Multivariate mixtures of Erlangs for density estimation under censoring

Multivariate mixtures of Erlang distributions form a versatile, yet analytically tractable, class of distributions making them suitable for multivariate density estimation. We present a flexible and effective fitting procedure for multivariate mixtures of Erlangs, which iteratively uses the EM algorithm, by introducing a computationally efficient initialization and adjustment strategy for the shape parameter vectors. We furthermore extend the EM algorithm for multivariate mixtures of Erlangs to be able to deal with randomly censored and fixed truncated data. The effectiveness of the proposed algorithm is demonstrated on simulated as well as real data sets.     

S–S method for stochastically ordered multinomial populations with missing data

Stochastic ordering is a useful concept in order restricted inferences. In this paper, we propose a new estimation technique for the parameters in two multinomial populations under stochastic orderings when missing data are present. In comparison with traditional maximum likelihood estimation method, our new method can guarantee the uniqueness of the maximum of the likelihood function. Furthermore, it does not depend on the choice of initial values for the parameters in contrast to the EM algorithm. Finally, we give the asymptotic distributions of the likelihood ratio statistics based on the new estimation method.     

Implementing particle swarm optimization algorithm to estimate the mixture of two Weibull parameters with censored data

We present the maximum likelihood estimation (MLE) via particle swarm optimization (PSO) algorithm to estimate the mixture of two Weibull parameters with complete and multiple censored data. A simulation study is conducted to assess the performance of the MLE via PSO algorithm, quasi-Newton method and expectation-maximization (EM) algorithm for different parameter settings and sample sizes in both uncensored and censored cases. The simulation results showed that the PSO algorithm outperforms the quasi-Newton method and the EM algorithm in most cases regarding bias and root mean square errors. Two numerical examples are used to demonstrate the performance of our proposed method.     

Mean-Based Iterative Procedures in Linear Models with General Errors and Grouped Data

Abstract.  We present in this paper iterative estimation procedures, using conditional expectations, to fit linear models when the distributions of the errors are general and the dependent data stem from a finite number of sources, either grouped or non-grouped with different classification criteria. We propose an initial procedure that is inspired by the expectation-maximization (EM) algorithm, although it does not agree with it. The proposed procedure avoids the nested iteration, which implicitly appears in the initial procedure and also in the EM algorithm. The stochastic asymptotic properties of the corresponding estimators are analysed.     

Estimation of Wood Fibre Length Distributions from Censored Data through an EM Algorithm

Abstract.  An expectation maximization (EM) algorithm is proposed to find fibre length distributions in standing trees. The available data come from cylindric wood samples (increment cores). The sample contains uncut fibres as well as fibres cut once or twice. The sample contains not only fibres, but also other cells, the so-called 'fines'. The lengths are measured by an automatic fibre-analyser, which is not able to distinguish fines from fibres and cannot tell if a cell has been cut. The data thus come from a censored version of a mixture of the fine and fibre length distributions in the tree. The parameters of the length distributions are estimated by a stochastic version of the EM algorithm, and an estimate of the corresponding covariance matrix is derived. The method is applied to data from northern Sweden. A simulation study is also presented. The method works well for sample sizes commonly obtained from increment cores.     

On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data

A progressive hybrid censoring scheme is a mixture of type-I and type-II progressive censoring schemes. In this paper, we mainly consider the analysis of progressive type-II hybrid-censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Since the maximum likelihood estimators (MLEs) of these parameters cannot be obtained in the closed form, we propose to use the expectation and maximization (EM) algorithm to compute the MLEs. Also, the Newton–Raphson method is used to estimate the model parameters. The asymptotic variance–covariance matrix of the MLEs under EM framework is obtained by Fisher information matrix using the missing information and asymptotic confidence intervals for the parameters are then constructed. This study will end up with comparing the two methods of estimation and the asymptotic confidence intervals of coverage probabilities corresponding to the missing information principle and the observed information matrix through a simulation study, illustrated examples and real data analysis.     

Parameter Estimation of Type-II Hybrid Censored Weighted Exponential Distribution

A hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. We study the estimation of parameters of weighted exponential distribution based on Type-II hybrid censored data. By applying the EM algorithm, maximum likelihood estimators are evaluated. Using Fisher information matrix, asymptotic confidence intervals are provided. By applying Markov chain Monte Carlo techniques, Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. Monte Carlo simulations are performed to compare the performances of the different methods, and one dataset is analyzed for illustrative purposes.     

The analysis of mixed interval-censored and complete data

The article focuses mainly on a conditional imputation algorithm of quantile-filling to analyze a new kind of censored data, mixed interval-censored and complete data related to interval-censored sample. With the algorithm, the imputed failure times, which are the conditional quantiles, are obtained within the censoring intervals in which some exact failure times are. The algorithm is viable and feasible for the parameter estimation with general distributions, for instance, a case of Weibull distribution that has a moment estimation of closed form by log-transformation. Furthermore, interval-censored sample is a special case of the new censored sample, and the conditional imputation algorithm can also be used to deal with the failure data of interval censored. By comparing the interval-censored data and the new censored data, using the imputation algorithm, in the view of the bias of estimation, we find that the performance of new censored data is better than that of interval censored.     

Likelihood inference for lognormal data with left truncation and right censoring with an illustration

The lognormal distribution is quite commonly used as a lifetime distribution. Data arising from life-testing and reliability studies are often left truncated and right censored. Here, the EM algorithm is used to estimate the parameters of the lognormal model based on left truncated and right censored data. The maximization step of the algorithm is carried out by two alternative methods, with one involving approximation using Taylor series expansion (leading to approximate maximum likelihood estimate) and the other based on the EM gradient algorithm (Lange, 1995). These two methods are compared based on Monte Carlo simulations. The Fisher scoring method for obtaining the maximum likelihood estimates shows a problem of convergence under this setup, except when the truncation percentage is small. The asymptotic variance-covariance matrix of the MLEs is derived by using the missing information principle (Louis, 1982), and then the asymptotic confidence intervals for scale and shape parameters are obtained and compared with corresponding bootstrap confidence intervals. Finally, some numerical examples are given to illustrate all the methods of inference developed here.     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

Robust and efficient parameter estimation based on censored data with stochastic covariates

Analysis of random censored life-time data along with some related stochastic covariables is of great importance in many applied sciences. The parametric estimation technique commonly used under this set-up is based on the efficient but non-robust likelihood approach. In this paper, we propose a robust parametric estimator for censored data with stochastic covariates based on the minimum density power divergence approach. The resulting estimator also has competitive efficiency with respect to the maximum likelihood estimator under pure data. The strong robustness property of the proposed estimator with respect to the presence of outliers is examined and illustrated through an appropriate real data example and simulation studies. Further, the theoretical asymptotic properties of the proposed estimator are also derived in terms of a general class of M-estimators based on the estimating equation.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

Nonparametric versus Parametric Estimation of the Cure Fraction Using Interval Censored Data

This article discusses estimation of the cure rate by means of the bounded cumulative hazard (BCH) model using interval censored data. The parametric and nonparametric estimation methods within the framework of the EM algorithm were employed for cure rate estimation and their results compared. The Turnbull estimator was used in the nonparametric estimation while in parametric method both the exponential and Weibull distributions were considered. We show via simulation that the nonparametric method is a viable alternative to the parametric one when the censoring rate is rapidly increasing.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

Estimation and prediction for a Burr type-III distribution with progressive censoring

We consider estimation of unknown parameters and reliability characteristics of a Burr type-III distribution under progressive censoring. Predictive estimates for censored observations and the associated prediction intervals are also obtained. We derive maximum-likelihood estimators of unknown quantities using the EM algorithm and then also obtain the observed Fisher information matrix. We provide various Bayes estimators for unknown parameters under the squared error loss function. Highest posterior density and asymptotic intervals are also constructed. We evaluate performance of proposed methods using simulations. Finally, an illustrative example is presented in support of the methods discussed.     

Multivariate normal mean-variance mixture distribution based on Lindley distribution

This article introduces a new asymmetric distribution constructed by assuming the multivariate normal mean-variance mixture model. Called normal mean-variance mixture of the Lindley distribution, we derive some mathematical properties of the new distribution. Also, a feasible maximum likelihood estimation procedure using the EM algorithm and the asymptotic standard errors of parameter estimates are developed. The performance of the proposed distribution is illustrated by means of real datasets and simulation analysis.     

Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring

In this paper, we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution under type II progressive censored samples. Estimation of reliability and hazard functions is also considered. Maximum likelihood estimators are obtained using the Expectation–Maximization (EM) algorithm. Further, we obtain expected Fisher information matrix using the missing value principle. Bayes estimators are derived under squared error and linex loss functions. We have used Lindley, and Tiernery and Kadane methods to compute these estimates. In addition, Bayes estimators are computed using importance sampling scheme as well. Samples generated from this scheme are further utilized for constructing highest posterior density intervals for unknown parameters. For comparison purposes asymptotic intervals are also obtained. A numerical comparison is made between proposed estimators using simulations and observations are given. A real-life data set is analyzed for illustrative purposes.