Regression Modelling with the Generalized Power Weibull Distribution under Progressive Censoring

This paper introduces the new novel four‐parameter Weibull distribution named as the Marshall–Olkin alpha power Weibull (MOAPW) distribution. Some statistical properties of the distribution are examined. Based on Type‐I censored and Type‐II censored samples, maximum likelihood estimation (MLE), maximum product spacing (MPS), and Bayesian estimation for the MOAPW distribution parameters are discussed. Numerical analysis using real data sets and Monte Carlo simulation are accomplished to compare various estimation methods. This novel model’s supremacy upon some famous distributions is explained using two real data sets and it is shown that the MOAPW model can achieve better fits than other competitive distributions.

This paper introduces a novel approach by implementing Type II censoring on both stages of the stage life testing (SLT) experiment. First, we have assumed that the lifetime of the experimental units follows a combination of Weibull distribution on initial stage and an exponential distribution on the accelerated stage. Furthermore, we have also considered a case where the lifetime of the experimental units on both stages follows Weibull distribution with same shape parameter and different scale parameters. The distributions under the two stress levels are connected through the cumulative exposure model. We derive maximum likelihood estimation (MLE) in the SLT model under different lifetime distributions. As an alternative to MLE, we also derive the maximum product of spacings (MPS) estimation method and compare the performance of the MPS and MLE methods through a simulation study. Additionally, we derive asymptotic confidence intervals (CIs) based on the asymptotic normality of MLE, and we also employ bootstrap methods to obtain the boot‐ and boot‐ CIs. Further, we proposed an optimal design for the SLT experiment based on two optimality criteria. Finally, an extensive simulation study and a real data analysis have been carried out to demonstrate the effectiveness of our approach.

In this work the analysis of interval-censored data, with Weibull distribution as the underlying lifetime distribution has been considered. It is assumed that censoring mechanism is independent and non-informative. As expected, the maximum likelihood estimators cannot be obtained in closed form. In our simulation experiments it is observed that the Newton-Raphson method may not converge many times. An expectation maximization algorithm has been suggested to compute the maximum likelihood estimators, and it converges almost all the times. The Bayes estimates of the unknown parameters under gamma priors are considered. If the shape parameter is known, the Bayes estimate of the scale parameter can be obtained in explicit form. When both the parameters are unknown, the Bayes estimators cannot be obtained in explicit form. Lindley’s approximation, importance sampling procedures and Metropolis Hastings algorithm are used to compute the Bayes estimates. Highest posterior density credible intervals of the unknown parameter are obtained using importance sampling technique. Small simulation experiments are conducted to investigate the finite sample performance of the proposed estimators, and the analysis of two data sets; one simulated and one real life, have been provided for illustrative purposes.

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

In this paper, constant-stress accelerated life test is assumed when the lifetime of test units follows an extension of the exponential distribution. Based on progressive censoring, the maximum likelihood estimates (MLEs) and Bayes estimates (BEs) of the model parameters are obtained. The BEs are obtained based on both non-informative and informative priors. In addition, the approximate, bootstrap and credible confidence intervals (CIs) of the estimators are constructed. Moreover, a real dataset is analyzed to illustrate the proposed procedures. Furthermore, the real dataset is used to show that extension of the exponential distribution can be a better model than Weibull distribution and generalized exponential distribution. Finally, simulation studies are carried out to investigate the accuracy of the MLEs and BEs for the parameters involved.

In this paper, we consider the Bayesian inference on the stress-strength parameter \(R = P(Y < X)\), when X and Y follow independent Weibull distributions. We have considered different cases. It is assumed that the random variables X and Y have different scale parameters and (a) a common shape parameter or (b) different shape parameters. Moreover, both stress and strength may depend on some known covariates also. When the two distributions have a common shape parameter, Bayesian inference on R is obtained based on the assumption that the shape parameter has a log-concave prior, and given the shape parameter, the scale parameters have Dirichlet-Gamma prior. The Bayes estimate cannot be obtained in closed form, and we propose to use Gibbs sampling method to compute the Bayes estimate and also to compute the associated highest posterior density (HPD) credible interval. The results have been extended when the covariates are also present. We further consider the case when the two shape parameters are different. Simulation experiments have been performed to see the effectiveness of the proposed methods. One data set has been analyzed for illustrative purposes and finally, we conclude the paper.

Using progressive Type-II censoring data, this study deals with the estimation of parameters of the Unit-Weibull distribution using two classical methods and the Bayesian method. In the classical methods, maximum likelihood and the maximum product of spacing (MPS) methods are used to obtain the parameters of the model by utilizing the Newton–Raphson method. On the basis of observed Fisher information matrix, approximate confidence intervals for the unknown parameters are obtained. In addition, two bootstrap methods are used to obtain confidence intervals for the unknown parameters of the model. In the Bayesian estimation, we have considered both likelihood function as well as product of spacing function to estimate the model parameters. Bayes estimators are obtained under squared error loss function using independent gamma density priors for the unknown model parameters. Since closed-form of the Bayes estimators is not available, the Metropolis–Hastings algorithm is proposed to approximate the Bayes estimates. In addition, highest posterior density credible intervals are obtained. Further, using different optimally criteria, an optimal scheme has been proposed. A simulation study is conducted to assess the statistical performance of all the estimators. To demonstrate the proposed methodology a real data analysis is provided to illustrate all the statistical inferential procedures developed in the paper.

As an important reliability metric in disc drive industries, the annual failure rate (AFR), estimated from reliability demonstration tests (RDTs), is often used to make business decisions. Due to limited test sample size and short test duration, estimating the AFR from direct test data using Weibull distribution fit may result in a wide confidence interval, raising concerns regarding the uncertainty of the AFR estimate. To improve the confidence of the estimate, this article presents a Bayesian posterior analysis approach to estimate the AFR. Prior distributions of the Weibull distribution's shape and scale parameters are presented based on historic test data. An analysis of real RDT data demonstrates that the Bayesian posterior estimate of the AFR can produce a significantly narrower confidence interval than direct Weibull fitting, promoting higher confidence to accept the AFR estimate. To evaluate the significance of the prior distribution parameters on the AFR estimate, a sensitivity analysis is presented with respect to each individual parameter. The results indicate that the standard deviation value of either β or η has a more significant impact on the AFR estimate than the mean value.

In this paper, we consider Dagum distribution which is capable of modeling various shapes of failure rates and aging criteria. Based on progressively type-I interval censoring data, we first obtain the maximum likelihood estimators and the approximate confidence intervals of the unknown parameters of the Dagum distribution. Next, we obtain the Bayes estimators of the parameters of Dagum distribution under the squared error loss (SEL) and balanced squared error loss (BSEL) functions using independent informative gamma and non informative uniform priors for both scale and two shape parameters. A Monte Carlo simulation study is performed to assess the performance of the proposed Bayes estimators with the maximum likelihood estimators. We also compute credible intervals and symmetric 100(1 − τ)% two-sided Bayes probability intervals under the respective approaches. Besides, based on observed samples, Bayes predictive estimates and intervals are obtained using one-and two-sample schemes. Simulation results reveal that the Bayes estimates based on SEL and BSEL performs better than maximum likelihood estimates in terms of bias and MSEs. Besides, credible intervals have smaller interval lengths than confidence interval. Further, predictive estimates based on SEL with informative prior performs better than non-informative prior for both one and two sample schemes. Further, the optimal censoring scheme has been suggested using a optimality criteria. Finally, we analyze a data set to illustrate the results derived.

This study focuses on estimating the shape and scale parameters of the Topp-Leone Exponential distribution. Bayes estimators are obtained using Exponential, Gamma, LogNormal, and Weibull distributions as the identical priors under asymmetric loss functions such as LINEX and Entropy and integrated with the Lindley approximation method. The simulation study was employed to determine identical prior and loss functions for the shape and scale parameters. It is observed that LINEX loss function with Exponential-Exponential prior for the shape parameter and the scale parameter Gamma-Gamma prior is most preferred for this distribution.

In this paper, we introduce and study a new extension of Lomax distribution with four-parameter named as the Marshall–Olkin alpha power Lomax (MOAPL) distribution. Some statistical properties of this distribution are discussed. Maximum likelihood estimation (MLE), maximum product spacing (MPS) and least Square (LS) method for the MOAPL distribution parameters are discussed. A numerical study using real data analysis and Monte-Carlo simulation are performed to compare between different methods of estimation. Superiority of the new model over some well-known distributions are illustrated by physics and economics real data sets. The MOAPL model can produce better fits than some well-known distributions as Marshall–Olkin Lomax, alpha power Lomax, Lomax distribution, Marshall–Olkin alpha power exponential, Kumaraswamy-generalized Lomax, exponentiated Lomax and power Lomax.

Statistical analysis of adaptive type-II progressively censored competing risks for Weibull models

Introduction and objectives The objective measurement of cough frequency has changed the standards by which we evaluate anti-tussive treatments and study the mechanisms driving cough. The distribution of individual cough events over time can vary substantially between patients with identical cough frequencies but this has received little attention. The aim of this study was to explore the differences in the temporal distribution of cough in 24 hour acoustic recordings across a range of respiratory conditions using the Weibull Distribution. Methods We studied 24 hour acoustic recordings made using the VitaloJAK™ cough monitoring system from 62 participants [healthy volunteers (n=11), chronic cough (n=15), asthma (n=10) chronic obstructive pulmonary disease (COPD) (n=16), Idiopathic Pulmonary Fibrosis (IPF) n=10)]. Using the exact location of the explosive phase of each cough sound, the lengths of inter-cough intervals were calculated in each recording and the fit to Weibull and exponential distributions assessed; the shape parameter which describes the ‘clustering’ of events was examined. Results The number of coughs per 24 recording ranged from 0 to 3133, median 158.0 coughs and was increased compared with healthy controls in all disease groups [healthy volunteers median 13.0(IQR 1.0–22.0) coughs/24 hour versus chronic cough 509.0 (296.0–916.0), asthma 35.0 (13.5–144.0) COPD 158.0 (41.0–791.5), Idiopathic Pulmonary Fibrosis 193.5 (101.3–542.8), all p Conclusions The intervals between cough events are not random and can be described by a Weibull distribution. Significant differences in the Weibull shape parameter suggest differences in the temporal clustering of coughs between different respiratory conditions and therefore may be an important novel parameter from objective cough monitoring.

In this article, a new two-parameter model called the truncated Cauchy power-inverted Topp–Leone (TCP-ITL) is constructed by merging the truncated Cauchy power -G (TCP-G) family with the inverted Topp–Leone (ITL) distribution. Some structural properties of the newly suggested model are obtained. Different types of entropies are proposed under the TCP-ITL distribution. Under the complete and hybrid censored data, the maximum likelihood (ML), maximum product of spacing (MPSP), and Bayesian estimate approaches are explored. A simulation study is developed to test the proposed distribution’s restricted sample attributes. In the majority of cases, the numerical data revealed that the Bayesian estimates provided more accurate outcomes than the equivalent alternative estimates. The adaptability of the proposed approach is proven using examples from dependability, medicine, and engineering. A real-world data set is utilized to demonstrate the potential of the TCP-ITL distribution in comparison to other well-known distributions. The results of the model selection revealed that the proposed distribution is the best choice for the data sets under consideration.

This paper presents a novel extension of the exponentiated inverse Rayleigh distribution called the half-logistic exponentiated inverse Rayleigh distribution. This extension improves the flexibility of the distribution for modeling lifetime data for both monotonic and non-monotonic hazard rates. The statistical properties of the half-logistic exponentiated inverse Rayleigh distribution, such as the quantiles, moments, reliability, and hazard function, are examined. In particular, we provide several techniques to estimate the half-logistic exponentiated inverse Rayleigh distribution parameters: weighted least squares, Cramér-Von Mises, maximum likelihood, maximum product spacings and ordinary least squares methods. Moreover, numerical simulations were performed to evaluate these estimation methods for both small and large samples through Monte Carlo simulations, and the finding reveals that the maximum likelihood estimation was the best among all estimation methods since it comprises small mean square error compared to other estimation methods. We employ real-world lifetime data to demonstrate the performance of the newly generated distribution compared to other distributions through practical application. The results show that the half-logistic exponentiated inverse Rayleigh distribution performs better than alternative versions of the Rayleigh distributions.