Classical and Bayesian Estimation of the Inverse Weibull Distribution: Using Progressive Type-I Censoring Scheme

The challenge of estimating the parameters for the inverse Weibull (IW) distribution employing progressive censoring Type-I (PCTI) will be addressed in this study using Bayesian and non-Bayesian procedures. To address the issue of censoring time selection, qauntiles from the IW lifetime distribution will be implemented as censoring time points for PCTI. Focusing on the censoring schemes, maximum likelihood estimators (MLEs) and asymptotic confidence intervals (ACI) for unknown parameters are constructed. Under the squared error (SEr) loss function, Bayes estimates (BEs) and concomitant maximum posterior density credible interval estimations are also produced. The BEs are assessed using two methods: Lindley’s approximation (LiA) technique and the Metropolis-Hasting (MH) algorithm utilizing Markov Chain Monte Carlo (MCMC). The theoretical implications of MLEs and BEs for specified schemes of PCTI samples are shown via a simulation study to compare the performance of the different suggested estimators. Finally, application of two real data sets will be employed.

Markov Chain Confidence Intervals and Biases

We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

Geometrically Convergent Simulation of the Extrema of Lévy Processes

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum, and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in Lp (for any [Formula: see text]) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct nonasymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e., of order [Formula: see text] if the mean squared error is at most [Formula: see text]) for locally Lipschitz and barrier-type functions of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.

Analysis for Xgamma Parameters of Life under Type-II Adaptive Progressively Hybrid Censoring with Applications in Engineering and Chemistry

Censoring mechanisms are widely used in various life tests, such as medicine, engineering, biology, etc., as they save (overall) test time and cost. In this context, we consider the problem of estimating the unknown xgamma parameter and some survival characteristics, such as reliability and failure rate functions in the presence of adaptive type-II progressive hybrid censored data. For this purpose, the maximum likelihood and Bayesian inferential approaches are used. Using the observed Fisher information under s-normal approximation, different asymptotic confidence intervals for any function of the unknown parameter were constructed. Using the gamma flexible prior, Bayes estimators against the squared-error loss were developed. Two procedures of Bayesian approximations—Lindley’s approximation and Metropolis–Hastings algorithm—were used to carry out the Bayes estimates and to construct the associated credible intervals. An extensive simulation study was implemented to compare the performance of the different methods. To validate the proposed methodologies of inference—two practical studies using datasets that form engineering and chemical fields are discussed.

Inferences for Joint Hybrid Progressive Censored Exponential Lifetimes under Competing Risk Model

The aim of this paper is devoted to the problem of comparative life tests under joint censoring samples from an exponential distribution with competing risks model. This problem is considered under the consideration that only two causes of failure are occurring and the units come from two production lines such that the exponential failure time of units is censored under a hybrid progressive Type-I censoring scheme. Maximum likelihood estimation and different Bayes methods of estimation are discussed. The asymptotic confidence intervals as well as the Bayes credible intervals are established. A real data set representing time to failure on two groups of strain male mice receiving radiation is analyzed for illustrative purposes. All theoretical results are assessed and compared through the Monte Carlo study.

The Analysis for the Recovery Cases of COVID-19 in Egypt using Odd Generalized Exponential Lomax Distribution

In this paper, an odd generalized exponential Lomax (OGEL, in short) distribution has been considered. Some mathematical properties of the distribution are studied. The methods of maximum likelihood and maximum product of spacing are used for estimating the model parameters. Moreover, 95% asymptotic confidence intervals for the estimates of the parameters are derived. The Monte Carlo simulation is conducted for the two proposed methods of estimation to evaluate the performance of the various proposed estimators. The proposed methods are utilized to find estimates of the parameters of OGEL distribution for the daily recovery cases of COIVD-19 in Egypt from 12 May to 30 September 2020.The practical applications show that the proposed model provides better fits than the other models.

Weighted Power Lomax Distribution and its Length Biased Version: Properties and Estimation Based on Censored Samples

In this paper, a weighted version of the power Lomax distribution referred to the weighted power Lomax distribution, is introduced. The new distribution comprises the length biased and the area biased of the power Lomax distribution as new models as well as containing an existing model as the length biased Lomax distribution as special model. Essential distributional properties of the weighted power Lomax distribution are studied. Maximum likelihood and maximum product spacing methods are proposed for estimating the population parameters in cases of complete and Type-II censored samples. Asymptotic confidence intervals of the model parameters are obtained. A sample generation algorithm along with a Monte Carlo simulation study is provided to demonstrate the pattern of the estimates for different sample sizes. Finally, a real-life data set is analyzed as an illustration and its length biased distribution is compared with some other lifetime distributions.

Estimation and Prediction for Gompertz Distribution under General Progressive Censoring

In this article, we discuss the estimation of the parameters for Gompertz distribution and prediction using general progressive Type-II censoring. Based on the Expectation–Maximization algorithm, we calculate the maximum likelihood estimates. Bayesian estimates are considered under different loss functions, which are symmetrical, asymmetrical and balanced, respectively. An approximate method—Tierney and Kadane—is used to derive the estimates. Besides, the Metropolis-Hasting (MH) algorithm is applied to get the Bayesian estimates as well. According to Fisher information matrix, we acquire asymptotic confidence intervals. Bootstrap intervals are also established. Furthermore, we build the highest posterior density intervals through the sample generated by the MH algorithm. Then, Bayesian predictive intervals and estimates for future samples are provided. Finally, for evaluating the quality of the approaches, a numerical simulation study is implemented. In addition, we analyze two real datasets.

Comparison of Accuracy Properties for Confidence Intervals of the Cross-Product Ratio of Binomial Proportions under Direct-Direct Sampling Scheme

We consider a general problem of the confidence interval for a cross-product ratio ρ=p1(1-p2)/p2(1-p1) according to data from two independent samples. Each sample may be obtained in the framework of direct Binomial sampling scheme. Asymptotic confidence intervals are constructed in accordance with direct Binomial sampling scheme, with parameter estimators demonstrating exponentially decreasing bias. Our goal is to investigate the cases when the normal approximations (which are relatively simple) for estimators of the cross-product ratio are reliable for the construction of confidence intervals. We use the closeness of the confidence coefficient to the nominal confidence level as our main evaluation criterion, and use the Monte-Carlo method to investigate the key probability characteristics of intervals corresponding to direct Binomial sampling schemes. We present estimations of the coverage probability, expectation and standard deviation of interval widths in tables and provide some recommendations for applying each obtained interval.

Generalized Weighted Rama Distribution: Properties and Application to Model Lifetime Data

In this paper, we propose a new lifetime distribution called the generalized weighted Rama (GWR) distribution, which extends the two-parameter Rama distribution and has the Rama distribution as a special case. The GWR distribution has the ability to model data sets that have positive skewness and upside-down bathtub shape hazard rate. Expressions for mathematical and reliability properties of the GWR distribution have been derived. Estimation of parameters was achieved using the method of maximum likelihood estimation and a simulation was performed to verify the stability of the maximum likelihood estimates of the model parameters. The asymptotic confidence intervals of the parameters of the proposed distribution are obtained. The applicability of the GWR distribution was illustrated with a real data set and the results obtained show that the GWR distribution is a better candidate for the data than the other competing distributions being investigated.