Parametric inference of the process capability index for exponentiated exponential distribution

Process capability indices (PCIs) are most effective devices/techniques used in industries for determining the quality of products and performance of manufacturing processes. In this article, we consider the PCI C_pc which is based on the proportion of conformance and is applicable to normally as well as non-normally and continuous as well as discrete distributed processes. In order to estimate the PCI C_pc when the process follows exponentiated exponential distribution, we have used five classical methods of estimation. The performances of these classical estimators are compared with respect to their biases and mean squared errors (MSEs) of the index C_pc through simulation study. Also, the confidence intervals for the index C_pc are constructed using five bootstrap confidence interval (BCIs) methods. Monte Carlo simulation study has been carried out to compare the performances of these five BCIs in terms of their average width and coverage probabilities. Besides, net sensitivity (NS) analysis for the given PCI C_pc is considered. We use two data sets related to electronic and food industries and two failure time data sets to illustrate the performance of the proposed methods of estimation and BCIs. Additionally, we have developed PCI C_pc using aforementioned methods for generalized Rayleigh distribution.     

Statistical inference about the shape parameter of the Weibull distribution by upper record values

Summary In this paper, we provide some pivotal quantities to test and establish confidence interval of the shape parameter on the basis of the firstn observed upper record values. Finally, we give some examples and the Monte Carlo simulation to assess the behaviors (including higher power and more shorter length of confidence interval) of these pivotal quantities for testing null hypotheses and establishing confidence interval concerning the shape parameter under the given significance level and the given confidence coefficient, respectively.     

Reliability analysis for stress-strength model from a general family of truncated distributions under censored data

Abstract

Under progressive Type-II censoring, inference of stress-strength reliability (SSR) is studied for a general family of lower truncated distributions. When the lifetime models of the strength and stress variables have arbitrary and common parameters, maximum likelihood and pivotal quantities based generalized estimators of SSR are established, respectively. Confidence intervals are constructed based on generalized pivotal quantities and bootstrap technique under different parameter cases as well. In addition, to compare the equivalence of the strength and stress parameters, likelihood ratio testing of interested parameters is provided as a complementary. Simulation studies and two real-life data examples are provided to investigate the performance of proposed methods.     

Estimation of Pr(Y < X) for the two-parameter generalized exponential records

This article considers the maximum likelihood and Bayes estimation of the stress–strength reliability based on two-parameter generalized exponential records. Here, we extend the results of Baklizi [Computational Statistics and Data Analysis 52 (2008), 3468–3473] to explain a wide variety of real datasets. We also consider the estimation of R when the same shape parameter is known. The results for exponential distribution can be obtained as a special case with different scale parameters.     

Simultaneous Confidence Intervals for the Parameters of Pareto Distribution under Progressive Censoring

In this article, we consider the progressive Type II right censored sample from Pareto distribution. We introduce a new approach for constructing the simultaneous confidence interval of the unknown parameters of this distribution under progressive censoring. A Monte Carlo study is also presented for illustration. It is shown that this confidence region has a smaller area than that introduced by Ku? and Kaya (2007 Ku? , C. , Kaya , M. F. ( 2007 ). Estimation for the parameters of the Pareto distribution under progressive censoring . Commun. Statist. Theor. Meth. 36 : 1359 – 1365 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring

Epstein (1954) introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. (2003) introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one-parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes.     

Markov chain Monte Carlo exact inference for binomial and multinomial logistic regression models

We develop Metropolis-Hastings algorithms for exact conditional inference, including goodness-of-fit tests, confidence intervals and residual analysis, for binomial and multinomial logistic regression models. We present examples where the exact results, obtained by enumeration, are available for comparison. We also present examples where Monte Carlo methods provide the only feasible approach for exact inference.     

E-Bayesian estimation of the exponentiated distribution family parameter under LINEX loss function

This paper is concerned with using the E-Bayesian method for computing estimates of the exponentiated distribution family parameter. Based on the LINEX loss function, formulas of E-Bayesian estimation for unknown parameter are given, these estimates are derived based on a conjugate prior. Moreover, property of E-Bayesian estimation—the relationship between of E-Bayesian estimations under different prior distributions of the hyper parameters are also provided. A comparison between the new method and the corresponding maximum likelihood techniques is conducted using the Monte Carlo simulation. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.     

Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.     

Theoretical results on the discrete Weibull distribution of Nakagawa and Osaki

In this paper, we discuss some theoretical results and properties of the discrete Weibull distribution, which was introduced by Nakagawa and Osaki [The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300–301]. We study the monotonicity of the probability mass, survival and hazard functions. Moreover, reliability, moments, p-quantiles, entropies and order statistics are also studied. We consider likelihood-based methods to estimate the model parameters based on complete and censored samples, and to derive confidence intervals. We also consider two additional methods to estimate the model parameters. The uniqueness of the maximum likelihood estimate of one of the parameters that index the discrete Weibull model is discussed. Numerical evaluation of the considered model is performed by Monte Carlo simulations. For illustrative purposes, two real data sets are analyzed.     

Interval estimation for a lower-truncated distribution based on the double Type-II censored sample

ABSTRACT

The interval estimation problem is investigated for the parameters of a general lower truncated distribution under double Type-II censoring scheme. The exact, asymptotic and bootstrap interval estimates are derived for the unknown model parameter and the lower truncated threshold bound. One real-life example and a numerical study are presented to illustrate performance of our methods.     

Bayesian inference for multivariate gamma distributions

The paper considers the multivariate gamma distribution for which the method of moments has been considered as the only method of estimation due to the complexity of the likelihood function. With a non-conjugate prior, practical Bayesian analysis can be conducted using Gibbs sampling with data augmentation. The new methods are illustrated using artificial data for a trivariate gamma distribution as well as an application to technical inefficiency estimation.     

Piecewise linear approximation of empirical distributions under a Wasserstein distance constraint

Big data applications and Monte Carlo simulation results can nowadays easily contain data sets in the size of millions of entries. We consider the situation when the information on a large univariate data set or sample needs to be preserved, stored or transferred. We suggest an algorithm to approximate a univariate empirical distribution through a piecewise linear distribution which requires significantly less memory to store. The approximation is chosen in a computationally efficient manner, such that it preserves the mean, and its Wasserstein distance to the empirical distribution is sufficiently small.     

Likelihood inference for a step-stress partially accelerated life test model with Type-I progressively hybrid censored data from Weibull distribution

Recently, progressively hybrid censoring schemes have become quite popular in life testing and reliability studies. In this article, the point and interval maximum-likelihood estimations of Weibull distribution parameters and the acceleration factor are considered. The estimation process is performed under Type-I progressively hybrid censored data for a step-stress partially accelerated test model. The biases and mean square errors of the maximum-likelihood estimators are computed to assess their performances in the presence of censoring developed in this article through a Monte Carlo simulation study.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Bayesian inference for generalized extreme value distributions via Hamiltonian Monte Carlo

In this article, we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC), and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets and simulation studies provide evidence that the extra analytical work involved in Hamiltonian Monte Carlo algorithms is compensated by a more efficient exploration of the parameter space.     

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.     

Bayesian inference for circular distributions with unknown normalising constants

Very often, the likelihoods for circular data sets are of quite complicated forms, and the functional forms of the normalising constants, which depend upon the unknown parameters, are unknown. This latter problem generally precludes rigorous, exact inference (both classical and Bayesian) for circular data.Noting the paucity of literature on Bayesian circular data analysis, and also because realistic data analysis is naturally permitted by the Bayesian paradigm, we address the above problem taking a Bayesian perspective. In particular, we propose a methodology that combines importance sampling and Markov chain Monte Carlo (MCMC) in a very effective manner to sample from the posterior distribution of the parameters, given the circular data. With simulation study and real data analysis, we demonstrate the considerable reliability and flexibility of our proposed methodology in analysing circular data.     

Prediction for future failures in Weibull distribution under hybrid censoring

In this paper, we consider the prediction of a future observation based on a type-I hybrid censored sample when the lifetime distribution of experimental units is assumed to be a Weibull random variable. Different classical and Bayesian point predictors are obtained. Bayesian predictors are obtained using squared error and linear-exponential loss functions. We also provide a simulation consistent method for computing Bayesian prediction intervals. Monte Carlo simulations are performed to compare the performances of the different methods, and one data analysis has been presented for illustrative purposes.