E-Bayesian estimation for the proportional hazard rate model based on record values

In this article, a new parameter estimation method, named E-Bayesian method, is considered to obtain the estimates of the unknown parameter and reliability function based on record values. The maximum likelihood, Bayesian, E-Bayesian, and hierarchical Bayesian estimates of the unknown parameter and reliability function are obtained when the underlying distribution belongs to the proportional hazard rate model. The Bayesian estimates are obtained based on squared error and linear-exponential loss functions. The previously obtained some relations for the E-Bayesian estimates are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations are obtained under the same loss functions. The comparison of the derived estimates are carried out by using Monte Carlo simulations. Real data are analyzed for an illustration of the findings.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.     

Bayesian estimation under a mixture of the Burr type XII distribution and its reciprocal

The Bayesian estimation for the parameters of the finite mixture of the Burr type XII distribution with its reciprocal are obtained based on the type-I censored data. The Bayes estimators are computed based on squared error and Linex loss functions and using the idea of Markov chain Monte Carlo algorithm. Based on the Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum-likelihood estimators.     

The Kumaraswamy Burr XII distribution: theory and practice

For the first time, a five-parameter distribution, called the Kumaraswamy Burr XII (KwBXII) distribution, is defined and studied. The new distribution contains as special models some well-known distributions discussed in lifetime literature, such as the logistic, Weibull and Burr XII distributions, among several others. We obtain the complete moments, incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and reliability of the KwBXII distribution. We provide two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. For different parameter settings and sample sizes, various simulation studies are performed and compared to the performance of the KwBXII distribution. Three applications to real data sets demonstrate the usefulness of the proposed distribution and that it may attract wider applications in lifetime data analysis.     

Recurrence relations for moments of progressive type-II right censored order statstics from burr distribution

In this paper some recurrence relations between moments of progressive Type-II right censored order statistics from doubly truncated Burr distribution are established. These recurrence relations would enable one to obtain all the single and product moments of Burr progressive Type-II right censored order statistics in a simple recursive manner.     

Prediction bounds for the burr model

Given a type 2 censored sample from the Burr life time distribution, Bayesian prediction bounds are derived for future observations. An approximate Bayesian method has been used to simplify the computation of the prediction bounds. Numerical examples are used to illustrate the procedures.     

Comparisons of the percentage points of distributions with the same first four moments,chosen from eight different systems of frequency curves

Probability integrals and percentage points of univariate distributions from up to eight different families, having common first four moments are compared. Among interesting observations is the remarkable consistency in the standardized upper and lower 5% points over considerable regions of the √β₁, β₂ plane; also the closeness of agreement between the log-normal and non-central t distributions and the Pearson Type VI and Type IV curves respectively.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Statistical Comparison of the Goodness of Fit Delivered by Five Families of Distributions Used in Distribution Fitting

A basic assumption in distribution fitting is that a single family of distributions may deliver useful representation to the universe of available distributions. To date, little study has been conducted to compare the relative effectiveness of these families. In this article, five families are compared by fitting them to a sample of 20 distributions, using 2 fitting objectives: minimization of the L ₂ norm and four-moment matching. Values of L ₂ norm associated with the fitted families are used as input data to test for significant differences. The Pearson family and the RMM (Response Modeling Methodology) family significantly outperforms all other families.