Bayesian Premium Estimators for NXLindley Model Under Different Loss Functions

Most of the data arising in reliability can be modelled by Pareto distribution. In the present paper, we proposed the Bayes estimation of dynamic cumulative residual entropy for the classical Pareto distribution under upper record values. This measure performs important roles in reliability and survival analysis to model and analyse the data. A class of informative and non-informative priors has been assumed to derive the corresponding posterior distributions. The Bayes estimators (BEs) and the associated posterior risks have been calculated under different symmetric and asymmetric loss functions. We demonstrate the use of the proposed Bayesian estimation procedure with the average July temperatures data of Neuenburg, Switzerland, during the period 1864–1993. The performance of the BEs has been evaluated and compared under a comprehensive simulation study. The purpose is to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Comparisons of estimates between the Bayes and frequentist methods are fascinating and challenging theme of interest in statistics with significant impact on professionals. For Bayes estimators, the performance depends on the form of the prior distribution and the assumed loss function. This paper illu strates the problem of estimation of the one-parameter generalized geometric series distribution, using conjugate and improper prior distributions under symmetric and asymmetric loss functions. Performance of the Bayes estimates with respect to different priors, loss functions, and maximum likelihood estimates are illustrated for a data set and through a simulation study.

Burr distribution is one of the most important types of distribution in Burr system and has gained special attention. It has an important role in various disciplines, such as reliability analysis, life testing, survival analysis, actuarial science, economics, forestry, hydrology and meteorology. Thus, the parameter estimation for Burr distribution becomes an important thing to do. The frequentist approach using the maximum likelihood method is the most commonly used way to estimate the parameters of a distribution. In this paper we considered using the Bayesian method to estimate the shape parameter k of Burr distribution using gamma prior which is a conjugate prior. The Bayes estimate for the shape parameter k is obtained under the squared-error loss function (SELF) which is one of the symmetric loss function and the precautionary loss function (PLF) which is one of the asymmetric loss function. Through a simulation study, the comparison was made on the performance of the Bayes estimate for the shape parameter k under these two loss functions with respect to the mean-squared error (MSE) and the posterior risk.

In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.

In this article we consider statistical inferences about the unknown parameters of the inverse Weibull distribution based on progressively first-failure censoring using Bayesian procedures. The Bayes estimators are obtained based on both the symmetric and asymmetric (Linex, General Entropy and Precautionary) loss functions. There are no explicit forms for the Bayes estimators; therefore, we propose the Lindley’s approximation method to compute the Bayes estimators. A comparison between these estimators and the maximum likelihood estimator (MLE) is provided by using extensive simulation and two criteria, namely, the bias and the mean squared error. It is concluded that the approximate Bayes estimators outperform the MLEs most of the time. Real life data example is provided to illustrate our proposed estimators.

Statistical inference based on lower record values for the inverse Weibull distribution under the modified loss function

This paper examines asymmetry of the loss function of professional forecasters for output growth, inflation, and exchange rate forecasts, based on the survey of professional forecasts (SPF) data of three South Asian countries (India, Indonesia, and Singapore). Our results provide India’s unbiased output growth forecasts and under-predicted inflation and exchange rate forecasts; Indonesia’s broadly unbiased forecasts; and Singapore’s under-predicted output growth forecasts and unbiased inflation and exchange rate forecasts. Testing the rationality of all three countries’ SPF forecasts, we find that all are rational under an asymmetric loss function but not under a symmetric loss function.

In this study, the estimation of parameters of a three‐parameter flexible reduced logarithmic‐inverse Lomax (FRL‐IL) distribution based on progressive type‐II right censored sample is studied. These methods include maximum likelihood estimations (MLEs) and Bayesian estimators. Approximate confidence intervals (ACIs) for the reliability and hazard functions are estimated based on the asymptotic distribution of maximum likelihood estimates (MLEs). In addition, two bootstrap CIs are also proposed. Bayesian estimates are obtained for symmetric and asymmetric loss functions such as squared error loss (SEL) and linear‐exponential (LINEX) loss functions. The Gibbs within Metropolis‐Hasting sampler procedure is applied using the Markov Chain Monte Carlo (MCMC) technique to get the Bayes estimates of the unknown parameters and their credible intervals (CRIs). Finally, a real‐life dataset that represents a group of patients with bladder cancer is considered an application of the proposed methods.

ABSTRACTThis article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.

In this study, we use the idea of the hierarchical model (HM) to estimate an unknown parameter of the hierarchical Poisson-Gamma model using the E-Bayesian (E-B) theory. We propose the idea of hierarchical probability function instead of the traditional hierarchical prior density function. We aim to infer E-B estimates with respect to the conjugate Gamma prior distribution along with the E-posterior risks on the basis of different symmetric and asymmetric loss functions (LFs) under restricted and unrestricted parameter spaces using uniform hyperprior. Whereas, E-B estimators are compared with maximum likelihood estimators (MLEs) using mean squared error (MSE). Monte Carlo simulations are prosecuted to study the efficiency of E-B estimators empirically. It is shown that the LFs under a restricted parameter space dominate to estimate the parameter of the hierarchical Poisson-Gamma model. It is also found that the E-B estimators are more precise than MLEs, and Stein’s LF has the least E-PR. Moreover, the application of outcomes to a real-life example has been made for analysis, comparison, and motivation.

The Bayesian approach offers a flexible, interpretable and powerful framework for statistical analysis, making it a valuable tool to help in making optimal decisions under uncertainty. It incorporates prior knowledge or beliefs about the parameters, which can lead to more accurate and informative results. Also, it offers credible intervals as a measure of uncertainty, which are often more interpretable than confidence intervals. Hence, the Bayesian approach is utilized to estimate the parameters, reliability function, hazard rate function and reversed hazard rate function of a new competing risks model. A squared error loss function as a symmetric loss function and a linear exponential loss function as an asymmetric loss function are employed to derive the Bayesian estimators. Credible intervals of the parameters, reliability function, hazard rate function and reversed hazard rate function are obtained. Predicting future observations is important in many fields, from finance and weather forecasting to healthcare and engineering. Thus, two-sample prediction (as a special case of the multi-sample prediction) for future observation is considered. An adaptive Metropolis algorithm is applied to conduct a simulation study to evaluate the performance of the Bayes estimates and predictors. Moreover, two applications of medical and engineering data sets are used to test and validate the theoretical results, ensuring that they are accurate, applicable to real-world scenarios and contribute to the understanding of the world and inform decision-making.

ABSTRACTOne of the most important topics in manufacturing industries is the evaluation of performance lifetimes of products. Based on a given lifetime performance index, this paper deals with evaluating the performance of a process subject to a given lower specification limit. We confine ourselves to the progressively first-failure-censored data coming from a common Pareto distribution. With both the Bayesian and the non-Bayesian approaches being investigated here, we pay more attention to Bayesian estimators under balanced type loss functions. The results are presented under the balanced versions of two well-known loss functions, namely the squared error loss and the Varian's linear-exponential (LINEX) loss. Moreover, based on the Bayesian and the non-Bayesian approaches, the problem of testing hypotheses on the lifetime performance index is studied. Also, a simulation study is performed to assess the obtained results. Finally, two illustrative examples are given.

Estimators and predictors that are optimal relative to Varian's asymmetric LINEX loss function are derived for a number of well-known models. Their risk functions and Bayes risks are derived and compared with those of usual estimators and predictors. It is shown that some usual estimators, for example, a scalar sample mean or a scalar least squares regression coefficient estimator, are inadmissible relative to asymmetric LINEX loss by providing alternative estimators that dominate them uniformly in terms of risk.

Taguchi's philosophy of quality improvement is quite unique and challenging. Traditionally, quality management focused on conformance to specifications within a given range. In Taguchi's philosophy, the aim of quality management should be achieving a specific target value. Any deviation from the predetermined target value results in a loss (e.g., due to customer dissatisfaction). Taguchi measures this loss with a quadratic function called quality loss function (QLF). An important component of QLF is the proportionality constant ( ). An inaccurate or erroneous value for can result in misallocation of the organizational resources and can easily lead to chaos in quality management efforts. Thus, an accurate estimate for is of utmost importance. This paper proposes a new methodology for calculating the proportionality constant . To this end, the analytical hierarchy process is used to estimate the value of the proportionality constant in QLF. It is argued that the proposed methodology takes into account the relative importance of various quality characteristics and captures different levels of sensitivity along the loss function. Both symmetric and asymmetric loss functions are considered.