Laird

N. M.

Rubin

D. B.

(1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x

Ghosh

Meeden

(1984). Bayesian estimation and prediction. Marcel Dekke.

Graybill

F. A.

Deal

R. D.

(1959). Combining unbiased estimators. Biometrics, 15, 543–550. https://doi.org/10.2307/2527652

Jin.

Pal

(1992). On common location of several exponentials under a class of convex loss functions. Calcutta Statistical Association Bulletin, 42(3–4), 191–200. https://doi.org/10.1177/0008068319920304

Khoolenjani

N. B.

Shahsanaie

(2016). Estimating the parameter of Exponential distribution under Type-II censoring from fuzzy data. Journal of Statistical Theory and Applications, 15(2), 181–195. https://doi.org/10.2991/jsta.2016.15.2.8

Kotz

Balakrishnan

Johnson

N. L.

(2019). Continuous multivariate distributions (Vol. 1). Wiley.

10.

Lindley

D. V.

(1980). Approximate Bayesian methods. In J. M. Bernardo et al. (Eds.), Bayesian statistics (pp. 223–245). University Press.

11.

Lingutla

V. K.

Nagamani

(2024). Bayesian approach for heavy-tailed model fitting in two Lomax populations. Reliability: Theory & Applications, 4, 132–150. https://doi.org/10.24412/1932-2321-2024-480-132-150

12.

Moore

Krishnamoorthy

(1997). Combining independent normal sample means by weighting with their standard errors. Journal of Statistical Computation and Simulation, 58, 145–153. https://doi.org/10.1080/00949659708811827

13.

Nadarajah

(2004). Information measures for gamma distributions. Statistics, 38, 199–213. https://doi.org/10.1080/02331880412331319305

14.

Nagamani

Tripathy

M. R.

(2017). Estimating common scale parameter of two gamma populations: A simulation study. Journal of Statistics and Management Systems, 36, 346–362. https://doi.org/10.1080/01966324.2017.1369473

15.

Nagamani

Tripathy

M. R.

(2018). Estimating common dispersion parameter of several inverse Gaussian populations: A simulation study. Journal of Statistics and Management Systems, 21, 1357–1389. https://doi.org/10.1080/09720510.2018.1503406

16.

Nagamani

Tripathy

M. R.

Kumar

(2020). Estimating common scale parameter of two logistic populations: A Bayesian study. American Journal of Mathematical and Management Sciences, 40, 44–67. https://doi.org/10.1080/01966324.2020.1833794

17.

Pak

(2020). Bayesian analysis of weighted distributions. Statistics and Probability Letters, 162, 108753. https://doi.org/10.1016/j.spl.2020.108753

18.

Pak

Parham

(2013). Inference for weighted exponential distributions. Communications in Statistics – Theory and Methods, 42, 1143–1158. https://doi.org/10.1080/03610926.2013.759778

19.

Rashad

Mohmud

Yusuf

(2016). Bayes estimation of the logistic distribution parameters based on progressive sampling. Applied Mathematics & Information Sciences, 10, 2293–2301. https://doi.org/10.18576/amis/100632

20.

Roohanizadeh

Zahra

Ezzatallah

(2022). Parameters and reliability estimation for the Weibull distribution based on intuitionistic fuzzy lifetime data. Complex & Intelligent Systems, 8, 4881–4896. https://doi.org/10.1007/s40747-022-00720-x

21.

Tripathy

M. R.

Kumar

(2015). Equivariant estimation of common mean of several normal populations. Journal of Statistical Computation and Simulation, 85, 3679–3699. https://doi.org/10.1080/00949655.2014.995658

22.

Tripathy

M. R.

Kumar

Misra

(2014). Estimating the common location of two exponential populations under order restricted failure rates. American Journal of Mathematical and Management Sciences, 33, 125–146. https://doi.org/10.1080/01966324.2014.908331

23.

Tripathy

M. R.

Nagamani

(2017). Estimating common shape parameter of two gamma populations: A simulation study. Journal of Statistics and Management Systems, 20, 369–398. https://doi.org/10.1080/09720510.2017.1292688

24.

Yang

(2007). Improved estimation of a common mean. Journal of Statistical Planning and Inference, 137, 1990–1999. https://doi.org/10.1016/j.jspi.2006.03.011

25.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8, 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

26.

Zadeh

L. A.

(1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications, 23, 421–427. https://doi.org/10.1016/0022-247X(68)90078-4

	Wald Test
Data	$\hat{s}$	V( $s$ )	Statistic	$P$ -value
Set 1	1.2324	0.1444	0.3552	0.5511
Set 2	1.5236	0.0944

Parameter Estimation for Two Logistic Populations with Fuzzy Data: A Comparative Study of MLE and Bayesian Methods

Abstract

Keywords

1. Introduction

1.1. Key Contributions

2. Theoretical Background

2.1. Logistic Distribution

– μ is the location parameter, which represents the median of the distribution. – s > 0 is the scale parameter, which controls the spread of the distribution. 2.2. Fuzzy Set Theory

3. Maximum Likelihood Estimation

4.2.1. ABC Procedure

4.3. Hamiltonian Monte Carlo (HMC)

4.3.1. Working Rule for HMC

4.4.1. Working Rule for MH

5. Data Analysis

6. Real Aata Analysis

6.2. Wald Test Results

Table 7. Wald Test for Equality of Scale Parameters. Wald Test Data s ^ V( s ) Statistic P -value Set 1 1.2324 0.1444 0.3552 0.5511 Set 2 1.5236 0.0944

Footnotes

ORCID iDs

Funding Statement

Declaration of Conflicting Interests

References

–
$μ$ is the location parameter, which represents the median of the distribution.
–
$s > 0$ is the scale parameter, which controls the spread of the distribution.

2.2. Fuzzy Set Theory

Table 7.
Wald Test for Equality of Scale Parameters.

Wald Test

Data $\hat{s}$ V( $s$ ) Statistic $P$ -value

Set 1 1.2324 0.1444 0.3552 0.5511

Set 2 1.5236 0.0944