Bayesian Approximation Techniques for Gompertz Distribution

Comparison of Bayesian survival analysis and Cox regression analysis in simulated and breast cancer data sets

The present study is concerned with the estimation of inverse Rayleighdistribution using various Bayesian approximation techniques like normal approximation, and Tierney and Kadane (T-K) approximation. Different informative and non-informative priors are used to obtain the Baye’s estimate of inverse Rayleighdistribution under different approximation techniques. The simulation study has been conducted for comparison of Baye’s estimates obtained under different approximations using different priors. A real life example has also been discussed to compare the performance of these estimates. Keywords: Bayesian estimation, prior distribution, normal approximation, T-K approximation Cite this Article Kawsar Fatima, Ahmad SP. Bayesian Approximation Techniques of Inverse Rayleigh Distribution. Research & Reviews: Journal of Statistics . 2018; 7(1): 50–59p.

Bayesian estimation of the mixture of Burr Type-XII distributions using doubly censored data

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.

The “Rule of Three” gives an approximation for an upper 95% confidence bound for a proportion in a zero-numerator problem, which occurs when the observed relative frequency is zero. We compare the results from the Rule of Three with those from a Bayesian approach with noninformative and informative priors. Informative priors are especially valuable in zero-numerator problems because they can represent the available information and because different noninformative priors can give conicting advice. Moreover, the use of upper 95% bounds and noninformative priors in an effort to be conservative may backfire when the values are used in further predictive or decision-theoretic calculations. It is better to be candid than conservative, using all of the information available in forming the prior and considering the uncertainty represented by the full posterior distribution.

This article focuses on the application of wind speed data of 4 coastal areas of Baluchistan, that is, Gawadar, Jiwani, Ormara, and Pasni on 4-Component Rayleigh Mixture Model (4-CRMM) under Bayesian context. Type I right censoring scheme is used because it is popular in reliability theory and survival analysis. To accomplish the objective, the Bayes estimates (BEs) of the parameter of the mixture model along with their posterior risks (PRs) using informative prior (IP) and noninformative prior (NIP) are obtained. Hyperparameters are obtained by employing the prior predictive method. BEs are calculated under two distinct loss functions, squared error loss function (SELF) and modified squared error loss function (MSELF). The statistical properties and performance of the BEs under said loss functions are also evaluated by simulation study for different sample sizes.

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.

Over the past 20years, there has been increasing interest in the use of Bayesian statistical methods for the analysis of clinical trials used to support regulatory decisions. Bayesian methods for the analysis of clinical trials are an attractive option when good prior information is available. Yet, in many cases, prior information is scarce and only tentative or proprietary prior information exists. In these situations, it is necessary to use noninformative type or skeptical-type priors. We undertook a systematic study of Bayesian methods and applied them to 13 randomized clinical trials in metastatic breast cancer submitted to the U.S. Food and Drug Administration for registrational purposes. Across all 13 studies, there were a total of 10,521 patients using 10 experimental agents. Our results demonstrated that Bayesian analyses with noninformative priors provided similar results to more traditional analyses. We also found that early interim looks at the study results can vary widely based upon the type of prior used. Finally, we found that pre-defined threshold stopping rules need to be relatively strong to prevent trials from stopping very early. Our results suggest that, when prior information is limited and a noninformative prior is used, there is little numerical difference between Bayesian methods and more traditional analysis methods. Bayesian methods, however, may provide additional summaries of the data that are more easily interpretable than means and confidence intervals.

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.

Effects of globally obtained informative priors on bayesian safety performance functions developed for Australian crash data

The purpose of this study is to determine the best estimator for estimating the shape parameters of the Pareto distribution with the known scale parameter. Estimation of these parameters is done by using the Gamma distribution as the prior distribution of the conjugate and the Uniform distribution as the non-conjugate prior distribution. A comparison of the two prior distributions is done through simulation studies with various sample sizes. The best estimator net is a method that produces the smallest posterior variance, absolute bias, and Bayes confidence interval. This study proves that the Bayes estimator by using the prior conjugate distribution produces all indicators of the goodness of the model with a smaller value than the non-conjugate prior distribution. Thus it can be concluded that the estimator with prior conjugate will produce a better predictive value than prior non-conjugate.

ABSTRACTBayesian analysis of 3-component mixture model, of exponentiated Weibull distribution under type-I right censoring scheme, is explored in this study. With the help of non-informative (uniform and Jeffreys) priors and loss functions (e.g. squared error loss function, quadratic loss function, precautionary loss function and DeGroot loss function), Bayes estimators and posterior risks are derived. The Bayes estimators and posterior risks are observed as a function of the test termination time. Simulation study as well as a practical example is given in this study.

Bayesian analysis of the 3-component mixture of an Exponential distribution under type-I right censoring scheme is considered in this paper. The Bayes estimators and posterior risks for the unknown parameters are derived under squared error loss function, precautionary loss function and DeGroot loss function assuming the non-informative (uniform and Jeffreys') priors. The Bayes estimators and posterior risks are viewed as a function of the test termination time. A simulation study is given to highlight and compare the properties of the Bayes estimates.

This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.

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.