Confidence Intervals for the Ratio of the Coefficients of Variation of Inverse-Gamma Distributions

Herein, we present four methods for constructing confidence intervals for the ratio of the coefficients of variation of inverse-gamma distributions using the percentile bootstrap, fiducial quantities, and Bayesian methods based on the Jeffreys and uniform priors. We compared their performances using coverage probabilities and expected lengths via simulation studies. The results show that the confidence intervals constructed with the Bayesian method based on the uniform prior and fiducial quantities performed better than those constructed with the Bayesian method based on the Jeffreys prior and the percentile bootstrap. Rainfall data from Thailand was used to illustrate the efficacies of the proposed methods.

Objective bayesian analysis for multiple repairable systems

This article focus on the analysis of the reliability of multiple identical systems that can have multiple failures over time. A repairable system is defined as a system that can be restored to operating state in the event of a failure. This work under minimal repair, it is assumed that the failure has a power law intensity and the Bayesian approach is used to estimate the unknown parameters. The Bayesian estimators are obtained using two objective priors know as Jeffreys and reference priors. We proved that obtained reference prior is also a matching prior for both parameters, i.e., the credibility intervals have accurate frequentist coverage, while the Jeffreys prior returns unbiased estimates for the parameters. To illustrate the applicability of our Bayesian estimators, a new data set related to the failures of Brazilian sugar cane harvesters is considered.

Bayesian Analysis of Cancer Data Using a 4-Component Exponential Mixture Model

Cancer is among the major public health problems as well as a burden for Pakistan. About 148,000 new patients are diagnosed with cancer each year, and almost 100,000 patients die due to this fatal disease. Lung, breast, liver, cervical, blood/bone marrow, and oral cancers are the most common cancers in Pakistan. Perhaps smoking, physical inactivity, infections, exposure to toxins, and unhealthy diet are the main factors responsible for the spread of cancer. We preferred a novel four-component mixture model under Bayesian estimation to estimate the average number of incidences and death of both genders in different age groups. For this purpose, we considered 28 different kinds of cancers diagnosed in recent years. Data of registered patients all over Pakistan in the year 2012 were taken from GLOBOCAN. All the patients were divided into 4 age groups and also split based on genders to be applied to the proposed mixture model. Bayesian analysis is performed on the data using a four-component exponential mixture model. Estimators for mixture model parameters are derived under Bayesian procedures using three different priors and two loss functions. Simulation study and graphical representation for the estimates are also presented. It is noted from analysis of real data that the Bayes estimates under LINEX loss assuming Jeffreys’ prior is more efficient for the no. of incidences in male and female. As far as no. of deaths are concerned again, LINEX loss assuming Jeffreys’ prior gives better results for the male population, but for the female population, the best loss function is SELF assuming Jeffreys’ prior.

Bayesian using Importance Sampling Technique of Weibull Regression with Type II Censored Data

Keeping in view the Bayesian approach, the study aims to develop methods through the utilization of Jeffreys prior and modified Jeffreys prior to the covariate obtained by using the Importance sampling technique. For maximum likelihood estimator, covariate parameters, and the shape parameter of Weibull regression distribution with the censored data of Type II will be estimated by the study. It is shown that the obtained estimators in closed forms are not available, but through the usage of appropriate numerical methods, they can be solved. The mean square error is the criterion of comparison. With the use of simulation, performances of these three estimates are assessed, bearing in mind different censored percentages, and various sizes of the sample.

Bayesian Reference Analysis for the Generalized Normal Linear Regression Model

This article proposes the use of the Bayesian reference analysis to estimate the parameters of the generalized normal linear regression model. It is shown that the reference prior led to a proper posterior distribution, while the Jeffreys prior returned an improper one. The inferential purposes were obtained via Markov Chain Monte Carlo (MCMC). Furthermore, diagnostic techniques based on the Kullback–Leibler divergence were used. The proposed method was illustrated using artificial data and real data on the height and diameter of Eucalyptus clones from Brazil.

Evaluating the performance of degrees of freedom estimation in asymmetric GARCH models with t-student innovations

This work investigates the effects of using the independent Jeffreys prior for the degrees of freedom parameter of a t-student model in the asymmetric generalised autoregressive conditional heteroskedasticity (GARCH) model. To capture asymmetry in the reaction to past shocks, smooth transition models are assumed for the variance. We adopt the fully Bayesian approach for inference, prediction and model selection We discuss problems related to the estimation of degrees of freedom in the Student-t model and propose a solution based on independent Jeffreys priors which correct problems in the likelihood function. A simulated study is presented to investigate how the estimation of model parameters in the t-student GARCH model are affected by small sample sizes, prior distributions and misspecification regarding the sampling distribution. An application to the Dow Jones stock market data illustrates the usefulness of the asymmetric GARCH model with t-student errors.

Population size estimation based upon zero-truncated, one-inflated and sparse count data

Estimating the size of a hard-to-count population is a challenging matter. In particular, when only few observations of the population to be estimated are available. The matter gets even more complex when one-inflation occurs. This situation is illustrated with the help of two examples: the size of a dice snake population in Graz (Austria) and the number of flare stars in the Pleiades. The paper discusses how one-inflation can be easily handled in likelihood approaches and also discusses how variances and confidence intervals can be obtained by means of a semi-parametric bootstrap. A Bayesian approach is mentioned as well and all approaches result in similar estimates of the hidden size of the population. Finally, a simulation study is provided which shows that the unconditional likelihood approach as well as the Bayesian approach using Jeffreys’ prior perform favorable.

The Bayesian Inference of Pareto Models Based on Information Geometry

Bayesian methods have been rapidly developed due to the important role of explicable causality in practical problems. We develope geometric approaches to Bayesian inference of Pareto models, and give an application to the analysis of sea clutter. For Pareto two-parameter model, we show the non-existence of α-parallel prior in general, hence we adopt Jeffreys prior to deal with the Bayesian inference. Considering geodesic distance as the loss function, an estimation in the sense of minimal mean geodesic distance is obtained. Meanwhile, by involving Al-Bayyati’s loss function we gain a new class of Bayesian estimations. In the simulation, for sea clutter, we adopt Pareto model to acquire various types of parameter estimations and the posterior prediction results. Simulation results show the advantages of the Bayesian estimations proposed and the posterior prediction.

Bayesian Approach in Estimation of Shape Parameter of an Exponential Inverse Exponential Distribution

Aims: This study aimed to obtain the shape parameter of an Exponential Inverted Exponential distribution using different prior distributions under different loss functions. Methodology: The Bayes’ theorem was adopted to obtain the posterior distribution of the shape parameter of an Exponential inverted Exponential distribution for both non-information prior (such as Jeffreys prior, Hartigen prior and Uniform prior) and an informative prior (such as Gamma distribution and chi-square distribution). Different loss functions (such as Entropy loss function, Square error loss function, Al-Bayyati’s loss function and Precautionary loss function) were employed to obtain the estimate parameter of the shape parameter with an assumption that the scale parameter is known. Results: The posterior distribution of the shape parameter of an Exponential Inverted Exponential distribution follows a Gamma distribution for all the prior distribution in the study. Also the Bayes estimate for the simulated datasets and real life dataset were obtained. Conclusion: The Bayes’ estimates for different prior distribution under different loss functions are close to the true parameter value of the shape parameter. The estimators are then compared in terms of their Mean Square Error (MSE) which is computed using R programming language. We deduce that the MSE reduces as the sample size (n) increases.