Reliability estimation in Maxwell distribution with progressively Type-II censored data

There may be situations in which either the reliability data do not fit to popular lifetime models or the estimation of the parameters is not easy, while there may be other distributions which are not popular but either they provide better goodness-of-fit or have a smaller number of parameters to be estimated, or they have both the advantages. This paper proposes the Maxwell distribution as a lifetime model and supports its usefulness in the reliability theory through real data examples. Important distributional properties and reliability characteristics of this model are elucidated. Estimation procedures for the parameter, mean life, reliability and failure-rate functions are developed. In view of cost constraints and convenience of intermediate removals, the progressively Type-II censored sample information is used in the estimation. The efficiencies of the estimates are studied through simulation. Apart from researchers and practitioners in the reliability theory, the study is also useful for scientists in physics and chemistry, where the Maxwell distribution is widely used.     

Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

Using jackknife to correct bias of MLE for the truncated Pareto distribution

The Pareto distribution is a well-known probability distribution in statistics, which has been widely used in many fields, such as finance, physics, hydrology, geology and astronomy. However, the parameter estimation for the truncated Pareto distribution is much more complicated than that for the Pareto distribution. In this paper, we demonstrate that the bias of the maximum likelihood estimation for the truncated Pareto distribution can be significantly reduced by its jackknife estimation, which has a very simple form.     

A consistent method of estimation for the three-parameter lognormal distribution based on Type-II right censored data

ABSTRACT

In this paper, we propose a parameter estimation method for the three-parameter lognormal distribution based on Type-II right censored data. In the proposed method, under mild conditions, the estimates always exist uniquely in the entire parameter space, and the estimators also have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs very well compared to a prominent method of estimation in terms of bias and root mean squared error (RMSE) in small-sample situations. Finally, two examples based on real data sets are presented for illustrating the proposed method.     

A bayesian estimator for the dependence function of a bivariate extreme‐value distribution

Any continuous bivariate distribution can be expressed in terms of its margins and a unique copula. In the case of extreme‐value distributions, the copula is characterized by a dependence function while each margin depends on three parameters. The authors propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the margins. They describe a nonparametric model for the dependence function and a reversible jump Markov chain Monte Carlo algorithm for the computation of the Bayesian estimator. They show through simulations that their estimator has a smaller mean integrated squared error than classical nonparametric estimators, especially in small samples. They illustrate their approach on a hydrological data set.     

A New Kernel Distribution Function Estimator Based on a Non-parametric Transformation of the Data

Abstract.  A new kernel distribution function (df) estimator based on a non-parametric transformation of the data is proposed. It is shown that the asymptotic bias and mean squared error of the estimator are considerably smaller than that of the standard kernel df estimator. For the practical implementation of the new estimator a data-based choice of the bandwidth is proposed. Two possible areas of application are the non-parametric smoothed bootstrap and survival analysis. In the latter case new estimators for the survival function and the mean residual life function are derived.     

A test of fit for a continuous distribution based on the empirical convex conditional mean function

Gupta and Kirmani (2008 Gupta, R.C., Kirmani, S.N.U.A. (2008). Characterization based on convex conditional mean function. J. Stat. Plann Inference. 138:964–970.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the convex conditional mean function (CCMF) characterizes the distribution function completely. In this paper, we introduce a consistent estimator of CCMF and call it empirical convex conditional mean function (ECCMF). Then we construct a simple consistent test of fit based on the integrated squared difference between ECCMF and CCMF. The theoretical and asymptotic properties of the estimator ECCMF and the proposed test statistic are studied. The performance of the constructed test is investigated under different distributions using simulations.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Reliability Characteristics of the Maxwell Distribution: A Bayes Estimation Study

ABSTRACT

This article presents maximum likelihood, Bayes, and empirical Bayes estimators of the truncated first moment and hazard function of the Maxwell distribution. A comparison of the relative efficiency of these three estimators is performed via a Monte Carlo simulation study.     

Testing the validity of the logistic model based on the empirical distribution function

The logistic distribution is one of the fundamental distribution and is widely used for describing model growth curves in survival analysis and biological studies. Applications of this distribution are presented in statistical literature. In this article, goodness of fit tests for the logistic distribution based on the empirical distribution function (EDF) are considered. In order to compute the test statistics, because the MLEs cannot be obtained explicitly, we use the approximate maximum likelihood estimates (AMLEs) suggested by Balakrishnan and Cohen (1990 Balakrishnan, N., Cohen, A. C. (1990). Order Statistics and Inference: Estimation Methods. Boston: Academic Press. [Google Scholar]), which are simple explicit estimators. Power comparisons of the considered tests are carried out via simulations. Finally, two illustrative examples are presented and analyzed.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

A Bayesian predictive approach to determining the number of components in a mixture distribution

This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.     

Maximum likelihood estimation of the parameters of a multiple step-stress model from the Birnbaum-Saunders distribution under time-constraint: A comparative study

The cumulative exposure model (CEM) is a commonly used statistical model utilized to analyze data from a step-stress accelerated life testing which is a special class of accelerated life testing (ALT). In practice, researchers conduct ALT to: (1) determine the effects of extreme levels of stress factors (e.g., temperature) on the life distribution, and (2) to gain information on the parameters of the life distribution more rapidly than under normal operating (or environmental) conditions. In literature, researchers assume that the CEM is from well-known distributions, such as the Weibull family. This study, on the other hand, considers a p-step-stress model with q stress factors from the two-parameter Birnbaum-Saunders distribution when there is a time constraint on the duration of the experiment. In this comparison paper, we consider different frameworks to numerically compute the point estimation for the unknown parameters of the CEM using the maximum likelihood theory. Each framework implements at least one optimization method; therefore, numerical examples and extensive Monte Carlo simulations are considered to compare and numerically examine the performance of the considered estimation frameworks.