Inference on the Weibull distribution based on record values

Goodness-of-fit test for a parametric mixture cure model with partly interval-censored data

Partly interval-censored event time data arise naturally in medical, biological, sociological and demographic studies. In practice, some patients may be immune from the event of interest, invoking a cure model for survival analysis. Choosing an appropriate parametric distribution for the failure time of susceptible patients is an important step to fully structure the mixture cure model. In the literature, goodness-of-fit tests for survival models are usually restricted to uncensored or right-censored data. We fill in this gap by proposing a new goodness-of-fit test dealing with partly interval-censored data under mixture cure models. Specifically, we investigate whether a parametric distribution can fit the susceptible part by using a Cramér-von Mises type of test, and establish the asymptotic distribution of the test . Empirically, the critical value is determined from the bootstrap resamples. The proposed test, compared to the traditional leveraged bootstrap approach, yields superior practical results under various settings in extensive simulation studies. Two clinical data sets are analyzed to illustrate our method.

Interval Estimation of Generalized Inverted Exponential Distribution under Records Data: A Comparison Perspective

In this paper, the problem of interval estimation is considered for the parameters of the generalized inverted exponential distribution. Based on upper record values, different pivotal quantities are proposed and the associated exact and generalized confidence intervals are constructed for the unknown model parameters and reliability indices, respectively. For comparison purposes, conventional likelihood based approximate confidence intervals are also provided by using observed Fisher information matrix. Moreover, prediction intervals are also constructed for future records based on proposed pivotal quantities and likelihood procedures as well. Finally, numerical studies are carried out to investigate and compare the performances of the proposed methods and a real data analysis is presented for illustrative purposes.

Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction

The interest in the study of record statistics has been increasing in recent years in the context of predicting stock markets and addressing global warming and climate change problems such as cyclones and floods. However, because record values are mostly rare observed, its probability distribution may be skewed or asymmetric. In this case, the Bayesian approach with a reasonable choice of the prior distribution can be a good alternative. This paper presents an objective Bayesian method for predicting future record values when observed record values have a two-parameter exponentiated Gumbel distribution with the scale and shape parameters. For objective Bayesian analysis, objective priors such as the Jeffreys and reference priors are first derived from the Fisher information matrix for the scale and shape parameters, and an analysis of the resulting posterior distribution is then performed to examine its properness and validity. In addition, under the derived objective prior distributions, a simple algorithm using a pivotal quantity is proposed to predict future record values. To validate the proposed approach, it was applied to a real dataset. For a closer examination and demonstration of the superiority of the proposed predictive method, it was compared to time-series models such as the autoregressive integrated moving average and dynamic linear model in an analysis of real data that can be observed from an infinite time series comprising independent sample values.

Statistical Inference for the Weibull Distribution Based on δ-Record Data

We consider the maximum likelihood and Bayesian estimation of parameters and prediction of future records of the Weibull distribution from δ -record data, which consists of records and near-records. We discuss existence, consistency and numerical computation of estimators and predictors. The performance of the proposed methodology is assessed by Montecarlo simulations and the analysis of monthly rainfall series. Our conclusion is that inferences for the Weibull model, based on δ -record data, clearly improve inferences based solely on records. This methodology can be recommended, more so as near-records can be collected along with records, keeping essentially the same experimental design.

On Reliability in a Multicomponent Stress-Strength Model with Power Lindley Distribution

In this paper  we study the reliability of a multicomponent stress-strength model assuming that the components follow power Lindley model.  The maximum likelihood estimate of the reliability parameter and its asymptotic confidence interval are obtained. Applying the parametric Bootstrap technique, interval estimation of the reliability is presented.  Also, the Bayes estimate and highest posterior density credible interval of the reliability parameter are derived using suitable priors on the parameters. Because there is no closed form for the Bayes estimate, we use the Markov Chain Monte Carlo method to obtain approximate Bayes  estimate of the reliability. To evaluate the performances of different procedures,  simulation studies are conducted and an example of real data sets is provided.

Perturbative method for maximum likelihood estimation of the Weibull distribution parameters

The two-parameter Weibull distribution is the predominant distribution in reliability and lifetime data analysis. The classical approach for estimating the scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha )$$\end{document}(α) and shape \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta )$$\end{document}(β) parameters employs the maximum likelihood estimation (MLE) method. However, most MLE based-methods resort to numerical or graphical techniques due to the lack of closed-form expressions for the Weibull \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β parameter. A Weibull \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β parameter estimator based on perturbation theory is proposed in this work. An explicit expression for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β is obtained, making the estimation of both parameters straightforward. Several right-censored lifetime data sets with different sample sizes and censoring percentages were analyzed in order to assess the performance of the proposed estimator. Study case results show that our parameter estimator provides solutions of high accuracy, overpassing limitations of other parameter estimators.