Burr Type Research Articles

Size-biased distributions in the generalized beta distribution family, with applications to forestry

Size-biased distributions arise in many forestry applications, as well as other environmental, econometric, and biomedical sampling problems. We examine the size-biased versions of the generalized beta of the first kind, generalized beta of the second kind and generalized gamma distributions. These distributions include, as special cases, the Dagum (Burr Type III), Singh-Maddala (Burr Type XII) and Fisk distributions as well as better-studied distributions such as the Weibull, lognormal, beta (of the first and second kind), gamma and exponential. Our results indicate that specification and estimation of the size-biased forms of these distributions can be viewed within a unified framework. This should facilitate broader application of size-biased distributions in forestry sampling, modeling and analysis.

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

Bayesian analysis of randomly censored Burr type XII distribution under different loss functions

The paper deals with Bayesian estimation of unknown parameters of Burr type XII distribution under the Koziol-Green model of random censorship assuming both the informative and noninformative priors. We use different symmetric and asymmetric loss functions to obtain the Bayes estimates. It is seen that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Gibbs sampling scheme to obtain the approximate Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed estimators and also to compare with the maximum likelihood estimators. Based on the simulation study, we propose a set of estimators of the model parameters. One real data analysis is performed and it is seen that the proposed set of estimators fit the data best than the rest.

BILEVEL PROGRAMMING BASED TIME CENSORED STEP-STRESS PALTSP USING TAMPERED FAILURE RATE MODEL AND WARRANTY

This paper deals with optimal design of time-censored step-stress partially accelerated life test sampling plan (PALTSP) using Burr type-XII life distribution. The Burr type XII distribution has been found appropriate for modeling failures that occur with less frequency and also when there is high occurrence of early failures. This distribution has been found appropriate for accelerated life testing experiments. The optimum sampling plan obtained using bilevel programming approach consists in finding optimum sample size and optimum stress change point by minimizing expected total cost per lot comprising warranty costs with respect to acceptance or rejection of the lot, sampling cost and testing cost such that the producer's and consumer's interested are safeguarded. The methods developed has been illustrated using an example and sensitivity analyses carried out.

Estimation of 2- and 3-parameter BURR type XII distribution using EM algorithm

The Burr Type XII distribution is one of the systems of continuous distributions and is widely known because the distribution includes the characteristics of various well known distributions such as Weibull and gamma distributions. Maximum likelihood estimation (MLE) has been a common method in estimating model parameters. In this paper, we present an alternative method that is expectation-maximization (EM) algorithm to estimate the two- and three- parameter Burr Type XII distributions in the presence of complete and censored data. Furthermore, simulation study is conducted to compare the efficiency and accuracy of MLE and EM algorithm. We discover that EM estimation is more efficient and accurate than those estimates obtained via MLE approach.________________________________________GRAPHICAL ABSTRACT

Optimum Time-Censored Constant-Stress PALTSP for the Burr Type XII Distribution Using Tampered Failure Rate Model

This paper presents optimum design of time-censored constant-stress partially accelerated life test sampling plan (PALTSP) in which each item runs either at use or at accelerated conditions and product life follows Burr type XII. The optimal plan consists in finding out sample proportions allocated to both use and accelerated conditions by minimizing the asymptotic variance of test statistic for deciding on acceptance/rejection of the lot such that producer’s and consumer’s interests are safeguarded. The method developed has been illustrated using an example. Sensitivity analysis has also been carried out.

M-estimator for estimating the Burr type III parameters with outliers

The Burr type III distribution allows a wider region for the skewness and kurtosis plane, which covers several distributions including: the log-logistic, the Weibull and the Burr type XII distributions. However, outliers may occur in the data set. The robust regression method has been successfully used to diminish the effect of outliers on statistical inference. This paper presents an M-estimator based on the quantile function to estimate the Burr type III parameters for complete data with outliers. We showed that all regularity conditions are satisfied for the normal approximation of the Burr type III distribution using the M-estimator. Thus, the confidence intervals for all parameters can be obtained. The simulation results showed that the M-estimator generally outperforms the maximum likelihood and least squares methods regarding bias, root mean square errors and a number of quantile values. One real example and one simulated data example are used to demonstrate the performance of our proposed method.

Data Processing with Computation in Bayes Reliability Analysis for Burr Type X Distribution under Different Loss Functions

This paper studies the estimation of the parameter of Burr Type X distribution. Maximum likelihood estimator is first derived, and then the Bayes and Empirical Bayes estimators of the unknown parameter are obtained under three loss functions, which are squared error loss, LINEX loss and entropy loss functions. The prior distribution of parmeter used in this paper is Gamma distribution. Finally, a Monte Carlo simulation is given to illustrate the application of these estimators.

Estimation of R=P[Y less than X] for Burr type XII distribution based on ranked set sampling

Ranked set sampling (RSS) is a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random sample (SRS). In the current paper, the estimation of R=P[Y<X] when Y and X are two independent Burr type XII distribution with common known shape parameter c will be considered. Maximum likelihood estimator is proposed to estimate R based on ranked set sampling data. These estimators will be compared in terms of their biases, mean square errors and efficiencies with known estimators based on SRS data. It is shown that the estimators based on RSS are more efficient than the corresponding SRS. The results are illustrated using simulated data. Keywords : Burr Type XII, Stress Strength Model, Ranked Set Sampling, Method of Maximum Likelihood.

Robust Estimation for Parameters of the Extended Burr Type III Distribution

We consider various robust estimators for the extended Burr Type III (EBIII) distribution for complete data with outliers. The considered robust estimators are M-estimators, least absolute deviations, Theil, Siegel's repeated median, least trimmed squares, and least median of squares. Before we perform the aforementioned estimators for the EBIII, we adapt the quantiles method to the estimation of the shape parameter k of the EBIII. The simulation results show that the considered robust estimators generally outperform the existing estimation approaches for data with upper outliers, with certain of them retaining a relatively high degree of efficiency for small sample sizes.

PREFERENCE OF PRIOR FOR BAYESIAN ANALYSIS OF THE MIXED BURR TYPE X DISTRIBUTION UNDER TYPE I CENSORED SAMPLES

The paper is concerned with the preference of prior for the Bayesian analysis of the shape parameter of the mixture of Burr type X distribution using the censored data. We modeled the heterogeneous population using two components mixture of the Burr type X distribution. A comprehensive simulation scheme, through probabilistic mixing, has been followed to highlight the properties and behavior of the estimates in terms of sample size, corresponding risks and the proportion of the component of the mixture. The Bayes estimators of the parameters have been evaluated under the assumption of informative and non-informative priors using symmetric and asymmetric loss functions. The model selection criterion for the preference of the prior has been introduced. The hazard rate function of the mixture distribution has been discussed. The Bayes estimates under exponential prior and precautionary loss function exhibit the minimum posterior risks with some exceptions.

Estimation under Burr type X distribution based on doubly type II censored sample of dual generalized order statistics

Abstract In this paper, maximum likelihood and Bayes estimates of the parameters for Burr type X distribution based on doubly type II censored sample of dual generalized order statistics are obtained. Two different cases are considered. In the first case, the shape parameter is estimated when the scale parameter is known while in the second case the estimators for scale and shape parameters are obtained when both parameters are assumed to be unknown. For Bayesian estimation, Monte Carlo Integration is used to improve the approximation of resulting integrals. Simulation studies are conducted to demonstrate the efficiency of the proposed methods through two special cases.

On the Three-Parameter Burr Type XII Distribution and its Application to Heavy Tailed Lifetime Data

This paper identifies the characteristics of three-parameter Burr Type XII distribution and discusses its utility in survivorship applications. It addresses the problem of estimating the three-parameter Burr XII distribution and its doubly truncated version. The results are applied on a real dataset by fitting the distribution to the survival time of breast cancer patients in the Gaza Strip. These data are known to have a heavy tailed distribution since patients in this area received different protocols of treatments in different levels of hospitals locally and abroad. The findings indicated that the estimates of the parameters of the truncated distribution are more efficient than those obtained from the original distribution since the distribution is heavy tailed and involves many highly extreme observations.

Inference for constant-stress accelerated life test with Type-I progressively hybrid censored data from Burr-XII distribution

This paper proposes a simple constant-stress accelerated life test (ALT) model from Burr type XII distribution when the data are Type-I progressively hybrid censored. The maximum likelihood estimation (MLE) of the parameters is obtained through the numerical method for solving the likelihood equations. Approximate confidence interval (CI), based on normal approximation to the asymptotic distribution of MLE and percentile bootstrap CI is derived. Finally, a numerical example is introduced and then a Monte Carlo simulation study is carried out to illustrate the proposed method.

Bayesian Estimation and Prediction of Burr Type XI Distribution under Singly and Doubly Censored Samples

The purpose of the paper is to address the problem of estimation and prediction of the Burr type XI distribution under Bayesian framework based on censored samples. Five informative and non-informative priors have been assumed under five different (symmetric and asymmetric) loss functions for posterior analysis. The expressions for Bayes estimators, posterior risks, credible intervals, posterior predictive intervals have been derived and evaluated. The simulation study has been carried out in order to assess and compare the performance of Bayesian point and interval estimators. The study indicated that for Bayesian estimation and prediction of the said distribution, the gamma prior along with quadratic loss function can efficiently be employed.

Inference about the Burr Type III Distribution under Type-II Hybrid Censored Data

This paper presents the statistical inference on the parameters of the Burr type III distribution, when the data are Type-II hybrid censored. The maximum likelihood estimators are developed for the unknown parameters using the EM algorithm method. We provided the observed Fisher information matrix using the missing information principle which is useful for constructing the asymptotic confidence intervals. The Bayesian estimates of the unknown parameters under the assumption of independent gamma priors are obtained using two approximations, namely Lindley's approximation and the Markov Chain Monte Carlo technique. Monte Carlo simulations are performed to observe the behavior of the proposed methods and a real dataset representing is used to illustrate the derived results.

Improved confidence intervals for the scale parameter of Burr XII model based on record values

In this paper some different sorts of confidence intervals are considered for the scale parameter of the Burr type XII distribution based on the upper record values. In this regard, the coverage probability is adopted as a measure of improvement when the endpoints are the same for all types of confidence intervals. Proposed confidence intervals are based on the preliminary test estimator, Thompson shrinkage estimator and Bayes estimator with conjugate prior information. It is nicely demonstrated that the confidence intervals based on the above methodologies are superior to the equal tail confidence interval on specific intervals. Subsequently, to construct a uniformly dominant confidence interval, the result of Kubokawa (Ann Stat 22(1):290---299, 1994) is extended for dependent observations by making use of the information that exists in a covariate record value.

On the Parameter of the Burr Type X under Bayesian Principles

On the Parameter of the Burr Type X under Bayesian Principles

Marshall-Olkin Extended Burr Type XII Distribution

Marshall and Olkin (1997) proposed a new method to establish more flexible new families of distributions by adding a parameter to a distribution. In this article, Marshall-Olkin extended Burr type XII (MOEBXII) distribution is introduced. Properties of MOEBXII distribution are studied and analyzed. Based on complete sample, maximum likelihood and Bayesian estimators of the parameters are derived. Application to a real data set is carried out to illustrate the flexibility of the model.

Testing Reliability Equivalence Factors of a Series-Parallel Systems in Burr Type X Distribution

In reliability analysis for improving the system performance, the scale parameter of the life time model has mainly considered to obtain equivalence factors for the system designs. In this paper, we propose a new approach through modifying the shape parameter of the Burr type X distribution. The proposed approach is applied to the general series parallel systems. Three different methods are used to improve the system reliability: (i) the reduction method, (ii) the hot duplication method and (iii) the cold duplication method. Numerical example is presented to compare performance of the applied methods, to find limitations for the equivalence factors and to illustrate the overall theoretical analysis.