On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

On finite mixture of two-component gompertz lifetime model

The Gompertz distribution has been used as a growth model, especially in epidemiological and biomedical studies. Based on Type I and II censored samples from a heterogeneous population that can be represented by a finite mixture of two-component Gompertz lifetime model, the maximum likelihood and Bayes estimates of the parameters, reliability and hazard rate functions are obtained. An approximation form due to Lindley (1980) is used in obtaining the corresponding Bayes estimates. The maximum likelihood and Bayes estimates are comparedvia a Monte Carlo simulation study.     

On MLE of a nonlinear discriminant function from a mixture of two Gompertz distributions based on small sample size

The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions. Also classification of a random variable based on a mixture can be meaning fully discussed only if the class of all finite mixtures is identifiable. The problem of identifiability of finite mixture of Gompertz distributions is studied. A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two Gompertz distributions, using classified and unclassified observations. Based on small sample size, estimation of a nonlinear discriminant function is considered. Throughout simulation experiments, the performance of the corresponding estimated nonlinear discriminant function is investigated.     

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT

The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

Statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data

In this article, the statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed. The estimation of the parameters for Gompertz distribution is obtained using maximum likelihood method (MLE) and Bayesian method under three different loss functions. We also proved the existence and uniqueness of the MLE. The one-sample Bayesian prediction intervals are obtained. The work is done for different values of the parameters. We apply the Monto Carlo simulation to compare the proposed methods, also an example is discussed to construct the Prediction intervals.     

Testing for the Presence of a Cure Fraction in Clustered Interval‐Censored Survival Data

Clustered interval‐censored survival data are often encountered in clinical and epidemiological studies due to geographic exposures and periodic visits of patients. When a nonnegligible cured proportion exists in the population, several authors in recent years have proposed to use mixture cure models incorporating random effects or frailties to analyze such complex data. However, the implementation of the mixture cure modeling approaches may be cumbersome. Interest then lies in determining whether or not it is necessary to adjust the cured proportion prior to the mixture cure analysis. This paper mainly focuses on the development of a score for testing the presence of cured subjects in clustered and interval‐censored survival data. Through simulation, we evaluate the sampling distribution and power behaviour of the score test. A bootstrap approach is further developed, leading to more accurate significance levels and greater power in small sample situations. We illustrate applications of the test using data sets from a smoking cessation study and a retrospective study of early breast cancer patients.     

On hybrid censored Weibull distribution

A hybrid censoring is a mixture of Type-I and Type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are hybrid censored. The maximum likelihood estimators (MLEs) and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the MLEs are used to construct approximate confidence intervals. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are obtained under suitable priors on the unknown parameters and using the Gibbs sampling procedure. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. Monte Carlo simulations are performed to compare the performances of the different methods and one data set is analyzed for illustrative purposes.     

Using BBPSO Algorithm to Estimate the Weibull Parameters with Censored Data

This article proposes the maximum likelihood estimates based on bare bones particle swarm optimization (BBPSO) algorithm for estimating the parameters of Weibull distribution with censored data, which is widely used in lifetime data analysis. This approach can produce more accuracy of the parameter estimation for the Weibull distribution. Additionally, the confidence intervals for the estimators are obtained. The simulation results show that the BB PSO algorithm outperforms the Newton–Raphson method in most cases in terms of bias, root mean square of errors, and coverage rate. Two examples are used to demonstrate the performance of the proposed approach. The results show that the maximum likelihood estimates via BBPSO algorithm perform well for estimating the Weibull parameters with censored data.     

Step partially accelerated life tests under finite mixture models

In this paper, step partially accelerated life tests are considered when the lifetime of an item under use condition follows a finite mixture of distributions. The analysis is performed when each of the components follows a general class of distributions, which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), power function, Gompertz and compound Gompertz distributions. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the mixing proportions, scale parameters and acceleration factor are obtained. Special attention is paid to a mixture of two exponential components. Simulation results are obtained to study the precision of MLEs.     

A Random Coefficient Degradation Model With Ramdom Sample Size

In testing product reliability, there is often a critical cutoff level that determines whether a specimen is classified as failed. One consequence is that the number of degradation data collected varies from specimen to specimen. The information of random sample size should be included in the model, and our study shows that it can be influential in estimating model parameters. Two-stage least squares (LS) and maximum modified likelihood (MML) estimation, which both assume fixed sample sizes, are commonly used for estimating parameters in the repeated measurements models typically applied to degradation data. However, the LS estimate is not consistent in the case of random sample sizes. This article derives the likelihood for the random sample size model and suggests using maximum likelihood (ML) for parameter estimation. Our simulation studies show that ML estimates have smaller biases and variances compared to the LS and MML estimates. All estimation methods can be greatly improved if the number of specimens increases from 5 to 10. A data set from a semiconductor application is used to illustrate our methods.     

Maximum Likelihood Estimation of Parameters in a Mixture Model

The estimation of parameters of the log normal distribution based on complete and censored samples are considered in the literature. In this article, the problem of estimating the parameters of log normal mixture model is considered. The Expectation Maximization algorithm is used to obtain maximum likelihood estimators for the parameters, as the likelihood equation does not yield closed form expression. The standard errors of the estimates are obtained. The methodology developed here is then illustrated through simulation studies. The confidence interval based on large-sample theory is obtained.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Unimodality of the pareto distribution likelihood function for multicensored samples and implications for estimation

This paper discusses maximum likelihood parameter estimation in the Pareto distribution for multicensored samples. In particu-

lar, the modality of the associated conditional log-likelihood function is investigated in order to resolve questions concerninc

the existence and uniqurneas of the lnarimum likelihood estimates.For the cases with one parameter known, the maximum likelihood

estimates of the remaining unknown parameters are shown to exist and to be unique. When both parameters are unknown, the maximum likelihood estimates may or may not exist and be unique. That is, their existence and uniqueness would seem to depend solely upon the information inherent in the sample data. In viav of the possible nonexistence and/or non-uniqueness of the maximum likelihood estimates when both parameters are unknown, alternatives to standard iterative numerical methods are explored.     

Inference for Burr XII distribution under Type I progressive hybrid censoring

We consider estimation of unknown parameters of a Burr XII distribution based on progressively Type I hybrid censored data. The maximum likelihood estimates are obtained using an expectation maximization algorithm. Asymptotic interval estimates are constructed from the Fisher information matrix. We obtain Bayes estimates under the squared error loss function using the Lindley method and Metropolis–Hastings algorithm. The predictive estimates of censored observations are obtained and the corresponding prediction intervals are also constructed. We compare the performance of the different methods using simulations. Two real datasets have been analyzed for illustrative purposes.