Estimating Common Scale Parameter of Two Logistic Populations: A Bayesian Study

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.

This article deals with the problem of reliability in a multicomponent stress-strength (MSS) model when both stress and strength variables are from exponentiated Teissier (ET) distributions. The reliability of the system is determined using both classical and Bayesian methods, based on two scenarios where the common scale parameter is unknown or known. In the first scenario, where the common scale parameter is unknown, the maximum likelihood estimation (MLE) and the approximate Bayes estimation are derived. In the second scenario, where the scale parameter is known, the MLE, the uniformly minimum variance unbiased estimator (UMVUE) and the exact Bayes estimation are obtained. In the both scenarios, the asymptotic confidence interval and the highest probability density credible interval are established. Furthermore, two other asymptotic confidence intervals are computed based on the Logit and Arcsin transformations. Monte Carlo simulations are implemented to compare the different proposed methods. Finally, one real example is presented in support of suggested procedures.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

Inference on adaptive progressive hybrid censored accelerated life test for Gompertz distribution and its evaluation for virus-containing micro droplets data

In this paper, we discuss the estimation problem of the location and scale parameters of the log-Cauchy distribution, a member of the super heavy-tailed distributions family. We consider several methods of estimation, including a new percentile method, maximum likelihood estimation and robust estimators. A Monte Carlo simulation experiment is conducted to compare the proposed estimation methods. Further, we illustrate the estimation methods via a real life example.

In this article, the reliability inference for a multicomponent stress-strength (MSS) model, when both stress and strength random variables follow inverse Topp-Leone distributions, was studied. The maximum likelihood and uniformly minimum variance unbiased estimates for the reliability of MSS model were obtained explicitly. The exact Bayes estimate of MSS reliability was derived the under squared error loss function. Also, the Bayes estimate was obtained using the Monte Carlo Markov Chain method for comparison with the aforementioned exact estimate. The asymptotic confidence interval was determined under the expected Fisher information matrix. Furthermore, the highest probability density credible interval was established through using Gibbs sampling method. Monte Carlo simulations were implemented to compare the different proposed methods. Finally, a real life example was presented in support of the suggested procedures.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

ABSTRACTWe consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

In this article, the approximate Bayes estimation problem for the log-Dagum distribution with three parameters is considered. Firstly, the maximum likelihood estimators and asymptotic confidence intervals based on these estimators for unknown parameters of log-Dagum distribution are constructed. In addition, approximate Bayes estimators under squared error loss function for unknown parameters of this distribution are obtained using Tierney and Kadane approximation. A Monte-Carlo simulation study is performed to compare performances of maximum likelihood and approximate Bayes estimators in terms of mean square errrors and biases. Finally, real data analysis for this distribution is performed.

We consider here the generalization of the Bilal distribution proposed by Abd-Elrahman (2017) by zeroing in on two measures of reliability, R(t) and P, based on type II censoring. We obtain point estimators namely, λ and θ, of the above said distribution, when both parameters of the distribution are unknown. Maximum likelihood estimators (MLEs), Bayes estimators (BEs) and Lindley’s approximation for the Bayes estimators are proposed. By using independent noninformative type of priors for the unknown parameters Bayes estimators are derived. Although the proposed estimators cannot be expressed in closed forms, these can be easily obtained through the use of numerical procedures. The performance of these estimators is studied on the basis of their mean squared error (MSE), computed separately under LINEX loss function (LLF) and squared error loss function (SELF) through Monte-Carlo simulation technique.

In this article we consider statistical inferences about the unknown parameters of the inverse Weibull distribution based on progressively first-failure censoring using Bayesian procedures. The Bayes estimators are obtained based on both the symmetric and asymmetric (Linex, General Entropy and Precautionary) loss functions. There are no explicit forms for the Bayes estimators; therefore, we propose the Lindley’s approximation method to compute the Bayes estimators. A comparison between these estimators and the maximum likelihood estimator (MLE) is provided by using extensive simulation and two criteria, namely, the bias and the mean squared error. It is concluded that the approximate Bayes estimators outperform the MLEs most of the time. Real life data example is provided to illustrate our proposed estimators.

The EM algorithm is a widely used technique for finding maximum likelihood (ML) estimates when the data are not fully observed. Despite its popularity for computing ML estimates in unrestricted problems and the need for simplified computations for problems with equality and inequality restrictions, there have been few applications of the algorithm to restricted ML estimation. The EM algorithm is presented for restricted ML estimation and provides its applications to the probit model under equality and inequality restrictions using two small data sets.

Receiver Operating Characteristic (ROC) curves are used to describe and compare the accuracy of diagnostic test or the ability of a continuous biomarker in discriminating between the subjects into healthy or diseased cases in medical field. The most familiar form of ROC curve is Bi-normal (Gaussian) ROC curve model, which assumes that the test scores or a monotone transformation of the test scores are from two normal populations (i.e. healthy and diseased). It may not be true all the time, it may violate the assumptions of normal distribution in some situations and also we cannot adopt the model as it is when the sample size is small. In this paper, we have proposed ROC curve model for Rayleigh distribution which can be used even when sample size is small. The properties of Bi-Rayleigh ROC model are studied and Area Under the ROC Curve (AUC) are derived. The proposed model is supported by real life example as well as simulation studies. The confidence interval for the population parameter is studied with simulation studies of varying sample sizes. It is found that Bi-Rayleigh ROC model provides better accuracy of classification than the conventional bi-normal ROC model. In the ROC model section, Bi-Rayleigh ROC model and its AUC are derived and its properties are discussed. In application section, two real life example viz. Head trauma data and Multiple Sclerosis data were used for the proposed model and then these are compared with the conventional bi- normal model. In the simulation section, simulation studies have been carried out for different values of parameters to support the Bi-Rayleigh ROC model. In the last section, the confidence intervals for the parameter σi, i=0,1 are discussed for real life example and simulated data sets. II. ROC MODEL AND AREA UNDER THE CURVE Let the scores (S) be a random variable in healthy (H) and diseased (D) populations respectively. The ROC curve is plotted over values of threshold value (t) and is given by y(t) = h (x(t)) where x(t) is the False positive rate which is given by x(t) = P(S>t|H) and y(t) is True positive rate which is given by y(t) = P (S>t|D) The general functional form of ROC can be represented as

This study investigates the performance of four parameter estimation methods for the Laplace distribution: Method of Moments (MM), Maximum Likelihood Estimation (MLE), Minimum Chi-Square Estimation using equiprobable cells (MCE-EQ), and Minimum Chi-Square Estimation using Representative Points (MCE-RP). Through comprehensive Monte Carlo simulations with sample sizes ranging from 50 to 400, we compare the root mean squared error (RMSE) of the location (μ) and scale (b) parameter estimates. Our results demonstrate that while MLE remains robust for location estimation, the MCE-RP method consistently outperforms other estimators—including MLE—for the scale parameter, particularly in small to moderate samples. The use of Representative Points, which provide an optimal discretization of the distribution, significantly enhances estimation precision. These findings are especially relevant for medical research, where accurate estimation of variability—such as in biomarker concentration levels or physiological response times—is critical for reliable sample size determination, risk assessment, and clinical decision-making. MCE-RP thus offers a superior, reliable estimator for the Laplace scale parameter, with direct implications for improving statistical inference in applied biomedical studies. Purpose: The purpose of this research is to empirically evaluate and compare the finite-sample performance of four estimation methods for the Laplace distribution’s parameters, with a focus on the novel application of Representative Points in minimum chi-square estimation. This work seeks to bridge the gap between theoretical estimation methods and practical applications, providing applied researchers with a more robust estimation tool when modeling data with Laplace characteristics, such as those commonly encountered in medical and biomedical studies. Methods: We conducted an extensive Monte Carlo simulation study to compare the four estimation methods: MM, MLE, MCE-EQ, and MCE-RP. For each method, we generated independent and identically distributed samples from a standard Laplace distribution (μ=0, b=1) with sample sizes n = 50, 100, 200, and 400. Each scenario was replicated 1,000 times. The performance of each estimator was assessed using the root mean squared error (RMSE) for both μ and b. The MCE-RP method utilized pre-computed Representative Points for the standard Laplace distribution, which were transformed according to preliminary MLE estimates to form an optimal cell structure for chi-square minimization. All nonlinear optimizations required for MCE-EQ and MCE-RP were implemented programmatically. Results: The simulation results indicate that MLE performs best for estimating the location parameter μ across all sample sizes. However, for the scale parameter b, the MCE-RP method consistently yields lower RMSE values compared to MLE, MM, and MCE-EQ. In many cases, particularly for smaller samples, the RMSE of MCE-RP is approximately half that of MLE for b. The advantage of MCE-RP is evident across varying numbers of Representative Points (m = 5, 10, 15, 20), with optimal performance often observed at m = 10 or 15. These findings confirm that MCE-RP provides a more precise and reliable estimator for the scale parameter, making it particularly advantageous in small-sample settings. Contribution: This paper contributes to the statistical methodology for the Laplace distribution by introducing and validating the use of Representative Points within a minimum chi-square estimation framework. The key contributions are: (1) demonstrating that MCE-RP significantly outperforms established methods for estimating the scale parameter; (2) providing empirical evidence that RP-based discretization enhances estimation efficiency, especially in finite samples; (3) offering practical guidance for applied researchers in fields such as medical statistics, where accurate scale estimation is crucial for variability assessment, power analysis, and reliable inference; and (4) laying a methodological foundation for extending the RP approach to other location-scale distributions.