scholarly journals Estimation of Unknown Parameters of Truncated Normal Distribution under Adaptive Progressive Type II Censoring Scheme

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 49
Author(s):  
Siqi Chen ◽  
Wenhao Gui
Keyword(s):  
Normal Distribution ◽  
Confidence Intervals ◽  
Censored Data ◽  
Likelihood Method ◽  
Type Ii ◽  
Truncated Normal Distribution ◽  
Unknown Parameters ◽  
Progressive Type ◽  
Truncated Normal ◽  
Highest Posterior Density

In reality, estimations for the unknown parameters of truncated distribution with censored data have wide utilization. Truncated normal distribution is more suitable to fit lifetime data compared with normal distribution. This article makes statistical inferences on estimating parameters under truncated normal distribution using adaptive progressive type II censored data. First, the estimates are calculated through exploiting maximum likelihood method. The observed and expected Fisher information matrices are derived to establish the asymptotic confidence intervals. Second, Bayesian estimations under three loss functions are also studied. The point estimates are calculated by Lindley approximation. Importance sampling technique is applied to discuss the Bayes estimates and build the associated highest posterior density credible intervals. Bootstrap confidence intervals are constructed for the purpose of comparison. Monte Carlo simulations and data analysis are employed to present the performances of various methods. Finally, we obtain optimal censoring schemes under different criteria.

scholarly journals Classical and Bayesian Inference for a Progressive First-Failure Censored Left-Truncated Normal Distribution

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 490
Author(s):  
Yuxin Cai ◽  
Wenhao Gui
Keyword(s):  
Bayesian Inference ◽  
Maximum Likelihood ◽  
Normal Distribution ◽  
Confidence Intervals ◽  
Loss Function ◽  
Interval Length ◽  
Truncated Normal Distribution ◽  
Lindley Approximation ◽  
Truncated Normal ◽  
Highest Posterior Density

Point and interval estimations are taken into account for a progressive first-failure censored left-truncated normal distribution in this paper. First, we derive the estimators for parameters on account of the maximum likelihood principle. Subsequently, we construct the asymptotic confidence intervals based on these estimates and the log-transformed estimates using the asymptotic normality of maximum likelihood estimators. Meanwhile, bootstrap methods are also proposed for the construction of confidence intervals. As for Bayesian estimation, we implement the Lindley approximation method to determine the Bayesian estimates under not only symmetric loss function but also asymmetric loss functions. The importance sampling procedure is applied at the same time, and the highest posterior density (HPD) credible intervals are established in this procedure. The efficiencies of classical statistical and Bayesian inference methods are evaluated through numerous simulations. We conclude that the Bayes estimates given by Lindley approximation under Linex loss function are highly recommended and HPD interval possesses the narrowest interval length among the proposed intervals. Ultimately, we introduce an authentic dataset describing the tensile strength of 50mm carbon fibers as an illustrative sample.

scholarly journals Parameter Estimation Based on the Frèchet Progressive Type II Censored Data with Binomial Removals

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Mohamed Mubarak
Keyword(s):  
Censored Data ◽  
Time Distribution ◽  
Maximum Likelihood Method ◽  
Failure Time ◽  
Likelihood Method ◽  
Type Ii ◽  
Failure Time Distribution ◽  
Sampling Distributions ◽  
Progressive Type ◽  
Type Ii Censored Data

This paper considers the estimation problem for the Frèchet distribution under progressive Type II censoring with random removals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood method to obtain the estimators of parameters and derive the sampling distributions of the estimators, and we also construct the confidence intervals for the parameters and percentile of the failure time distribution.

Revisit to progressively Type-II censored competing risks data from Lomax distributions

2021 ◽  
pp. 1748006X2098397
Author(s):  
Rui Hua ◽  
Wenhao Gui
Keyword(s):  
Confidence Intervals ◽  
Competing Risks ◽  
Maximum Likelihood Estimates ◽  
Type Ii ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Linex Loss ◽  
Credible Intervals ◽  
Competing Risks Data ◽  
Highest Posterior Density

In survival analysis, more than one factor typically contributes to individual failure. In addition, censoring is inevitable in lifespan tests or reliability studies due to external causes or experimental purposes. In this article, the competing risks model is considered and investigated under progressively Type-II censoring where data is from Lomax distributions. Assumptions are further made that these competitive factors are independently distributed, and the latent lifetimes of these factors follow Lomax distributions where both scale parameters and shape parameters are different. For all unknown parameters, maximum likelihood estimates have been attained by Newton-Raphson (NR) method as well as expectation maximization (EM) method, and then the approximate confidence intervals are acquired, in addition to bootstrap confidence intervals. Furthermore, under square error and LINEX loss functions, Bayes estimates and corresponding highest posterior density credible intervals are successively constructed. Finally, simulation experiments are implemented to access performance of several proposed methods in this article, and laboratory dataset is presented and analyzed for illustrative purposes.

scholarly journals Bayesian Nonlinear Latent variable Models with Mixed Non-normal Variables and Covariates for Multi-sample Psychological Data

2019 ◽  
pp. 627-645
Author(s):  
Thanoon Y. Thanoon ◽  
Athar Talal Hamed ◽  
Robiah Adnan
Keyword(s):  
Normal Distribution ◽  
Latent Variable ◽  
Latent Variable Models ◽  
Truncated Normal Distribution ◽  
Dichotomous Data ◽  
Psychological Data ◽  
Truncated Normal ◽  
Highest Posterior Density ◽  
Ordered Categorical ◽  
Nonlinear Latent Variable Models

The purpose of this paper is to develop a latent variable model with nonlinear covariates and latent variables. Mixed ordered categorical and dichotomous variables and covariates with two different types of thresholds (with equal and unequal spaces) are used in Bayesian multi-sample nonlinear latent variable models and the Gibbs sampling method is applied for estimation and model comparison. Hidden continuous normal distribution (censored normal distribution) and (truncated normal distribution with known parameters) are used to handle the problem of mixed ordered categorical and dichotomous data. Hidden continuous normal distribution (truncated normal distribution with known parameters) is used to handle the problem of mixed ordered categorical and dichotomous data in covariates. Statistical analysis, which involves the estimation of parameters, standard deviations and their highest posterior density, are discussed. The proposed procedure is illustrated using psychological data with the results obtained from the OpenBUGS program.

scholarly journals Classical and Bayesian Inference of an Exponentiated Half-Logistic Distribution under Adaptive Type II Progressive Censoring

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1558
Author(s):  
Ziyu Xiong ◽  
Wenhao Gui
Keyword(s):  
Confidence Intervals ◽  
Bootstrap Method ◽  
Information Matrix ◽  
Logistic Distribution ◽  
Type Ii ◽  
Progressive Censoring ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Data Set ◽  
Highest Posterior Density

The point and interval estimations for the unknown parameters of an exponentiated half-logistic distribution based on adaptive type II progressive censoring are obtained in this article. At the beginning, the maximum likelihood estimators are derived. Afterward, the observed and expected Fisher’s information matrix are obtained to construct the asymptotic confidence intervals. Meanwhile, the percentile bootstrap method and the bootstrap-t method are put forward for the establishment of confidence intervals. With respect to Bayesian estimation, the Lindley method is used under three different loss functions. The importance sampling method is also applied to calculate Bayesian estimates and construct corresponding highest posterior density (HPD) credible intervals. Finally, numerous simulation studies are conducted on the basis of Markov Chain Monte Carlo (MCMC) samples to contrast the performance of the estimations, and an authentic data set is analyzed for exemplifying intention.

scholarly journals Bayesian Estimation of Entropy for Burr Type XII Distribution under Progressive Type-II Censored Data

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 313
Author(s):  
Xinjing Wang ◽  
Wenhao Gui
Keyword(s):  
Maximum Likelihood ◽  
Bayesian Estimation ◽  
Censored Data ◽  
Loss Functions ◽  
Type Ii ◽  
Important Indicator ◽  
Progressive Type ◽  
Type Ii Censored Data ◽  
Highest Posterior Density ◽  
Burr Type Xii

With the rapid development of statistics, information entropy is proposed as an important indicator used to quantify information uncertainty. In this paper, maximum likelihood and Bayesian methods are used to obtain the estimators of the entropy for a two-parameter Burr type XII distribution under progressive type-II censored data. In the part of maximum likelihood estimation, the asymptotic confidence intervals of entropy are calculated. In Bayesian estimation, we consider non-informative and informative priors respectively, and asymmetric and symmetric loss functions are both adopted. Meanwhile, the posterior risk is also calculated to evaluate the performances of the entropy estimators against different loss functions. In a numerical simulation, the Lindley approximation and the Markov chain Monte Carlo method were used to obtain the Bayesian estimates. In turn, the highest posterior density credible intervals of the entropy were derived. Finally, average absolute bias and mean square error were used to evaluate the estimators under different methods, and a real dataset was selected to illustrate the feasibility of the above estimation model.

scholarly journals Estimation and Prediction for Nadarajah-Haghighi Distribution under Progressive Type-II Censoring

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 999
Author(s):  
Mingjie Wu ◽  
Wenhao Gui
Keyword(s):  
Loss Function ◽  
Maximum Likelihood Estimates ◽  
Type Ii ◽  
Unknown Parameters ◽  
Estimation And Prediction ◽  
Error Loss ◽  
Squared Error ◽  
Progressive Type ◽  
Metropolis Hasting Algorithm ◽  
Highest Posterior Density

The paper discusses the estimation and prediction problems for the Nadarajah-Haghighi distribution using progressive type-II censored samples. For the unknown parameters, we first calculate the maximum likelihood estimates through the Expectation–Maximization algorithm. In order to choose the best Bayesian estimator, a loss function must be specified. When the loss is essentially symmetric, it is reasonable to use the square error loss function. However, for some estimation problems, the actual loss is often asymmetric. Therefore, we also need to choose an asymmetric loss function. Under the balanced squared error and symmetric squared error loss functions, the Tierney and Kadane method is used for calculating different kinds of approximate Bayesian estimates. The Metropolis-Hasting algorithm is also provided here. In addition, we construct a variety of interval estimations of the unknown parameters including asymptotic intervals, bootstrap intervals, and highest posterior density intervals using the sample derived from the Metropolis-Hasting algorithm. Furthermore, we compute the point predictions and predictive intervals for a future sample when facing the one-sample and two-sample situations. At last, we compare and appraise the performance of the provided techniques by carrying out a simulation study and analyzing a real rainfall data set.

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