Bayesian Prediction Intervals Research Articles

Estimation and Bayesian Prediction of the Generalized Pareto Distribution in the Context of a Progressive Type-II Censoring Scheme

The generalized Pareto distribution plays a significant role in reliability research. This study concentrates on the statistical inference of the generalized Pareto distribution utilizing progressively Type-II censored data. Estimations are performed using maximum likelihood estimation through the expectation–maximization approach. Confidence intervals are derived using the asymptotic confidence intervals. Bayesian estimations are conducted using the Tierney and Kadane method alongside the Metropolis–Hastings algorithm, and the highest posterior density credible interval estimation is accomplished. Furthermore, Bayesian predictive intervals and future sample estimations are explored. To illustrate these inference techniques, a simulation and practical example are presented for analysis.

A statistical model predicts N mineralization of various organic amendments for paddy fields

ABSTRACT To predict the mineralization of nitrogen(N) in organic amendments in paddy fields, we applied data obtained from incubation experiments under waterlogged conditions by a hierarchical Bayesian model using acid-detergent-soluble organic N (ADSON) as a model input. Each of 20 organic amendments (6 cattle manure composts, 3 swine manure composts, 3 poultry manure composts, 2 oil cakes, 2 rice brans, 1 fish meal, 1 sludge fertilizer, and 2 mixed commercial organic fertilizers) was mixed with paddy soil, and the N mineralization curves obtained after 1, 2, 4, 8, and 12 weeks of waterlogging were incorporated into the model to obtain values of the model parameters k i , α1, and Q 10. All model parameters converged with a mean absolute error (MAE) of 0.044 g N kg−1, a root mean square error (RMSE) of 0.060 g N kg−1, and a model efficiency (EF) of 0.96. These values were comparable to or slightly better than those of a previous model for upland soil conditions. Most of the observed values were within the 95% Bayesian prediction interval, indicating that the model performed well with the observed data. Although it is generally believed that livestock manure composts decompose more slowly than organic fertilizers such as oil cake, the livestock manure composts had a large value of the rate constant k i , presumably because of the initial rapid decomposition of the small amounts of organic N present in them. In paddy fields, the soil conditions are aerobic between the application of organic amendments and water entry. In such a case, we propose the use of the upland field model to predict the mineralization of organic N during the period of aerobic soil conditions and of this study’s model during the period of waterlogging, subtracting the amount of already mineralized N from the parameters N in and ADSON. The model devised here is expected to be useful for fertilization design in organic agriculture and in the reduction of the use of inorganic fertilizer.

Bayesian Estimation and Prediction for Inverse Power Maxwell Distribution with Applications to Tax Revenue and Health Care Data

This article discusses the problem of estimating the unknown model parameters as well as prediction of future observations from inverse power Maxwell distribution. The maximum likelihood method is applied for estimating the model parameters using Newton-Rapson iterative procedures. The existence and uniqueness of maximum likelihood estimates are established using Cauchy-Schwartz inequality. Approximate confidence intervals are constructed using Fisher information matrix. Using independent gamma informative priors, the Bayes estimates of unknown model parameters are obtained under squared error and Linex loss functions. Two approximation techniques namely: Lindley’s approximation and Metropolis-Hastings within Gibbs sampler algorithm have been employed to derive the Bayes estimators and also to construct the associate highest posterior density credible intervals. Based on the informative (observed) sample, Bayesian prediction, predictive density, and predictive intervals are derived for future observation and decision. The performance of proposed methods are evaluated though a Monte Carlo simulation experiment. Two real-life datasets related to tax revenue and heath are incorporated to show the practical utility of proposed methodology in real phenomenon.

Generalized Bayes Prediction Study Based on Joint Type-II Censoring

In this paper, the problem of predicting future failure times based on a jointly type-II censored sample from k exponential populations is considered. The Bayesian prediction intervals and point predictors were then obtained. Generalized Bayes is a Bayesian study based on a learning rate parameter. This study investigated the effects of the learning rate parameters on the prediction results. The loss functions of squared error, Linex, and general entropy were used as point predictors. Monte Carlo simulations were performed to show the effectiveness of the learning rate parameter in improving the results of prediction intervals and point predictors.

An improved Bayesian approach to estimating the reference interval from a meta-analysis: Directly monitoring the marginal quantiles and characterizing their uncertainty.

Reference intervals, or reference ranges, aid medical decision-making by containing a pre-specified proportion (e.g., 95%) of the measurements in a representative healthy population. We recently proposed three approaches for estimating a reference interval from a meta-analysis based on a random effects model: a frequentist approach, a Bayesian posterior predictive interval, and an empirical approach. Because the Bayesian posterior predictive interval becomes wider to incorporate estimation uncertainty, it may systematically contain greater than 95% of measurements when the number of studies is small or the between study heterogeneity is large. The frequentist and empirical approaches also captured a median of less than 95% of measurements in this setting, and 95% confidence or credible intervals for the reference interval limits were not developed. In this update, we describe how one can instead use Bayesian methods to summarize the appropriate quantiles (e.g., 2.5th and 97.5th) of the marginal distribution of individuals across studies and construct a credible interval describing the estimation uncertainty in the lower and upper limits of the reference interval. We demonstrate through simulations that this method performs well in capturing 95% of values from the marginal distribution and maintains a median coverage of near 95% of the marginal distribution even when the number of studies is small, or the between-study heterogeneity is large. We also compare the results of this method to those obtained from the three previously proposed methods in the original case study of the meta-analysis of frontal subjective postural vertical measurements.

Conformal Prediction Credibility Intervals

In the predictive modeling context, the credibility estimator is a point predictor; it is easy to calculate and avoids the model misspecification risk asymptotically, but it provides no quantification of inferential uncertainty. A Bayesian prediction interval quantifies uncertainty of prediction, but it often requires expensive computation and is subject to model misspecification risk even asymptotically. Is there a way to get the best of both worlds? Based on a powerful machine learning strategy called conformal prediction, this article proposes a method that converts the credibility estimator into a conformal prediction credibility interval. This conformal prediction credibility interval contains the credibility estimator, has computational simplicity, and guarantees finite-sample validity at a pre-assigned coverage level.

Bayesian Interval Estimation for the Two-Parameter Exponential Distribution Based on the Right Type II Censored Sample

The Bayesian interval estimation of the scale parameter for two-parameter exponential distribution is proposed based on the right type II censored sample. Under this type of censoring, two methods of Bayesian joint confidence region of the two parameters are also proposed. The simulation results show that the Bayesian method has a higher coverage probability than the existing method, so the Bayesian method is recommended for use. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. The predictive interval of the future observation based on the right type II censored sample is also provided. One biometrical example is given to illustrate the proposed methods for the Bayesian interval estimations and prediction interval.

Bayesian Prediction Intervals Based on Type‐I Hybrid Censored Data from the Lomax Distribution under Step‐Stress Model

The Bayesian prediction of future failures from Lomax distribution is the subject of this research. The observed data is censored using a Type‐I hybrid censoring scheme under a step‐stress partially accelerated life test. There are two types of sampling schemes considered: one‐sample and two‐sample. We create predictive intervals for failure observations in the future. Bayesian prediction intervals are constructed using MCMC algorithms. After all, two numerical examples, simulation study and a real‐life example are provided for both one‐sample and two‐sample methods for the purpose of illustration.

Prediction from Transmuted Rayleigh Distribution in the Presence of Outliers

The quality of the procedures used in statistical analysis depends largely on the assumed probability distribution. However, there are still many problems with data that do not follow any of the classical distributions; therefore, researchers have developed many standardized probability distributions by generalizing or transforming them. Transmuted Rayleigh distribution extends the Rayleigh distribution in the analysis of data and provides larger flexibility in modeling real data. In this article, Bayesian predictive intervals for order statistics of future observations from this distribution are obtained in the presence of outliers when the scale parameter is unknown. The slippage outlier model is utilized in addition to the two‐sample prediction scheme. We shall consider two cases: (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample. Numerical computations are obtained to illustrate the effect of outliers on the Bayesian predictive intervals.

On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study.

Automatic Indexation of the Pension Age to Life Expectancy: When Policy Design Matters

Increasing retirement ages in an automatic or scheduled way with increasing life expectancy at retirement is a popular pension policy response to continuous longevity improvements. The question addressed here is: to what extent is simply adopting this approach likely to fulfill the overall goals of policy? To shed some light on the answer, we examine the policies of four countries that have recently introduced automatic indexation of pension ages to life expectancy–The Netherlands, Denmark, Portugal and Slovakia. To this end, we forecast an alternative period and cohort life expectancy measures using a Bayesian Model Ensemble of heterogeneous stochastic mortality models comprised of parametric models, principal component methods, and smoothing approaches. The approach involves both the selection of the model confidence set and the determination of optimal weights. Model-averaged Bayesian credible prediction intervals are derived accounting for various stochastic process, model, and parameter risks. The results show that: (i) retirement ages are forecasted to increase substantially in the coming decades, particularly if a constant period in retirement is targeted; (ii) retirement age policy outcomes may substantially deviate from the policy goal(s) depending on the design adopted and its implementation; and (iii) the choice of a cohort over period life expectancy measure matters. In addition, the distributional issues arising with the increasing socio-economic gap in life expectancy remain largely unaddressed.

Estimation and Prediction for Gompertz Distribution under General Progressive Censoring

In this article, we discuss the estimation of the parameters for Gompertz distribution and prediction using general progressive Type-II censoring. Based on the Expectation–Maximization algorithm, we calculate the maximum likelihood estimates. Bayesian estimates are considered under different loss functions, which are symmetrical, asymmetrical and balanced, respectively. An approximate method—Tierney and Kadane—is used to derive the estimates. Besides, the Metropolis-Hasting (MH) algorithm is applied to get the Bayesian estimates as well. According to Fisher information matrix, we acquire asymptotic confidence intervals. Bootstrap intervals are also established. Furthermore, we build the highest posterior density intervals through the sample generated by the MH algorithm. Then, Bayesian predictive intervals and estimates for future samples are provided. Finally, for evaluating the quality of the approaches, a numerical simulation study is implemented. In addition, we analyze two real datasets.

Addressing the life expectancy gap in pension policy

Understanding the systematic relationship between period and cohort life expectancy and how the relationship evolves over time are critical issues in formulating the design of retirement income products, evaluating the actuarial balance of pension schemes, and more generally for all analyses where demographic projections are involved. In this study, estimates of the life expectancy gap at all ages are performed using data for 1960–2018 from the Human Mortality Database and projections are generated through 2050 for the 42 national populations, disaggregated by sex. Contrary to previous research that often uses a single deemed to be «best» model to forecast mortality rates, we use a novel adaptive Bayesian Model Ensemble of heterogeneous parametric generalized age-period-cohort stochastic mortality models, principal component methods, and smoothing approaches. The procedure involves both the selection of the model confidence set and the determination of optimal weights. Model-averaged Bayesian credible prediction intervals are derived accounting for both the uncertainty arising from model error and parameter uncertainty. With intergenerational actuarial fairness and neutrality as the guiding principles the study then explores potential policy interventions to address the consequences of the life expectancy gap - spanning over adjustments in the accumulation, benefit determination, and payout stages. Comprehensive numerical results are provided for two policy options: (i) introducing a sustainability factor; and (ii) conditional pension indexation. The results show that: (i) the life expectancy gap is positive and significant for almost all countries and years studied, (ii) it will continue to increase, (iii) the magnitude of the subsidy rates between generations can be sizeable demanding important initial pension benefit reduction and/or a gradual diminution in the annual indexation rate of pensions to correct them.

Bayesian prediction interval for a constant-stress partially accelerated life test model under censored data

The present communication develops the tools for Bayesian prediction of the Gompertz distribution based on CSPALT. The Metropolis-Hastings algorithm is applied to evaluate the BPIs for a censored sample based on unified hybrid censoring scheme. In order to investigate the impact of methodologies adopted, two numerical examples are performed. The simulated results show that reducing the censoring percentages causes smaller BPIs. The flexibility of the UHCS in evaluating the BPIs can be helped to overcome many difficulties in engineering problems.

Prediction Intervals for the First and Last Point in Future Sample Having from a New Bathtub Shape Failure Rate Life Time Model in the Presence of Outliers

Acquiring Bayesian prediction intervals for the first and final points of observation with the bathtub-shape distribution of the failure rate life-time type under the conditions of available outliers is the focus of the research. These bounds of predication acquired on the basis of the right Type-II censored sample. The procedure is presented with the help of a wide range of illustrative example.

Frequentist performances of Bayesian prediction intervals for random-effects meta-analysis.

The prediction interval has been increasingly used in meta-analyses as a useful measure for assessing the magnitude of treatment effect and between-studies heterogeneity. In calculations of the prediction interval, although the Higgins-Thompson-Spiegelhalter method is used most often in practice, it might not have adequate coverage probability for the true treatment effect of a future study under realistic situations. An effective alternative candidate is the Bayesian prediction interval, which has also been widely used in general prediction problems. However, these prediction intervals are constructed based on the Bayesian philosophy, and their frequentist validities are only justified by large-sample approximations even if noninformative priors are adopted. There has been no certain evidence that evaluated their frequentist performances under realistic situations of meta-analyses. In this study, we conducted extensive simulation studies to assess the frequentist coverage performances of Bayesian prediction intervals with 11 noninformative prior distributions under general meta-analysis settings. Through these simulation studies, we found that frequentist coverage performances strongly depended on what prior distributions were adopted. In addition, when the number of studies was smaller than 10, there were no prior distributions that retained accurate frequentist coverage properties. We also illustrated these methods via applications to two real meta-analysis datasets. The resultant prediction intervals also differed according to the adopted prior distributions. Inaccurate prediction intervals may provide invalid evidence and misleading conclusions. Thus, if frequentist accuracy is required, Bayesian prediction intervals should be used cautiously in practice.

Statistical Inference for Gompertz Distribution Using the Adaptive-General Progressive Type-II Censored Samples

In this article, we combine the adaptive progressive Type-II censoring model with the general progressive model, to obtain the estimates for the parameters of Gompertz distribution, and the Bayesian prediction intervals. Estimation is executed using the maximum likelihood method (MLE) and the Bayesian method. Bayesian estimates are constructed depending on four types of loss functions. The credible intervals and the asymptotic confidence intervals are determined for the parameters of Gompertz distribution based on the Bayesian estimates and the MLEs, respectively. Finally, a real data example and the simulation study are discussed to compare the proposed methods.

Bayesian Estimation and Prediction Based on Progressively First Failure Censored Scheme from a Mixture of Weibull and Lomax Distributions

This paper develops Bayesian estimation and prediction, for a mixture of Weibull and Lomax distributions, in the context of the new life test plan called progressive first failure censored samples. Maximum likelihood estimation and Bayes estimation, under informative and non-informative priors, are obtained using Markov Chain Monte Carlo methods, based on the symmetric square error Loss function and the asymmetric linear exponential (LINEX) and general entropy loss functions. The maximum likelihood estimates and the different Bayes estimates are compared via a Monte Carlo simulation study. Finally, Bayesian prediction intervals for future observations are obtained using a numerical example

Prediction of Kumaraswamy distribution in constant‐stress model based on type‐I hybrid censored data

AbstractIn this work, a particular problem of Bayesian prediction concerning future observation from Kumaraswamy distribution under constant‐stress partially accelerated life test is treated. Type‐I hybrid censored data of the observed data are utilized. One‐ and two‐sample Bayesian prediction intervals for an unobserved future sample from Kumaraswamy distribution are settled. Markov chain Monte Carlo (MCMC) procedure is used to get Bayesian predictive intervals. Lastly, simulation study and a numerical example are given to illustrate the consequences of the research.

Prediction for Progressively Type-II Censored Competing Risks Data from the Half-Logistic Distribution

Point and interval predictions of the s-th order statistic in a future sample are discussed. The informative sample is assumed to be drawn from a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz and half-logistic distributions. The informative and future samples are progressively type-II censored, under competing risks model, and assumed to be obtained from the same population. A special attention is paid to the half-logistic distribution. Using six different progressive censoring schemes, numerical computations are carried out to illustrate the performance of the procedure. An illustrative example based on real data is also considered. The biases, mean squared prediction errors of the maximum likelihood predictors, coverage probabilities and average interval lengths of the Bayesian prediction intervals are computed via a simulation study.