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This paper deals with multicomponent stress–strength system reliability (MSR) and its maximum likelihood (ML) as well as Bayesian estimation. We assume that X 1 , X 2 , ⋯ , X k being the random strengths of k- components of a system and Y is the applied common random stress on them, which independently follows gamma distribution with parameters α 1 , λ 1 and α 2 , λ 2 respectively. The system works only if s 1 ≤ s ≤ k or more of the strengths exceed the common load/stress is called s-out-of-k: G system. Maximum likelihood and asymptotic interval estimators of MSR are obtained. Bayes estimates are computed under symmetric and asymmetric loss functions assuming informative and non-informative priors. ML and Bayes estimators are numerically evaluated and compared based on mean square errors and absolute biases through simulation study employing the Metropolis–Hastings algorithm. [ABSTRACT FROM AUTHOR]

The inverse of the coefficient of variation (ICV), otherwise known as the signal to-noise ratio, is the ratio of the population standard deviation to the population mean. It has often been used in the fields of finance and image processing, among others. In this study, various methods were applied to estimate the confidence intervals (CIs) for the difference between and the ratio of the ICVs of two delta-gamma distributions. The fiducial quantity method, Bayesian CI estimates based on the Jeffreys, uniform, or normal-gamma-beta (NGB) prior, and highest posterior density (HPD) intervals based on the Jeffreys, uniform, or NGB priors were used in this endeavor. A Monte Carlo simulation study was conducted to assess the performances of the proposed CI estimation methods in terms of their coverage probabilities and average lengths. The results indicate that the HPD interval based on the NGB prior or the Jeffreys prior performed well for a small probability of the samples containing zero observations () whereas the fiducial quantity method performed well for large values of . Furthermore, we demonstrate the practicability of the proposed methods using rainfall data from Lampang province, Thailand. [ABSTRACT FROM AUTHOR]

Since rainfall data often contain zero observations, the ratio of the variances of delta-gamma distributions can be used to compare the rainfall dispersion between two rainfall datasets. To this end, we constructed the confidence interval for the ratio of the variances of two delta-gamma distributions by using the fiducial quantity method, Bayesian credible intervals based on the Jeffreys, uniform, or normal-gamma-beta priors, and highest posterior density (HPD) intervals based on the Jeffreys, uniform, or normal-gamma-beta priors. The performances of the proposed confidence interval methods were evaluated in terms of their coverage probabilities and average lengths via Monte Carlo simulation. Our findings show that the HPD intervals based on Jeffreys prior and the normal-gamma-beta prior are both suitable for datasets with a small and large probability of containing zeros, respectively. Rainfall data from Phrae province, Thailand, are used to illustrate the practicability of the proposed methods with real data. [ABSTRACT FROM AUTHOR]

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as compared to simple models. This paper considers mixture of Topp-Leone distributions under classical and Bayesian perspective based on complete sample. The new distribution which exhibits decreasing and upside down bathtub shaped density while the distribution has the ability to model lifetime data with decreasing, increasing and upside down bathtub shaped failure rates. We derive several properties of the new distribution such as moments, moment generating function, conditional moment, mean deviation, Bonferroni and Lorenz curves and the order statistics of the proposed distribution. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) and uniform as well as gamma priors are considered to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. Monte Carlo simulation study is done to access the behavior of these estimators. For the illustrative purposes, a real-life application of the proposed distribution to a tensile strength data set is provided. [ABSTRACT FROM AUTHOR]

The gamma distribution is commonly used to model environmental data. However, rainfall data often contain zero observations, which violates the assumption that all observations must be positive in a gamma distribution, and so a gamma model with excess zeros treated as a binary random variable is required. Rainfall dispersion is important and interesting, the confidence intervals for the variance of a gamma distribution with excess zeros help to examine rainfall intensity, which may be high or low risk. Herein, we propose confidence intervals for the variance of a gamma distribution with excess zeros by using fiducial quantities and parametric bootstrapping, as well as Bayesian credible intervals and highest posterior density intervals based on the Jeffreys', uniform, or normal-gamma-beta prior. The performances of the proposed confidence interval were evaluated by establishing their coverage probabilities and average lengths via Monte Carlo simulations. The fiducial quantity confidence interval performed the best for a small probability of the sample containing zero observations (°) whereas the Bayesian credible interval based on the normal-gamma-beta prior performed the best for large°. Rainfall data from the Kiew Lom Dam in Lampang province, Thailand, are used to illustrate the efficacies of the proposed methods in practice. [ABSTRACT FROM AUTHOR]

The gamma distribution is commonly used to model environmental data. However, rainfall data often contain zero observations, which violates the assumption that all observations must be positive in a gamma distribution, and so a gamma model with excess zeros treated as a binary random variable is required. Rainfall dispersion is important and interesting, the confidence intervals for the variance of a gamma distribution with excess zeros help to examine rainfall intensity, which may be high or low risk. Herein, we propose confidence intervals for the variance of a gamma distribution with excess zeros by using fiducial quantities and parametric bootstrapping, as well as Bayesian credible intervals and highest posterior density intervals based on the Jeffreys’, uniform, or normal-gamma-beta prior. The performances of the proposed confidence interval were evaluated by establishing their coverage probabilities and average lengths via Monte Carlo simulations. The fiducial quantity confidence interval performed the best for a small probability of the sample containing zero observations (δ) whereas the Bayesian credible interval based on the normal-gamma-beta prior performed the best for large δ. Rainfall data from the Kiew Lom Dam in Lampang province, Thailand, are used to illustrate the efficacies of the proposed methods in practice.

Specific bivariate classes of distributions with given marginals can be used for contribution of the linking distribution between conditional and unconditional effectiveness using copulas. In this paper, a Bayesian model is proposed for meta-analysis of treatment effectiveness data which are generally discrete Binomial and sparse. A bivariate class of priors is imposed to accommodate a wide range of heterogeneity between the multicenter clinical trials involved in the study. Applications to real data are provided. [ABSTRACT FROM AUTHOR]

Estimating incidence of rare cancers is challenging for exceptionally rare entities and in small populations. In a previous study, investigators in the Information Network on Rare Cancers (RARECARENet) provided Bayesian estimates of expected numbers of rare cancers and 95% credible intervals for 27 European countries, using data collected by population-based cancer registries. In that study, slightly different results were found by implementing a Poisson model in integrated nested Laplace approximation/WinBUGS platforms. In this study, we assessed the performance of a Poisson modeling approach for estimating rare cancer incidence rates, oscillating around an overall European average and using small-count data in different scenarios/computational platforms. First, we compared the performance of frequentist, empirical Bayes, and Bayesian approaches for providing 95% confidence/credible intervals for the expected rates in each country. Second, we carried out an empirical study using 190 rare cancers to assess different lower/upper bounds of a uniform prior distribution for the standard deviation of the random effects. For obtaining a reliable measure of variability for country-specific incidence rates, our results suggest the suitability of using 1 as the lower bound for that prior distribution and selecting the random-effects model through an averaged indicator derived from 2 Bayesian model selection criteria: the deviance information criterion and the Watanabe-Akaike information criterion. [ABSTRACT FROM AUTHOR]

Natural rainfall is necessary for agriculture in Thailand. Often, rainfall data contain zero and positive right-skewed observations. Within a given region, the mean rainfall can be used to evaluate how rainfall has changed over a period of time, and so, in this article, we propose interval estimates and an adjustment process based on the Bayesian approach to compute the rainfall amount mean. This includes highest posterior density intervals (HPDs) based on the beta (HPD-B), normal inverse chi-squared (HPD-NIC) and uniform (HPD-U) priors, which were compared with the existing methods. Coverage probability and relative average length were used to assess the performance of the methods by comparing their computation. A numerical evaluation showed that for a single mean and even chance of having zero observations, HPD-beta achieved the given target with small to moderate sample sizes, while HPD-U tended to perform very well with large sample size. To compare the difference between two means, HPD-U demonstrated excellent performance in almost all cases. Daily rainfall data from provinces in northern Thailand were used to confirm the efficacy of the new methods. [ABSTRACT FROM AUTHOR]

Since rainfall data often contain zero observations, the ratio of the variances of delta-gamma distributions can be used to compare the rainfall dispersion between two rainfall datasets. To this end, we constructed the confidence interval for the ratio of the variances of two delta-gamma distributions by using the fiducial quantity method, Bayesian credible intervals based on the Jeffreys, uniform, or normal-gamma-beta priors, and highest posterior density (HPD) intervals based on the Jeffreys, uniform, or normal-gamma-beta priors. The performances of the proposed confidence interval methods were evaluated in terms of their coverage probabilities and average lengths via Monte Carlo simulation. Our findings show that the HPD intervals based on Jeffreys prior and the normal-gamma-beta prior are both suitable for datasets with a small and large probability of containing zeros, respectively. Rainfall data from Phrae province, Thailand, are used to illustrate the practicability of the proposed methods with real data.

In engineering, it is usual to model the data so as to make a decision under the problem of uncertainty. Commonly, the data in engineering is skewed to the right, and the skewed distributions in statistics are the appropriate models for making a decision under the Bayesian paradigm. To model the lifetime of an electronic device, an engineer can use the Bayesian estimators to compute the effect of the evidence in increasing the probability for the lifetime of an electronic device by using the prior information. This study presents an estimation of the shape parameter of Lomax distribution under Uniform and Jeffery prior by adopting SELF, QELF, WSELF, and the PELF. The significance of various estimators is compared and presented in graphs using simulated data under the Bayesian paradigm. It was determined that under a uniform prior, Bayes estimator under weighted error loss function (BWEL) provides a better result than others. Under Jeffery prior, the precautionary error loss function (BPEL) leads to a better result than others. Moreover, an application to engineering is also presented for illustration purposes. [ABSTRACT FROM AUTHOR]

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.

Bayesian estimation in the mixture models under Type - I censored samples have been done by several authors. In this paper the Bayesian estimation of the parameters of the mixture of power function distribution under Type - II censoring is considered. The estimation is carried out using uniform priors and Jeffrey's priors for the parameters of the model under K - loss function and Precautionary loss function. The posterior risk and the root mean square error of the estimators are also obtained and to study the performance of these estimators a simulation study is conducted. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The exponential Lomax distribution is an extension of the Lomax distribution proposed by El-Bassiouny et al. (2015). This distribution is very useful and has been found to outperform other extensions of the Lomax distribution such as the exponentiated Lomax, Marshall-Olkin extended-Lomax, beta-Lomax, Kumaraswamy-Lomax, McDonald-Lomax and gamma-Lomax based on some applications to lifetime datasets. In this article, the scale parameter of the exponential Lomax is estimated using the Bayesian method of estimation under two non-informative (Jeffery and Uniform prior) and Informative prior (Gamma prior) distribution and compared to the estimates of maximum Likelihood using three loss functions (Square error, Quadratic, and Precautionary loss function). The posterior distributions of the said parameter were derived and also the Estimators and risks were also obtained using the priors and loss functions. Furthermore, a simulation study was carried out using R software package to assess the performance of the two methods by means of their MSEs. [ABSTRACT FROM AUTHOR]

The age of the root of the Indo-European language family has received much attention since the application of Bayesian phylogenetic methods by Gray and Atkinson (2003). With the application of new models, the root age of the Indo-European family has tended to decrease from an age that supported the Anatolian origin hypothesis to an age that supports the Steppe origin hypothesis (Chang et al., 2015). However, none of the published work in Indo-European phylogenetics has studied the effect of tree priors on phylogenetic analyses of the Indo-European family. In this paper, I intend to fill this gap by exploring the effect of tree priors on different aspects of the Indo-European family's phylogenetic inference. I apply three tree priors—Uniform, Fossilized Birth-Death (FBD), and Coalescent—to five publicly available datasets of the Indo-European language family. I evaluate the posterior distribution of the trees from the Bayesian analysis using Bayes Factor, and find that there is support for the Steppe origin hypothesis in the case of two tree priors. I report the median and 95 % highest posterior density (HPD) interval of the root ages for all three tree priors. A model comparison suggests that either the Uniform prior or the FBD prior is more suitable than the Coalescent prior to the datasets belonging to the Indo-European language family. [ABSTRACT FROM AUTHOR]

Censoring is an unavoidable feature of reliability and survival analysis due to time and cost constraints. In this paper, we numerically study the effect of sample size, censoring rate and parameter size of Weibull distribution on its estimates. Some interesting properties and comparison of the Bayes and ML estimates along with their standard errors have been studied. The sample expressions for the Bayes and ML estimates and their variances are derived as a function of test termination time. The difference between ML estimates and uninformative Bayesian estimates becomes negligible for large sample size. [ABSTRACT FROM AUTHOR]

Abstract: The inverse of normal covariance matrix is called precision matrix and often plays an important role in statistical estimation problem. This paper deals with the problem of estimating the precision matrix under a quadratic loss, where the precision matrix is restricted to a bounded parameter space. Gauss’ divergence theorem with matrix argument shows that the unbiased and unrestricted estimator is dominated by a posterior mean associated with a flat prior on the bounded parameter space. Also, an improving method is given by considering an expansion estimator. A hierarchical prior is shown to improve on the posterior mean. An application is given for a Bayesian prediction in a random-effects model. [Copyright &y& Elsevier]

Abstract: Many parameters and positive-definiteness are two major obstacles in estimating and modeling a correlation matrix for longitudinal data. In addition, when longitudinal data is incomplete, incorrectly modeling the correlation matrix often results in bias in estimating mean regression parameters. In this paper, we introduce a flexible and parsimonious class of regression models for a covariance matrix parameterized using marginal variances and partial autocorrelations. The partial autocorrelations can freely vary in the interval while maintaining positive definiteness of the correlation matrix so the regression parameters in these models will have no constraints. We propose a class of priors for the regression coefficients and examine the importance of correctly modeling the correlation structure on estimation of longitudinal (mean) trajectories and the performance of the DIC in choosing the correct correlation model via simulations. The regression approach is illustrated on data from a longitudinal clinical trial. [Copyright &y& Elsevier]

A growing body of research is demonstrating increased accuracy in aging from a relatively new method, transition analysis. Although transition analysis was developed for paleodemographic research, a majority of subsequent studies have been in the forensic arena, with very little work in bioarchaeological contexts. Using the Suchey-Brooks pubic symphysis phases, scored on a target sample of historic Italians from the island of Sardinia, we compare accuracy of aging between transition analysis combined with a Bayesian approach and the standard Suchey-Brooks age ranges. Because of the difficulty in identifying a reasonable informative prior for bioarchaeological samples, we also compared results of both an informative prior and a uniform prior for age estimation. Published ages-of-transition for the Terry Collection and Balkan genocide victims were used in conjunction with parameters generated from Gompertz hazard models derived from the priors. The ages-of-transition and hazard parameters were utilized to calculate the highest posterior density regions, otherwise known as 'coverages' or age ranges, for each Suchey-Brooks phase. Each prior, along with the parameters, were input into cumulative binomial tests. The results indicate that the Bayesian approach outperformed the Suchey-Brooks technique alone. The Terry Collection surpassed the Balkans as a reasonable sample from which to derive transition analysis parameters. This discrepancy between populations is due to different within phase age-at-death distributions that reflect differences in aging between the populations. These results indicate bioarchaeologists should strive to apply a Bayesian analysis when aging historic and archaeological populations by employing an informative prior. Am J Phys Anthropol 149:259-265, 2012. © 2012 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the correlation in bivariate normal data when the means and variances are assumed known, with emphasis on the small sample case. We consider eight different estimators, several of them considered here for the first time in the literature. In a simulation study, we found that Bayesian estimators using the uniform and arc-sine priors outperformed several empirical and exact or approximate maximum likelihood estimators in small samples. The arc-sine prior did better for large values of the correlation. For testing whether the correlation is zero, we found that Bayesian hypothesis tests outperformed significance tests based on the empirical and exact or approximate maximum likelihood estimators considered in small samples, but that all tests performed similarly for sample size 50. These results lead us to suggest using the posterior mean with the arc-sine prior to estimate the correlation in small samples when the variances are assumed known. [ABSTRACT FROM AUTHOR]

Abstract: A new class of Bayesian estimators for a proportion in multistage binomial designs is considered. Priors belong to the beta-J distribution family, which is derived from the Fisher information associated with the design. The transposition of the beta parameters of the Haldane and the uniform priors in fixed binomial experiments into the beta-J distribution yields bias-corrected versions of these priors in multistage designs. We show that the estimator of the posterior mean based on the corrected Haldane prior and the estimator of the posterior mode based on the corrected uniform prior have good frequentist properties. An easy-to-use approximation of the estimator of the posterior mode is provided. The new Bayesian estimators are compared to Whitehead''s and the uniformly minimum variance estimators through several multistage designs. Last, the bias of the estimator of the posterior mode is derived for a particular case. [Copyright &y& Elsevier]

Abstract: In this paper, we present a Bayesian inference methodology for Box–Cox transformed linear mixed model with ARMA() errors using approximate Bayesian and Markov chain Monte Carlo methods. Two priors are proposed and put into comparisons in parameter estimation and prediction of future values. The advantages of Bayesian approach over maximum likelihood method are demonstrated by both real and simulated data. [Copyright &y& Elsevier]

In this article, we investigate the properties of the posterior distribution under the uniform improper prior for two commonly used proportional hazards models; the Weibull regression model and the extreme value regression model. We allow the observations to be right censored. We obtain sufficient conditions for the existence of the posterior moment generating function of the regression coefficients. A dataset involving a lung cancer clinical trial and a simulation are presented to illustrate our results. [ABSTRACT FROM AUTHOR]