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

Type Ii ◽

Posterior Density ◽

Unknown Parameters ◽

Linex Loss ◽

Credible Intervals ◽

Competing Risks Data ◽

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.

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

Entropy ◽

10.3390/e23121558 ◽

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 ◽

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.

Statistical Inferences of Type-II Progressively Hybrid Censored Fuzzy Data with Rayleigh Distribution

Austrian Journal of Statistics ◽

10.17713/ajs.v47i3.752 ◽

2018 ◽

Vol 47 (3) ◽

pp. 40-62 ◽

Cited By ~ 2

Author(s):

Ankita Chaturvedi ◽

Sanjay Kumar Singh ◽

Umesh Singh

Keyword(s):

Real Data ◽

Rayleigh Distribution ◽

Model Parameters ◽

Type Ii ◽

Posterior Density ◽

Numerical Illustration ◽

Data Set ◽

Bayesian Procedures ◽

Credible Intervals ◽

This article presents the procedures for the estimation of the parameter of Rayleighdistribution based on Type-II progressive hybrid censored fuzzy lifetime data. Classicalas well as the Bayesian procedures for the estimation of unknown model parameters has been developed. The estimators obtained here are Maximum likelihood (ML) estimator, Method of moments (MM) estimator, Computational approach (CA) estimator and Bayes estimator. Highest posterior density (HPD) credible intervals of the unknown parameter are obtained by using Markov Chain Monte Carlo (MCMC) technique. For numerical illustration, a real data set has been considered.

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

Mathematics ◽

10.3390/math9010049 ◽

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 ◽

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.

On Progressive Censored Competing Risks Data: Real Data Application and Simulation Study

Mathematics ◽

10.3390/math9151805 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1805

Author(s):

Abd M. Abd El-Raheem ◽

Mona Hosny ◽

Mahmoud H. Abu-Moussa

Keyword(s):

Competing Risks ◽

Simulation Study ◽

Expectation Maximization Algorithm ◽

Real Data ◽

Estimation Methods ◽

Data Sets ◽

Unknown Parameters ◽

Competing Risks Data ◽

Competing Risk Models

Competing risks are frequently overlooked, and the event of interest is analyzed with conventional statistical techniques. In this article, we consider the analysis of bi-causes of failure in the context of competing risk models using the extension of the exponential distribution under progressive Type-II censoring. Maximum likelihood estimates for the unknown parameters via the expectation-maximization algorithm are obtained. Moreover, the Bayes estimates of the unknown parameters are approximated using Tierney-Kadane and MCMC techniques. Interval estimates using Bayesian and classical techniques are also considered. Two real data sets are investigated to illustrate the different estimation methods, and to compare the suggested model with Weibull distribution. Furthermore, the estimation methods are compared through a comprehensive simulation study.

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

Symmetry ◽

10.3390/sym13060999 ◽

2021 ◽

Vol 13 (6) ◽

pp. 999

Author(s):

Mingjie Wu ◽

Wenhao Gui

Keyword(s):

Loss Function ◽

Type Ii ◽

Unknown Parameters ◽

Estimation And Prediction ◽

Error Loss ◽

Squared Error ◽

Progressive Type ◽

Metropolis Hasting Algorithm ◽

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.

Sample Size Requirements for Calibrated Approximate Credible Intervals for Proportions in Clinical Trials

International Journal of Environmental Research and Public Health ◽

10.3390/ijerph18020595 ◽

2021 ◽

Vol 18 (2) ◽

pp. 595

Author(s):

Fulvio De Santis ◽

Stefania Gubbiotti

Keyword(s):

Clinical Trials ◽

Posterior Probability ◽

Interval Estimation ◽

Small Sample ◽

Sample Sizes ◽

Posterior Density ◽

Unknown Parameters ◽

Credible Intervals ◽

Small Sample Sizes ◽

In Bayesian analysis of clinical trials data, credible intervals are widely used for inference on unknown parameters of interest, such as treatment effects or differences in treatments effects. Highest Posterior Density (HPD) sets are often used because they guarantee the shortest length. In most of standard problems, closed-form expressions for exact HPD intervals do not exist, but they are available for intervals based on the normal approximation of the posterior distribution. For small sample sizes, approximate intervals may be not calibrated in terms of posterior probability, but for increasing sample sizes their posterior probability tends to the correct credible level and they become closer and closer to exact sets. The article proposes a predictive analysis to select appropriate sample sizes needed to have approximate intervals calibrated at a pre-specified level. Examples are given for interval estimation of proportions and log-odds.

Computation of posterior distribution in Bayesian analysis – application in an intermittently used reliability system

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie ◽

10.4102/satnt.v21i3.231 ◽

2002 ◽

Vol 21 (3) ◽

pp. 78-82

Author(s):

V. S.S. Yadavalli ◽

P. J. Mostert ◽

A. Bekker ◽

M. Botha

Keyword(s):

Posterior Distribution ◽

Jeffreys Prior ◽

Posterior Density ◽

Unknown Parameters ◽

Stationary Rate ◽

Analysis Application ◽

Definition Of ◽

Hpd Intervals ◽

Highest Posterior Density ◽

Bayes Procedure

Bayesian estimation is presented for the stationary rate of disappointments, D∞, for two models (with different specifications) of intermittently used systems. The random variables in the system are considered to be independently exponentially distributed. Jeffreys’ prior is assumed for the unknown parameters in the system. Inference about D∞ is being restrained in both models by the complex and non-linear definition of D∞. Monte Carlo simulation is used to derive the posterior distribution of D∞ and subsequently the highest posterior density (HPD) intervals. A numerical example where Bayes estimates and the HPD intervals are determined illustrates these results. This illustration is extended to determine the frequentistical properties of this Bayes procedure, by calculating covering proportions for each of these HPD intervals, assuming fixed values for the parameters.

Reliable confidence intervals for RelTime estimates of evolutionary divergence times

10.1101/677286 ◽

2019 ◽

Cited By ~ 1

Author(s):

Qiqing Tao ◽

Koichiro Tamura ◽

Beatriz Mello ◽

Sudhir Kumar

Keyword(s):

Confidence Intervals ◽

Divergence Time ◽

Simulated Data ◽

Molecular Dating ◽

Divergence Times ◽

Rate Variation ◽

Evolutionary Divergence ◽

Posterior Density ◽

Divergence Time Estimates ◽

AbstractConfidence intervals (CIs) depict the statistical uncertainty surrounding evolutionary divergence time estimates. They capture variance contributed by the finite number of sequences and sites used in the alignment, deviations of evolutionary rates from a strict molecular clock in a phylogeny, and uncertainty associated with clock calibrations. Reliable tests of biological hypotheses demand reliable CIs. However, current non-Bayesian methods may produce unreliable CIs because they do not incorporate rate variation among lineages and interactions among clock calibrations properly. Here, we present a new analytical method to calculate CIs of divergence times estimated using the RelTime method, along with an approach to utilize multiple calibration uncertainty densities in these analyses. Empirical data analyses showed that the new methods produce CIs that overlap with Bayesian highest posterior density (HPD) intervals. In the analysis of computer-simulated data, we found that RelTime CIs show excellent average coverage probabilities, i.e., the true time is contained within the CIs with a 95% probability. These developments will encourage broader use of computationally-efficient RelTime approach in molecular dating analyses and biological hypothesis testing.

Estimation of the Inverse Weibull Distribution Parameters under Type-I Hybrid Censoring

Austrian Journal of Statistics ◽

10.17713/ajs.v50i5.1134 ◽

2021 ◽

Vol 50 (5) ◽

pp. 38-51

Author(s):

Mohammad Kazemi ◽

Mina Azizpoor

Keyword(s):

Maximum Likelihood ◽

Weibull Distribution ◽

Type I ◽

Unknown Parameters ◽

Bayes Estimates ◽

Hybrid Censoring ◽

Inverse Weibull Distribution ◽

Distribution Parameters ◽

The hybrid censoring is a mixture of type-I and type-II censoring schemes. This paper presents the statistical inferences of the inverse Weibull distribution parameters when the data are type-I hybrid censored. First, we consider the maximum likelihood estimates of the unknown parameters. It is observed that the maximum likelihood estimates can not be obtained in closed form. We further obtain the Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also compute the approximate Bayes estimates using Lindley's approximation technique. The performance of the Bayes estimates have been compared with maximum likelihood estimates through the Monte Carlo Markov chain techniques. Finally, a real data set have been analysed for illustration purpose.

The Bayesian confidence intervals for measuring the difference between dispersions of rainfall in Thailand

PeerJ ◽

10.7717/peerj.9662 ◽

2020 ◽

Vol 8 ◽

pp. e9662

Author(s):

Noppadon Yosboonruang ◽

Sa-Aat Niwitpong ◽

Suparat Niwitpong

Keyword(s):

Confidence Interval ◽

Confidence Intervals ◽

Rainfall Data ◽

Posterior Density ◽

Lognormal Distributions ◽

Coefficients Of Variation ◽

Coverage Probabilities ◽

The Difference ◽

Fiducial Generalized Confidence Interval ◽

The coefficient of variation is often used to illustrate the variability of precipitation. Moreover, the difference of two independent coefficients of variation can describe the dissimilarity of rainfall from two areas or times. Several researches reported that the rainfall data has a delta-lognormal distribution. To estimate the dynamics of precipitation, confidence interval construction is another method of effectively statistical inference for the rainfall data. In this study, we propose confidence intervals for the difference of two independent coefficients of variation for two delta-lognormal distributions using the concept that include the fiducial generalized confidence interval, the Bayesian methods, and the standard bootstrap. The performance of the proposed methods was gauged in terms of the coverage probabilities and the expected lengths via Monte Carlo simulations. Simulation studies shown that the highest posterior density Bayesian using the Jeffreys’ Rule prior outperformed other methods in virtually cases except for the cases of large variance, for which the standard bootstrap was the best. The rainfall series from Songkhla, Thailand are used to illustrate the proposed confidence intervals.