Confirmatory factor analysis with missing data in a small sample: cognitive reserve in people with Down Syndrome

The presence of missing data and small sample sizes are very common in social and health sciences. Concurrently to present a methodology to solve the small sample size and missing data, we aim to present a definition of Cognitive Reserve for people with Down Syndrome. This population has become an appealing focus to study this concept because of the high incidence of dementia. The accidental sample comprised 35 persons with DS (16–35 years). A total of 12 variables were acquired, four of them had missing data. Two types of multiple imputation were made. Confirmatory factor analysis with Bayesian estimations was performed on the final database with non-informative priors. However, to solve the sample size problem, two additional corrections were made: first, we followed the Jiang and Yuan (2017) schema, and second, we made a Jackknife correlation correction. The estimations of the confirmatory factor analysis, as well as the global fit, are adequate. As an applied perspective, the acceptable fit of our model suggests the possibility of operationalizing the latent factor Cognitive Reserve in a simple way to measure it in the Down Syndrome population.

Determination of Quantile Range of Optimal Hyperparameters Using Bayesian Estimation

Bayesian estimations have the advantages of taking into account the uncertainty of all parameter estimates which allows virtually the use of vague priors. This study focused on determining the quantile range at which optimal hyperparameter of normally distributed data with vague information could be obtained in Bayesian estimation of linear regression models. A Monte Carlo simulation approach was used to generate a sample size of 200 data-set. Observation precisions and posterior precisions were estimated from the regression output to determine the posterior means estimate for each model to derive the new dependent variables. The variances were divided into 10 equal parts to obtain the hyperparameters of the prior distribution. Average absolute deviation for model selection was used to validate the adequacy of each model. The study revealed the optimal hyperparameters located at 5th and 7th deciles. The research simplified the process of selecting the hyperparameters of prior distribution from the data with vague information in empirical Bayesian inferences. Keywords: Optimal Hyperparameters; Quantile Ranges; Bayesian Estimation; Vague prior

Bayesian and E-Bayesian Estimations of Bathtub-Shaped Distribution under Generalized Type-I Hybrid Censoring

For the purpose of improving the statistical efficiency of estimators in life-testing experiments, generalized Type-I hybrid censoring has lately been implemented by guaranteeing that experiments only terminate after a certain number of failures appear. With the wide applications of bathtub-shaped distribution in engineering areas and the recently introduced generalized Type-I hybrid censoring scheme, considering that there is no work coalescing this certain type of censoring model with a bathtub-shaped distribution, we consider the parameter inference under generalized Type-I hybrid censoring. First, estimations of the unknown scale parameter and the reliability function are obtained under the Bayesian method based on LINEX and squared error loss functions with a conjugate gamma prior. The comparison of estimations under the E-Bayesian method for different prior distributions and loss functions is analyzed. Additionally, Bayesian and E-Bayesian estimations with two unknown parameters are introduced. Furthermore, to verify the robustness of the estimations above, the Monte Carlo method is introduced for the simulation study. Finally, the application of the discussed inference in practice is illustrated by analyzing a real data set.

The E-Bayesian Estimation for Lomax Distribution Based on Generalized Type-I Hybrid Censoring Scheme

This article studies the E-Bayesian estimation of the unknown parameter of Lomax distribution based on generalized Type-I hybrid censoring. Under square error loss and LINEX loss functions, we get the E-Bayesian estimation and compare its effectiveness with Bayesian estimation. To measure the error of E-Bayesian estimation, the expectation of mean square error (E-MSE) is introduced. With Markov chain Monte Carlo technology, E-Bayesian estimations are computed. Metropolis–Hastings algorithm is applied within the process. Similarly, the credible interval for the parameter is calculated. Then, we can compare the MSE and E-MSE to evaluate whose result is more effective. For the purpose of illustration in real datasets, cases of generalized Type-I hybrid censored samples are presented. In order to judge whether the sample data can be directly fitted by the Lomax distribution, we adopt the Kolmogorov–Smirnov tests for evaluation. Finally, we can get the conclusion after comparing the results of E-Bayesian and Bayesian estimation.

Estimation for Entropy and Parameters of Generalized Bilal Distribution under Adaptive Type II Progressive Hybrid Censoring Scheme

Entropy measures the uncertainty associated with a random variable. It has important applications in cybernetics, probability theory, astrophysics, life sciences and other fields. Recently, many authors focused on the estimation of entropy with different life distributions. However, the estimation of entropy for the generalized Bilal (GB) distribution has not yet been involved. In this paper, we consider the estimation of the entropy and the parameters with GB distribution based on adaptive Type-II progressive hybrid censored data. Maximum likelihood estimation of the entropy and the parameters are obtained using the Newton–Raphson iteration method. Bayesian estimations under different loss functions are provided with the help of Lindley’s approximation. The approximate confidence interval and the Bayesian credible interval of the parameters and entropy are obtained by using the delta and Markov chain Monte Carlo (MCMC) methods, respectively. Monte Carlo simulation studies are carried out to observe the performances of the different point and interval estimations. Finally, a real data set has been analyzed for illustrative purposes.

Reliability computation for an uncertain PVC window production system using a modified bayesian estimation

Nowadays, Industries have been receiving much attention in Failure modelling and reliability assessment of repairable systems due to the fact that it plays a crucial role in risk and safety management of process. The primary purpose of this article is to present a methodology for discussing uncertainty in the reliability assessment if the production system. In fact, we discuss the fuzzy E-Bayesian estimation of reliability for PVC window production system. This approach is used to create the fuzzy E-Bayesian estimations of system reliability by introducing and applying a theorem called “Resolution Identity” for fuzzy sets. To be more specific, the model parameters are assumed to be fuzzy random variables. For this purpose, the original problem is transformed into a nonlinear programming problem which is divided into four sub-problems to simplify the computations. Finally, the results obtained for the sub-problems can be used to determine the membership functions of the fuzzy E-Bayesian estimation of system reliability. To clarify the proposed model, a practical example for PVC window production system is conducted.

The Bayesian Inference of Pareto Models Based on Information Geometry

Bayesian methods have been rapidly developed due to the important role of explicable causality in practical problems. We develope geometric approaches to Bayesian inference of Pareto models, and give an application to the analysis of sea clutter. For Pareto two-parameter model, we show the non-existence of α-parallel prior in general, hence we adopt Jeffreys prior to deal with the Bayesian inference. Considering geodesic distance as the loss function, an estimation in the sense of minimal mean geodesic distance is obtained. Meanwhile, by involving Al-Bayyati’s loss function we gain a new class of Bayesian estimations. In the simulation, for sea clutter, we adopt Pareto model to acquire various types of parameter estimations and the posterior prediction results. Simulation results show the advantages of the Bayesian estimations proposed and the posterior prediction.