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This paper proposed a new family of loss function, as a generalization of weighted squared error loss function, aiming to construct more new loss functions. We proposed the definition of the generalized weighted squared error loss function and derived the Bayesian estimation and its posterior risk under the generalized weighted squared error loss function based on the definition. Moreover, some members of the family of the generalized weighted squared error loss function are discussed. The results show that the proposed generalized weighted squared error loss function contains some existing loss functions in special cases and can obviously be employed to generate more new ones. The expressions of Bayesian estimations and their posterior risks of Rayleigh distribution parameter under the squared error loss function, weighted squared-negative exponential error loss function and LINEX loss function are derived respectively. For ease of explanation, Monte Carlo simulation example and application example are provided, and the results are compared based on posterior risk. [ABSTRACT FROM AUTHOR]

Abstract Mixture distributions are naturally extra attractive to model the heterogeneous environment of processes in reliability analysis than simple probability models. This focus of the study is to develop and Bayesian inference on the 3-component mixture of power distributions. Under symmetric and asymmetric loss functions, the Bayes estimators and posterior risk using priors are derived. The presentation of Bayes estimators for various sample sizes and test termination time (a fact of time after that test is terminated) is examined in this article. To assess the performance of Bayes estimators in terms of posterior risks, a Monte Carlo simulation along with real data study is presented.

Objective: Effective first‐trimester screening for pre‐eclampsia (PE) can be achieved using a competing‐risks model that combines risk factors from the maternal history with multiples of the median (MoM) values of biomarkers. A new model using artificial intelligence through machine‐learning methods has been shown to achieve similar screening performance without the need for conversion of raw data of biomarkers into MoM. This study aimed to investigate whether this model can be used across populations without specific adaptations. Methods: Previously, a machine‐learning model derived with the use of a fully connected neural network for first‐trimester prediction of early (< 34 weeks), preterm (< 37 weeks) and all PE was developed and tested in a cohort of pregnant women in the UK. The model was based on maternal risk factors and mean arterial blood pressure (MAP), uterine artery pulsatility index (UtA‐PI), placental growth factor (PlGF) and pregnancy‐associated plasma protein‐A (PAPP‐A). In this study, the model was applied to a dataset of 10 110 singleton pregnancies examined in Spain who participated in the first‐trimester PE validation (PREVAL) study, in which first‐trimester screening for PE was carried out using the Fetal Medicine Foundation (FMF) competing‐risks model. The performance of screening was assessed by examining the area under the receiver‐operating‐characteristics curve (AUC) and detection rate (DR) at a 10% screen‐positive rate (SPR). These indices were compared with those derived from the application of the FMF competing‐risks model. The performance of screening was poor if no adjustment was made for the analyzer used to measure PlGF, which was different in the UK and Spain. Therefore, adjustment for the analyzer used was performed using simple linear regression. Results: The DRs at 10% SPR for early, preterm and all PE with the machine‐learning model were 84.4% (95% CI, 67.2–94.7%), 77.8% (95% CI, 66.4–86.7%) and 55.7% (95% CI, 49.0–62.2%), respectively, with the corresponding AUCs of 0.920 (95% CI, 0.864–0.975), 0.913 (95% CI, 0.882–0.944) and 0.846 (95% CI, 0.820–0.872). This performance was achieved with the use of three of the biomarkers (MAP, UtA‐PI and PlGF); inclusion of PAPP‐A did not provide significant improvement in DR. The machine‐learning model had similar performance to that achieved by the FMF competing‐risks model (DR at 10% SPR, 82.7% (95% CI, 69.6–95.8%) for early PE, 72.7% (95% CI, 62.9–82.6%) for preterm PE and 55.1% (95% CI, 48.8–61.4%) for all PE) without requiring specific adaptations to the population. Conclusions: A machine‐learning model for first‐trimester prediction of PE based on a neural network provides effective screening for PE that can be applied in different populations. However, before doing so, it is essential to make adjustments for the analyzer used for biochemical testing. © 2023 International Society of Ultrasound in Obstetrics and Gynecology. [ABSTRACT FROM AUTHOR]

In this present work, we are going to show the various useful properties of the existing distribution known as MGExp(ϵ)-distribution which have not quoted by the host authors like moments, mean deviation about mean, mean deviation about median, order statistics, count of uncertainty. Estimation procedures have been adopted under Bayesian estimation for progressive Type-II censored case. Simulation study has also been carried out to judge the behavior of the Bayes estimator at the long-run. Performance of the Bayes estimators and their posterior risks of the considered loss functions have been obtained, reported and compared for the considered values of sample size, effective sample size, parameter and removals. The comparison of Bayes estimators of all 6 chosen loss functions have been done on the ground of lowest posterior risks. [ABSTRACT FROM AUTHOR]

• Corrected an error (posterior risk of Bayesian estimation under the entropy loss function) in Ali et al. [10]. • Derived the corrected posterior risk of Bayesian estimation under the entropy loss function. • Derived posterior risk of Bayesian estimation for binomial distribution under the entropy and squared error loss function. • Monte Carlo simulation example and application example are provided for illustrative purposes. Scholars have discussed the Bayesian estimation and their corresponding posterior risks for the Lindley distribution parameter, respectively, using seven different loss functions, where the posterior risk under entropy loss function (ELF) is not correct. In this paper, we will derive the Bayesian estimation and its corrected posterior risk under the entropy loss function. Moreover, the Bayesian estimations and their posterior risks of binomial distribution parameter under the entropy loss function and squared error loss function are respectively developed. Monte Carlo simulation example and application example are provided for illustrative purposes, and results are compared based on posterior risk. [ABSTRACT FROM AUTHOR]

We introduce a new family via the log mean of an underlying distribution and as baseline the proportional hazards model and derive some important properties. A special model is proposed by taking the Weibull for the baseline. We derive several properties of the sub-model such as moments, order statistics, hazard function, survival regression and certain characterization results. We estimate the parameters using frequentist and Bayesian approaches. Further, Bayes estimators, posterior risks, credible intervals and highest posterior density intervals are obtained under different symmetric and asymmetric loss functions. A Monte Carlo simulation study examines the biases and mean square errors of the maximum likelihood estimators. For the illustrative purposes, we consider heart transplant and bladder cancer data sets and investigate the efficiency of proposed model. [ABSTRACT FROM AUTHOR]

Natural resources extraction involves continuous exposure to cooling meteorological factors typical for open production grounds. This necessitates relevant health risk assessment and management of health risks caused by exposure to these harmful occupational factors. However, the available risk assessment models do not provide a possibility to perform complete assessment of the existing risks created by exposure to meteorological hazards. The study design included the following. We performed hygienic assessment of working conditions and health of workers employed by “Samotlorneftegaz” Joint Stock Company (JSC) who had to perform their work tasks under exposure to cooling meteorological factors on open production grounds; the assessment involved calculating the group health risk. Individual peculiarities were assessed using subjective (547 people took part in questioning) and objective assessment methods (76 people took part in estimating thermal state of their bodies and 54 people participated in thermometry with cold stress). Finally, we assessed prior and posterior risks. The prior group risk assessment made it possible to identify risk groups who had a significant risk of developing occupational and non-occupational diseases and to rank working places as per health hazards. The posterior risk assessment confirmed the results produced by the prior risk assessment regarding potentiating negative effects produced by cooling meteorological factors. The assessment of developing general and local thermoregulation disorders revealed that certain individual peculiarities made a substantial contribution into their development. Among them, we can mention long-term outdoor work (60 % of work time or more) under exposure to cooling meteorological factors; a chronic pathology; tobacco smoking. The results produced by this study allowed us to suggest an integrated model for risk assessment, management and communication about health risks caused by working under exposure to cooling meteorological factors

Objective: To evaluate the accuracy of predicting the risk of developing pre-eclampsia (PE) according to first-trimester maternal demographic characteristics, medical history and biomarkers using artificial-intelligence and machine-learning methods.Methods: The data were derived from prospective non-interventional screening for PE at 11-13 weeks' gestation at two maternity hospitals in the UK. The data were divided into three subsets. The first set, including 30 437 subjects, was used to develop the training process, the second set of 10 000 subjects was utilized to optimize the machine-learning hyperparameters and the third set of 20 352 subjects was coded and used for model validation. An artificial neural network was used to predict from the demographic characteristics and medical history the prior risk that was then combined with biomarker values to determine the risk of PE and preterm PE with delivery at < 37 weeks' gestation. An additional network was trained without including race as input. Biomarkers included uterine artery pulsatility index (UtA-PI), mean arterial blood pressure (MAP), placental growth factor (PlGF) and pregnancy-associated plasma protein-A. All markers were entered using raw values without conversion into standardized multiples of the median. The prediction accuracy was estimated using the area under the receiver-operating-characteristics curve (AUC). We further computed the detection rate at 10%, 20% and 40% false-positive rates (FPR). The impact of taking aspirin was also added. Shapley values were calculated to evaluate the contribution of each parameter to the prediction of risk. We used a non-parametric test to compare the expected AUC with the one obtained when we randomly scrambled the labels and kept the predictions. For the general prediction, we performed 10 000 permutations of the labels. When the AUC was higher than the one obtained in all 10 000 permutations, we reported a P-value of < 0.0001. For the race-specific analysis, we performed 1000 permutations. When the AUC was higher than the AUC in permutations, we reported a P-value of < 0.001.Results: The detection rate for preterm PE vs no PE, at a 10% FPR, was 53.3% when screening by maternal factors only, and the corresponding AUC was 0.816; these increased to 75.3% and 0.909, respectively, with the addition of biomarkers into the model. Information on race was important for the prediction accuracy; when race was not used to train the model, at a 10% FPR, the detection rate of preterm PE vs no PE decreased to 34.5-45.5% (for different races) when screening by maternal factors only and to 55.0-62.1% when biomarkers were added. The major predictors of PE were high MAP and UtA-PI, and low PlGF. The accuracy of prediction of all PE cases was lower than that for preterm PE. Aspirin use was recommended for cases who were at high risk of preterm PE. The AUC of all PE vs no PE was 0.770 when screening by maternal factors and 0.817 when the biomarkers were added; the respective detection rates, at a 10% FPR, were 41.3% and 52.9%.Conclusions: Screening for PE using a non-linear machine-learning-based approach does not require a population-based normalization, and its performance is similar to that of logistic regression. Removing race information from the model reduces its prediction accuracy, especially for the non-white populations when only maternal factors are considered. © 2022 International Society of Ultrasound in Obstetrics and Gynecology. [ABSTRACT FROM AUTHOR]

Transmuted distributions have been centered of focus for researchers recently due to their flexibility and applicability in statistics. However, the only few contributions have considered estimation for mixture of transmuted lifetime models especially under Bayesian methods has been explored more recently. We have considered the Bayesian estimation of transmuted Lomax mixture model (TLMM) for type-I censored samples. The Bayes estimates (BEs) and posterior risks (PRs) for informative and noninformative priors are evaluated using four different loss functions (LFs), two symmetric and two asymmetric, namely the squared error loss function (SELF), precautionary loss function (PLF), weighted balance loss function (WBLF), and general entropy loss function (GELF). Simulations are run using Lindley Approximation method to compare the BEs under various sample sizes and censoring rates. The estimates under informative prior and GELF were found superior to their counterparts. The applicability of the proposed estimates has been illustrated using the analysis of a real data regarding type-I censored failure times of windshields airplanes. [ABSTRACT FROM AUTHOR]

The effect of different loss functions on Bayesian estimation and its posterior risk is studied in this paper. The definition of quasi-modified squared error loss function is proposed based on the modified squared error loss function, and the formulas of Bayesian estimation and its posterior risk under the quasi-modified squared error loss function are developed, respectively. Moreover, the Bayesian estimations and its posterior risks of binomial distribution parameter under different loss functions are introduced. Monte Carlo simulation and application examples are provided for illustrative purpose, and the results are compared on the basis of posterior risk and mean square error (MSE). Finally, the Bayesian estimations and the Bayesian credible intervals are obtained, respectively, by MCMC method (also using OpenBUGS). [ABSTRACT FROM AUTHOR]

Recently, El-Sherpieny et al., (2020), suggested Type-II hybrid censoring method for parametric estimation of Lomax distribution (LD) without due regard being given to the choice of priors and posterior risk associated with the model. This paper fills this gap and derived the new LD model with minimum posterior risk for the selection of priors. It derives a closed form expression for Bayes estimates and posterior risks using square error loss function (SELF), weighted loss function (WLF), quadratic loss function (QLF) and DeGroot loss function (DLF). Prior predictive approach is used to elicit the hyperparameters of mixture model. Analysis of Bayes estimates and posterior risks is presented in terms of sample size n, mixing proportion p and censoring rate t0, with the help of simulation study. Usefulness of the model is demonstrated on applying it to simulated and real-life data which show promising results in terms of better estimation and risk reduction.

We perform a Bayesian analysis of the upper truncated Zeghdoudi distribution based on type II censored data. Using various loss functions including the generalized quadratic, entropy and Linex functions, we obtain Bayes estimators and the corresponding posterior risks. As tractable analytical forms of these estimators are out of reach, we propose Markov chain Monte-Carlo (MCMC) based simulation approach to study their performance. Moreover, given initial values for the parameters of the model, we obtain maximum likelihood estimators. Furthermore, we compare their performance with those of the Bayesian estimators using Pitman’s closeness criterion and integrated mean square error. Finally, we illustrate our approach through an example with real data. [ABSTRACT FROM AUTHOR]

In this paper, mathematical properties of Length biased Exponential distribution via Bayesian approach are derived under various loss functions. These properties include Bayes estimators and posterior risks for the simulation study. The comparison was made based on the performance of the Bayes estimate for the parameter under different loss functions with respect to the posterior risk. Also, obtained the reliability characteristics of this distribution. [ABSTRACT FROM AUTHOR]

This study discusses the Bayesian and maximum likelihood estimation methods for analyzing the data from a 3-component mixture of Burr Type-XII probability distributions. The maximum likelihood estimators with their variances cannot be obtained in an explicit form and thus an iterative procedure is used to calculate them numerically. Contrary to this, elegant closed form algebraic expressions of Bayes estimators and their posterior risks are derived. Using the informative and noninformative priors, the posterior predictive distributions along with predictive intervals are also discussed. A method of eliciting hyperparameters using prior predictive distribution is also a part of this study. Some interesting properties (including, posterior risks and Bayesian predictive intervals) of Bayes estimators and their behavior across different sample sizes, left and right test termination times, informative prior versus Jeffreys noninformative prior, are provided via a detailed Monte Carlo simulation study. To assess the suitability and application of the proposed model, a real life data example is also discussed in this article. Based on the simulated results and real data application, it is concluded that the IP paired with DLF (SELF) is a more suitable for estimating mixing component. [ABSTRACT FROM AUTHOR]

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As compared to simple models, the mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII distributions are discussed. These mathematical properties include Bayes estimators and posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.

This study considers the Bayesian inference for the mixture of two components of half-normal distribution using non-informative and informative prior. Several of its structural properties were derived, including explicit expression for mean, median, mode, reliability and hazard rate functions. Due to cost and time constraints, in most lifetime testing experiments censoring is an obligatory feature of lifetime datasets. We investigated Bayesian estimation of the parameters using various loss functions. The prior belief of the mixture model is represented by the uniform and square-root inverted gamma priors. Some properties of the model with graphs of the mixture density and hazard function are also discussed. The efficiencies of the proposed set of estimates of the mixture model parameters were studied through simulation and a real life dataset. Posterior risks of the Bayes estimators are evaluated and compared to explore the effect of prior belief and loss functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the length biased form of the Nakagami distribution known as length biased Nakagami distribution (LBND). Some properties of the model were studied such as moments, reliability function, and the hazard rate function. Maximum likelihood and Bayes estimators of the scale parameter are derived. Also, the Posterior risk under different loss functions is obtained. A real life data set is used and the results are compared through R-software. [ABSTRACT FROM AUTHOR]

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- andm-cycle. A simulation study and real data are applied to show the proposed results. [ABSTRACT FROM AUTHOR]

In recent years, the finite mixtures of distributions have been proved to be of considerable attention in terms of their practical applications. This paper focuses on studying the problem of estimating the parameters of a 3-component mixture of exponential, Rayleigh and Burr Type-XII distributions using Type-I right censoring scheme in Bayesian framework. The expressions for the Bayes estimators and their variances using the non-informative and the informative priors are derived for censored sample as well as for complete sample. The hyperparameters are elicited using prior predictive distribution. The posterior predictive distribution with different priors is derived and the equations necessary to find the lower and upper limits of the Bayesian predictive intervals are constructed. A detailed simulation study is carried out to investigate the performance (in terms of variances) of the Bayes estimators. Finally, the model is illustrated using the real life data. Bayes estimators using the informative prior have been observed performing superior. [ABSTRACT FROM AUTHOR]

In a Rayleigh distribution, We are interested in the estimation of the parameter and some reliability characteristics, as the reliability and the failure rate functions. We used the Bayesian approach under different loss function (squared loss and Linex loss) with a type II censored data. The prior law of the parameter is non-informative prior then a natural conjugated prior. The estimators of σ, S(t) and h(t) are obtained with the exact analytic expression, the posterior risks are calculated in each case. A simulation study was carried out as well as real data analysis. A comparison between the different estimators from there posterior risks leads us to conclude that the best estimator is obtained under the Linex loss function. [ABSTRACT FROM AUTHOR]

Since the last few decades, constructing flexible parametric classes of probability distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3- component mixture of Rayleigh distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and posterior risks assuming the non-informative (uniform and Jeffreys’) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data.

This paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making. [ABSTRACT FROM PUBLISHER]

This study deals with Bayesian analysis of shifted Gompertz mixture model under type-I censored samples assuming both informative and noninformative priors. We have discussed the Bayesian estimation of parameters of shifted Gompertz mixture model under the uniform, and gamma priors assuming three loss functions. Further, some properties of the model with some graphs of the mixture density are discussed. These properties include Bayes estimators, posterior risks and reliability function under simulation scheme. Bayes estimates are obtained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. We analyzed some simulated sets in order to investigate the effect of prior belief, loss functions, and performance of the proposed set of estimators of the mixture model parameters. [ABSTRACT FROM AUTHOR]

As compared to simple models, the mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII distributions are discussed. These mathematical properties include Bayes estimators and posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data. [ABSTRACT FROM AUTHOR]

This study is concerned with the problem of estimating the parameters of a 3-component mixture of Burr distributions using type-I right censored data. The closedform expressions for the Bayes estimators and their posterior risks assuming the noninformative (uniform and Jeffreys') priors under squared-error loss function, precautionary loss function, and DeGroot loss function are derived. Performance of the Bayes estimators for different sample sizes, test termination times (a point of time after which all other tests are terminated), and parametric values under different loss functions is investigated. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and posterior risks are derived. Simulated data sets are designed for the comparisons and the model is finally illustrated using the real data. [ABSTRACT FROM AUTHOR]

This article considers generalized maximum likelihood and Bayesian estimations of reliability parameters under some balanced loss functions when the data are progressively Type II censored from a compound Rayleigh distribution. This is done with respect to a conjugate prior on scale parameter and a discrete prior on shape parameter of the distribution. Posterior risks of generalized maximum likelihood and Bayes estimates are also obtained under balanced loss functions. A real life application to the survival times of patients is also described for the developed estimation methods. A Monte Carlo simulation study has been carried out to examine and compare the performance of estimates on the basis of posterior risks. The simulation study indicates that Bayesian estimation should be preferred over generalized maximum likelihood estimation for estimation of the said parameters. This study will be very useful to researchers, practitioners, and statisticians where such type of life test is needed and especially where a compound Rayleigh model is used. [ABSTRACT FROM AUTHOR]

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose. [ABSTRACT FROM AUTHOR]

An engineering process is an output from a set of combined processes which may be homogeneous or heterogeneous. To study the lifetime of such processes, we need a model which can accommodate the nature of such processes. Single probability models are not capable of capturing the heterogeneity of nature. However, mixture models of some suitable lifetime distributions, have the potential to highlight such interesting feature. Due to time and cost constraint, in the most lifetime testing experiments, censoring is an unavoidable feature of most lifetime data sets. This article deals with the modeling of the heterogeneity existing in the lifetime processes using the mixture of the inverse Rayleigh distribution, and the spotlight is the Bayesian inference of the mixture model using non-informative (the Jeffreys and the uniform) and informative (gamma) priors. We are considering this particular distribution due to two reasons; the first one is due to its skewed behavior, i.e. in engineering processes, an engineer suspects that high failure rate in the beginning, but after continuous inspection, the failure goes down and the second reason is due to its vast application in many applied fields. A Gibbs sampling algorithm based on adaptive rejection sampling is designed for the posterior computation. A detailed simulation study is carried out to investigate the performance of the estimators based on different prior distributions. The posterior risks are evaluated under the squared error, the weighted, the quadratic, the entropy, the modified squared error and the precautionary loss functions. Posterior risks of the Bayes estimates are compared to explore the effect of prior information and loss functions. [ABSTRACT FROM AUTHOR]

This paper seeks to focus on the Bayesian estimation of scale parameter for Frechet distribution with known shape. Bayes estimators have been obtained for scale parameter for Frechet when sample is available from complete, Time (type I) and Failure (type II) censoring scheme. The results of different censoring schemes have been compared with those under complete samples. Bayes estimators have been developed under squared error loss function as well as under precautionary loss function using non-informative (uniform and Jeffreys) priors for the parameter. The performance of Bayes estimators has been compared under a Monte Carlo simulation study. [ABSTRACT FROM AUTHOR]

The development of flexible parametric classes of probability models in Bayesian analysis is a very popular approach. This study is designed for heterogeneous population for a two-component mixture of the Laplace probability distribution. When a process initially starts, the researcher expects that the failure components will be very high but after some improvement/inspection it is assumed that the failure components will decrease sufficiently. That is why in such situation the Laplace model is more suitable as compared to the normal distribution due to its fatter tails behaviour. We considered the derivation of the posterior distribution for censored data assuming different conjugate informative priors. Various kinds of loss functions are used to derive these Bayes estimators and their posterior risks. A method of elicitation of hyperparameter is discussed based on a prior predictive approach. The results are also compared with the non-informative priors. To examine the performance of these estimators we have evaluated their properties for different sample sizes, censoring rates and proportions of the component of the mixture through the simulation study. To highlight the practical significance we have included an illustrative application example based on real-life mixture data. [ABSTRACT FROM PUBLISHER]

In this paper, we discuss optimum sample size determination for a bounded linex loss function (blinex). The linex loss function is often used when losses are asymmetric, but it has the disadvantage that it can only be used if the moment-generating function of the posterior distribution exists. Blinex loss has the advantage that it can always be calculated. Also many authors have argued that a bounded loss function is to be preferred. We have obtained the optimum sample size for a number of distributions when the cost of sampling is linear. The form of the posterior risk function does not allow an analytical solution so simulation is necessary. [ABSTRACT FROM PUBLISHER]

Process capability (PC) indices measure the ability of a process of interest to meet the desired specifications under certain restrictions. There are a variety of capability indices available in literature for different interest variables such as weights, lengths, thickness, and the life time of items among many others. The goal of this article is to study the generalized capability indices from the Bayesian view point under different symmetric and asymmetric loss functions for the simple and mixture of generalized lifetime models. For our study purposes, we have covered a simple and two component mixture of Maxwell distribution as a special case of the generalized class of models. A comparative discussion of the PC with the mixture models under Laplace and inverse Rayleigh are also included. Bayesian point estimation of maintenance performance of the system is also part of the study (considering the Maxwell failure lifetime model and the repair time model). A real-life example is also included to illustrate the procedural details of the proposed method. [ABSTRACT FROM AUTHOR]

Berliner (1987) discussed the issue of controlling the output (response) towards the specified value by choosing the values for independent variables in a regression mixture model, taking it as a Bayesian Decision Problem. The quantification of the potential loss was done with the help of quadratic loss function, which was a symmetric loss function. We have tried to quantify this loss with the help of Precautionary Loss Function and Modified Squared Error Loss Function, in linear Scheffé (1958) mixture model and comparison is established between these loss function. Results are improved as compared to Berliner (1987). [ABSTRACT FROM AUTHOR]

In most cases, we use a symmetric loss such as the quadratic loss in a usual estimation problem. But, in the non-regular case when the regularity conditions do not necessarily hold, it seems to be more reasonable to choose an asymmetric loss than the symmetric one. In this paper, we consider the Bayes estimation under the linear exponential (LINEX) loss which is regarded as a typical example of asymmetric loss. We also compare the Bayes risks of estimators under the LINEX loss for a family of truncated distributions and a location parameter family of truncated distributions. [ABSTRACT FROM AUTHOR]

A large number of models have been derived from the two-parameter Weibull distribution including the inverse Weibull (IW) model which is found suitable for modeling the complex failure data set. In this paper, we present the Bayesian inference for the mixture of two IW models. For this purpose, the Bayes estimates of the parameters of the mixture model along with their posterior risks using informative as well as the non-informative prior are obtained. These estimates have been attained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the former case, Bayes estimates are obtained under three loss functions while for the latter case only the squared error loss function is used. Simulation study is carried out in order to explore numerical aspects of the proposed Bayes estimators. A real-life data set is also presented for both cases, and parameters obtained under case when shape parameter is known are tested through testing of hypothesis procedure. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, mathematical properties of Lindley distribution via Bayesian approach are derived under different loss functions. These properties include: Bayes Estimators, posterior risks and failure rate function for simulation scheme. Elicitation of hyperparameters is also discussed. A real life application to waiting time data at the bank is also described for the developed procedures (also using WinBUGS). Results are compared on the basis of posterior risk. [Copyright &y& Elsevier]

In this paper, the class of life-time distributions is considered for Bayesian analysis. The properties of Bayes estimators of the parameter are studied under different loss functions via simulated and real life data. For the particular case of the Maxwell distribution, a comprehensive simulation scheme is conducted using non-informative and informative priors. The loss functions are compared through posterior risk. Attractive closed form expressions are derived for the complete sample. Some other interesting comparison like credible interval (CI) and Highest Posterior Density (HPD) intervals are made and properties of the estimates are observed and presented in a very smooth way. [ABSTRACT FROM AUTHOR]

Constructing a flexible parametric classes of probability distributions appeared as a quite popular approach in Bayesian analysis for the last few decades. This study is planned in the same direction for two component Mixture of Laplace probability distribution considering heterogeneous population. We have considered censored and complete sample environments and derived the closed form expressions for the Bayes estimators and their posterior risks. In addition we have worked out complete sample expressions for the Maximum Likelihood (ML) estimates along with their Posterior Risk and constructed components of the information matrix. To examine the performance of these estimators we have evaluated their properties for different sample sizes, censoring rates, proportions of the component of mixture and a variety of loss functions. To highlight the practical significance we have included an illustrative application example based on a real-life mixture data. [ABSTRACT FROM AUTHOR]

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form $${L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}$$, as well as the weighted version $${q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}$$, where ρ( θ, δ) is an arbitrary loss function, δ is a chosen a priori 'target' estimator of $${\theta, \omega \in[0,1)}$$, and q(·) is a positive weight function. we develop Bayesian estimators under $${L_{\rho, \omega, \delta_0}}$$ with ω > 0 by relating such estimators to Bayesian solutions under $${L_{\rho, \omega, \delta_0}}$$ with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses. [ABSTRACT FROM AUTHOR]

Choice-based conjoint analysis is a popular marketing research technique to learn about consumers' preferences and to make market share forecasts under various scenarios for product offerings. Managers expect these forecasts to be "realistic" in terms of being able to replicate market shares at some prespecified or "basecase" scenario. Frequently, there is a discrepancy between the recovered and base-case market share. This paper presents a Bayesian decision theoretic approach to incorporating base-case market shares into conjoint analysis via the loss function. Because defining the base-case scenario typically involves a variety of management decisions, we treat the market shares as constraints on what are acceptable answers, as opposed to informative prior information. Our approach seeks to minimize the adjustment of parameters by using additive factors from a normal distribution centered at 0, with a variance as small as possible, but such that the market share constraints are satisfied. We specify an appropriate loss function, and all estimates are formally derived via minimizing the posterior expected loss. We detail algorithms that provide posterior distributions of constrained and unconstrained parameters and quantities of interest. The methods are demonstrated using discrete choice models with simulated data and data from a commercial market research study. These studies indicate that the method recovers base-case market shares without systematically distorting the preference structure from the conjoint experiment. [ABSTRACT FROM AUTHOR]