Bayesian Versus Maximum Likelihood Estimation of Multitrait–Multimethod Confirmatory Factor Models

Latent growth models (LGMs) are an application of structural equation modeling and frequently used in developmental and clinical research to analyze change over time in longitudinal outcomes. Maximum likelihood (ML), the most common approach for estimating LGMs, can fail to converge or may produce biased estimates in complex LGMs especially in studies with modest samples. Bayesian estimation is a logical alternative to ML for LGMs, but there is a lack of research providing guidance on when Bayesian estimation may be preferable to ML or vice versa. This study compared the performance of Bayesian versus ML estimators for LGMs by evaluating their accuracy via Monte Carlo (MC) simulations. For the MC study, longitudinal data sets were generated and estimated using LGM via both ML and Bayesian estimation with three different priors, and parameter recovery across the two estimators was evaluated to determine their relative performance. The findings suggest that ML estimation is a reasonable choice for most LGMs, unless it fails to converge, which can occur with limiting data situations (i.e., just a few time points, no covariate or outcome, modest sample sizes). When models do not converge using ML, we recommend Bayesian estimation with one caveat that the influence of the priors on estimation may have to be carefully examined, per recent recommendations on Bayesian modeling for applied researchers.

Latent variable models with many categorical items and multiple latent constructs result in many dimensions of numerical integration, and the traditional frequentist estimation approach, such as maximum likelihood (ML), tends to fail due to model complexity. In such cases, Bayesian estimation with diffuse priors can be used as a viable alternative to ML estimation. This study compares the performance of Bayesian estimation with ML estimation in estimating single or multiple ability factors across 2 types of measurement models in the structural equation modeling framework: a multidimensional item response theory (MIRT) model and a multiple-indicator multiple-cause (MIMIC) model. A Monte Carlo simulation study demonstrates that Bayesian estimation with diffuse priors, under various conditions, produces results quite comparable with ML estimation in the single- and multilevel MIRT and MIMIC models. Additionally, an empirical example utilizing the Multistate Bar Examination is provided to compare the practical utility of the MIRT and MIMIC models. Structural relationships among the ability factors, covariates, and a binary outcome variable are investigated through the single- and multilevel measurement models. The article concludes with a summary of the relative advantages of Bayesian estimation over ML estimation in MIRT and MIMIC models and suggests strategies for implementing these methods.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

This paper studies estimation of the parameters of the generalized Gompertz distribution based on ranked-set sample (RSS). Maximum likelihood (ML) and Bayesian approaches are considered. Approximate confidence intervals for the unknown parameters are constructed using both the normal approximation to the asymptotic distribution of the ML estimators and bootstrapping methods. Bayes estimates and credible intervals of the unknown parameters are obtained using differential evolution Markov chain Monte Carlo and Lindley’s methods. The proposed methods are compared via Monte Carlo simulations studies and an example employing real data. The performance of both ML and Bayes estimates is improved under RSS compared with simple random sample (SRS) regardless of the sample size. Bayes estimates outperform the ML estimates for small samples, while it is the other way around for moderate and large samples.

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.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

The article is concerned with parameter estimation for Gaussian Markov random fields observed with additive independent identically distributed noise. In particular, we consider maximum likelihood and maximum pseudo-likelihood estimation for the noise-free case and make adjustments to the estimators in the presence of noise. The adjusted maximum likelihood estimator is computed in O(n2) time for a square image with n pixels. The estimation method is useful when only the moments of the noise are specified or when the exact maximum likelihood estimator is difficult to compute (e.g., for certain non-Gaussian noise distributions). The adjusted maximum pseudo-likelihood estimator is straightforward to calculate, is useful as a starting value in maximization routines, and is often a reasonable estimator in its own right. We discuss asymptotic properties of the adjusted estimators including consistency. We also consider constrained maximum pseudo-likelihood estimation and a Bayesian estimator. We compare the adjusted estimators with the exact Gaussian maximum likelihood estimator and toroidal boundary approximation Gaussian maximum likelihood estimator in a simulation study. The adjusted estimators are also robust to the specification of the noise distribution.

Latent growth models (LGMs) with a distal outcome allow researchers to assess longer-term patterns, and to detect the need to start a (preventive) treatment or intervention in an early stage. The aim of the current simulation study is to examine the performance of an LGM with a continuous distal outcome under maximum likelihood (ML) and Bayesian estimation with default and informative priors, under varying sample sizes, effect sizes and slope variance values. We conclude that caution is needed when predicting a distal outcome from an LGM when the: (1) sample size is small; and (2) amount of variation around the latent slope is small, even with a large sample size. We recommend against the use of ML and Bayesian estimation with Mplus default priors in these situations to avoid severely biased estimates. Recommendations for substantive researchers working with LGMs with distal outcomes are provided based on the simulation results.

In this paper, Bayes estimators of the parameter of Maxwell distribution have been derived along with maximum likelihood estimator. The non-informative priors; Jeffreys and the extension of Jeffreys prior information has been considered under two different loss functions, the squared error loss function and the modified squared error loss function for comparison purpose. A simulation study has been developed in order to gain an insight into the performance on small, moderate and large samples. The performance of these estimators has been explored numerically under different conditions. The efficiency for the estimators was compared according to the mean square error MSE. The results of comparison by MSE show that the efficiency of Bayes estimators of the shape parameter of the Maxwell distribution decreases with the increase of Jeffreys prior constants. The results also show that values of Bayes estimators are almost close to the maximum likelihood estimator when the Jeffreys prior constants are small, yet they are identical in some certain cases. Comparison with respect to loss functions show that Bayes estimators under the modified squared error loss function has greater MSE than the squared error loss function especially with the increase of r.

Abstract. Longitudinal developmental research is often focused on patterns of change or growth across different (sub)groups of individuals. Particular to some research contexts, developmental inquiries may involve one or more (sub)groups that are small in nature and therefore difficult to properly capture through statistical analysis. The current study explores the lower-bound limits of subsample sizes in a multiple group latent growth modeling by means of a simulation study. We particularly focus on how the maximum likelihood (ML) and Bayesian estimation approaches differ when (sub)sample sizes are small. The results show that Bayesian estimation resolves computational issues that occur with ML estimation and that the addition of prior information can be the key to detect a difference between groups when sample and effect sizes are expected to be limited. The acquisition of prior information with respect to the smaller group is especially influential in this context.

Problem statement: The study and research of survival or reliability or life time belong to the same area of study but they may belong to a different area of application. In survival analysis one can use several life time distribution, exponential distribution with mean life time θ is one of them. To estimate this parameter and survival function we must be used estimation procedures with less MSE and MPE. Approach: The only statistical theory that combined modeling inherent uncertainty and statistical uncertainty is Bayesian statistics. The theorem of Bayes provided a solution to how learn from data. Bayes theorem was depending on prior and posterior distribution and standard Bayes estimator depends on Jeffery prior information. In this study we annexed Jeffery prior information to get the modify Bayes estimator and then compared it with standard Bayes estimator and maximum likelihood estimator to find the best (less MSE and MPE). Results: when we derived Bayesian and Maximum likelihood of the scale parameter and survival functions. Simulation study was used to compare between estimators and Mean Square Error (MSE) and Mean Percentage Error (MPE) of estimators are computed. Conclusion: The new proposed estimator of modify Bayes estimator in parameter and survival function was the best estimator (less MSE and MPE) when we compared it with standard Bayes and maximum likelihood estimator.

Estimating the midpoints of interval-valued data poses significant challenges, particularly when uncertainty and variability are inherent. This paper advances midpoint estimation by leveraging both Maximum Likelihood (ML) and Bayesian methodologies, incorporating various forms of additional information through parameter constraints. In practical applications, such information often naturally arises, enabling the imposition of constraints such as order, rectangular, spherical, and bounded-order conditions on the parameter space. By integrating these constraints, the estimators can exploit structural relationships among parameters to achieve improved precision and accuracy. Through comprehensive simulation studies, we assess the performance of ML and Bayesian estimators under these constrained settings, demonstrating that Bayesian estimators consistently outperform ML estimators, particularly when employing noninformative priors. The constrained Bayesian approach not only results in lower mean squared errors (MSEs) but also yields more precise credible intervals compared to the confidence intervals obtained via ML, showcasing its robustness and efficiency. Furthermore, a real-world climatological case study validates our theoretical and simulation findings, demonstrating the practical utility of the proposed methods for interval-valued temperature data. This example illustrates how integrating additional information–often readily available–enhances the accuracy and reliability of midpoint estimation in multivariate contexts. Our work underscores the potential of structural constraints in symbolic data analysis, providing actionable insights across diverse scientific domains.

In this paper, by including a trigonometric function, we propose a family of heavy-tailed distribution called the arcsine Kumaraswamy generalized-X family of distributions. Based on the proposed approach, a four-parameter extension of the Lomax distribution called the arcsine Kumaraswamy generalized Lomax (ASKUG-LOMAX) distribution is discussed in detail. Maximum likelihood, bootstrap, and Bayesian estimation are used to estimate the model parameters. A simulation study is used to evaluate ASKUG-LOMAX performance. The flexibility and usefulness of the proposed ASKUG-LOMAX distribution to predict unique symmetric and asymmetric patterns is demonstrated by analyzing real data. The results show that the ASKUG-LOMAX model is a good candidate for analyzing claims based on heavy-tailed data.

Multilevel latent class profile analysis (MLCPA) is a recently developed technique for understanding latent class dynamics in longitudinal studies; however, conventional maximum likelihood (ML) estimation may face challenges, particularly with small sample sizes or boundary solutions. As an alternative method, we propose a Bayesian estimation for MLCPA by employing non-informative prior distributions. In addition, we shed light on the underflow problem, which denotes a phenomenon such that the logarithm of the likelihood is negative infinity due to the multilevel structure. We perform extensive numerical studies to compare the behaviors of the MLE and the Bayesian estimates and investigate the accuracies of approximated model selection criteria. The simulation study revealed that Bayesian estimates are preferred to ML estimates when the underlying latent classes are well-separated, while the ML estimates are preferred when the underlying latent classes overlap. Utilizing the Progress Monitoring and Reporting Network data, which includes longitudinal academic performance metrics, our analysis uncovers distinct pathways of latent classes for students, further differentiated by latent groups of schools. These findings shed light on the considerable variations in academic proficiency trajectories and thus may offer new perspectives on academic proficiency patterns, with important implications for policy development and targeted educational interventions.

In this paper, constant-stress accelerated life test is assumed when the lifetime of test units follows an extension of the exponential distribution. Based on progressive censoring, the maximum likelihood estimates (MLEs) and Bayes estimates (BEs) of the model parameters are obtained. The BEs are obtained based on both non-informative and informative priors. In addition, the approximate, bootstrap and credible confidence intervals (CIs) of the estimators are constructed. Moreover, a real dataset is analyzed to illustrate the proposed procedures. Furthermore, the real dataset is used to show that extension of the exponential distribution can be a better model than Weibull distribution and generalized exponential distribution. Finally, simulation studies are carried out to investigate the accuracy of the MLEs and BEs for the parameters involved.