Estimation on Lomax progressive censoring using the EM algorithm

Based on progressively type-II censored data, the maximum-likelihood estimators (MLEs) for the Lomax parameters are derived using the expectation–maximization (EM) algorithm. Moreover, the expected Fisher information matrix based on the missing value principle is computed. Using extensive simulation and three criteria, namely, bias, root mean squared error and Pitman closeness measures, we compare the performance of the MLEs via the EM algorithm and the Newton–Raphson (NR) method. It is concluded that the EM algorithm outperforms the NR method in all the cases. Two real data examples are used to illustrate our proposed estimators.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

The competing risks analysis for parallel and series systems using Type-II progressive censoring

Abstract

Recently, the study of the lifetime of systems in reliability and survival analysis in the presence of several causes of failure (competing risks) has attracted attention in the literature. In this paper, series and parallel systems with exponential lifetime for each item of the system are considered. Several causes of failure independently affect lifetime distributions and observations of failure times of the systems are considered under progressive Type-II censored scheme. For series systems, the maximum likelihood estimates of parameters are computed and confidence intervals for parameters of the model are obtained using Fisher information matrix. For parallel systems, the generalized EM algorithm which uses the Newton-Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. Also, the standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. The simulation study confirms the good performance of the introduced approach.     

Estimation in a truncated bivariate poisson distribution using the EM algorithm

An EM algorithm (Dempster et al., 1977) is derived for the estimation of parameters of the truncated bivariate Poisson distribution with zeros rnissing from both margins. The observed inforrnation matrix is obtained and a numerical exarnple is given where the convergence of the EM algorithm is accelerated by the methods of Louis (1982) and conjugate gradients (Jamshidian antl Jennrich, 1993).     

Linear Quantile Regression Based on EM Algorithm

This article aims to put forward a new method to solve the linear quantile regression problems based on EM algorithm using a location-scale mixture of the asymmetric Laplace error distribution. A closed form of the estimator of the unknown parameter vector β based on EM algorithm, is obtained. In addition, some simulations are conducted to illustrate the performance of the proposed method. Simulation results demonstrate that the proposed algorithm performs well. Finally, the classical Engel data is fitted and the Bootstrap confidence intervals for estimators are provided.     

Quantile Regression via the EM Algorithm

The three-parameter asymmetric Laplace distribution (ALD) has received increasing attention in the field of quantile regression due to an important feature between its location and asymmetric parameters. On the basis of the representation of the ALD as a normal-variance–mean mixture with an exponential mixing distribution, this article develops EM and generalized EM algorithms, respectively, for computing regression quantiles of linear and nonlinear regression models. It is interesting to show that the proposed EM algorithm and the MM (Majorization–Minimization) algorithm for quantile regressions are really the same in terms of computation, since the updating formula of them are the same. This provides a good example that connects the EM and MM algorithms. Simulation studies show that the EM algorithm can successfully recover the true parameters in quantile regressions.     

Fitting mixtures of linear regressions

In most applications, the parameters of a mixture of linear regression models are estimated by maximum likelihood using the expectation maximization (EM) algorithm. In this article, we propose the comparison of three algorithms to compute maximum likelihood estimates of the parameters of these models: the EM algorithm, the classification EM algorithm and the stochastic EM algorithm. The comparison of the three procedures was done through a simulation study of the performance (computational effort, statistical properties of estimators and goodness of fit) of these approaches on simulated data sets.

Simulation results show that the choice of the approach depends essentially on the configuration of the true regression lines and the initialization of the algorithms.     

Ml estimation in the poisson binomial distribution with grouped data via the em algorithm

The maximum likelihood estimation of parameters of the Poisson binomial distribution, based on a sample with exact and grouped observations, is considered by applying the EM algorithm (Dempster et al, 1977). The results of Louis (1982) are used in obtaining the observed information matrix and accelerating the convergence of the EM algorithm substantially. The maximum likelihood estimation from samples consisting entirely of complete (Sprott, 1958) or grouped observations are treated as special cases of the estimation problem mentioned above. A brief account is given for the implementation of the EM algorithm when the sampling distribution is the Neyman Type A since the latter is a limiting form of the Poisson binomial. Numerical examples based on real data are included.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

The Rayleigh–Lindley model: properties and applications

In this paper, the Rayleigh–Lindley (RL) distribution is introduced, obtained by compounding the Rayleigh and Lindley discrete distributions, where the compounding procedure follows an approach similar to the one previously studied by Adamidis and Loukas in some other contexts. The resulting distribution is a two-parameter model, which is competitive with other parsimonious models such as the gamma and Weibull distributions. We study some properties of this new model such as the moments and the mean residual life. The estimation was approached via EM algorithm. The behavior of these estimators was studied in finite samples through a simulation study. Finally, we report two real data illustrations in order to show the performance of the proposed model versus other common two-parameter models in the literature. The main conclusion is that the model proposed can be a valid alternative to other competing models well established in the literature.     

Heterogeneous data modeling with two-component Weibull–Poisson distribution

The mixture distribution models are more useful than pure distributions in modeling of heterogeneous data sets. The aim of this paper is to propose mixture of Weibull–Poisson (WP) distributions to model heterogeneous data sets for the first time. So, a powerful alternative mixture distribution is created for modeling of the heterogeneous data sets. In the study, many features of the proposed mixture of WP distributions are examined. Also, the expectation maximization (EM) algorithm is used to determine the maximum-likelihood estimates of the parameters, and the simulation study is conducted for evaluating the performance of the proposed EM scheme. Applications for two real heterogeneous data sets are given to show the flexibility and potentiality of the new mixture distribution.     

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.     

Parabolic acceleration of the EM algorithm

A new acceleration scheme for optimization procedures is defined through geometric considerations and applied to the EM algorithm. In many cases it is able to circumvent the problem of stagnation. No modification of the original algorithm is required. It is simply used as a software component. Thus the new scheme can be easily implemented to accelerate a fixed point algorithm maximizing some objective function. Some practical examples and simulations are presented to show its ability to accelerate EM-type algorithms converging slowly.     

On the choice of the number of blocks with the incremental EM algorithm for the fitting of normal mixtures

The EM algorithm is a popular method for parameter estimation in situations where the data can be viewed as being incomplete. As each E-step visits each data point on a given iteration, the EM algorithm requires considerable computation time in its application to large data sets. Two versions, the incremental EM (IEM) algorithm and a sparse version of the EM algorithm, were proposed recently by Neal R.M. and Hinton G.E. in Jordan M.I. (Ed.), Learning in Graphical Models, Kluwer, Dordrecht, 1998, pp. 355–368 to reduce the computational cost of applying the EM algorithm. With the IEM algorithm, the available n observations are divided into B (B n) blocks and the E-step is implemented for only a block of observations at a time before the next M-step is performed. With the sparse version of the EM algorithm for the fitting of mixture models, only those posterior probabilities of component membership of the mixture that are above a specified threshold are updated; the remaining component-posterior probabilities are held fixed. In this paper, simulations are performed to assess the relative performances of the IEM algorithm with various number of blocks and the standard EM algorithm. In particular, we propose a simple rule for choosing the number of blocks with the IEM algorithm. For the IEM algorithm in the extreme case of one observation per block, we provide efficient updating formulas, which avoid the direct calculation of the inverses and determinants of the component-covariance matrices. Moreover, a sparse version of the IEM algorithm (SPIEM) is formulated by combining the sparse E-step of the EM algorithm and the partial E-step of the IEM algorithm. This SPIEM algorithm can further reduce the computation time of the IEM algorithm.     

The performance of standard and hybrid em algorithms for ml estimates of the normal mixture model with censoring

Many studies have been made of the performance of standard algorithms used to estimate the parameters of a mixture density, where data arise from two or more underlying populations. While these studies examine uncensored data, many mixture processes are right-censored. Therefore, this paper addresses the accuracy and efficiency of standard and hybrid algorithms under different degrees of right-censored data. While a common belief is that the EM algorithm is slow and inaccurate, we find that the EM generally exhibits excellent efficiency and accuracy. While extreme right censoring causes the EM to frequently fail to converge, a hybrid-EM algorithm is found to be superior at all levels of right-censoring.s     

Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling

In this article, we propose mixtures of skew Laplace normal (SLN) distributions to model both skewness and heavy-tailedness in the neous data set as an alternative to mixtures of skew Student-t-normal (STN) distributions. We give the expectation–maximization (EM) algorithm to obtain the maximum likelihood (ML) estimators for the parameters of interest. We also analyze the mixture regression model based on the SLN distribution and provide the ML estimators of the parameters using the EM algorithm. The performance of the proposed mixture model is illustrated by a simulation study and two real data examples.     

EM algorithm for mixture of skew-normal distributions fitted to grouped data

Grouped data are frequently used in several fields of study. In this work, we use the expectation-maximization (EM) algorithm for fitting the skew-normal (SN) mixture model to the grouped data. Implementing the EM algorithm requires computing the one-dimensional integrals for each group or class. Our simulation study and real data analyses reveal that the EM algorithm not only always converges but also can be implemented in just a few seconds even when the number of components is large, contrary to the Bayesian paradigm that is computationally expensive. The accuracy of the EM algorithm and superiority of the SN mixture model over the traditional normal mixture model in modelling grouped data are demonstrated through the simulation and three real data illustrations. For implementing the EM algorithm, we use the package called ForestFit developed for R environment available at https://cran.r-project.org/web/packages/ForestFit/index.html.     

Stochastic EM algorithm for generalized exponential cure rate model and an empirical study

In this paper, we consider two well-known parametric long-term survival models, namely, the Bernoulli cure rate model and the promotion time (or Poisson) cure rate model. Assuming the long-term survival probability to depend on a set of risk factors, the main contribution is in the development of the stochastic expectation maximization (SEM) algorithm to determine the maximum likelihood estimates of the model parameters. We carry out a detailed simulation study to demonstrate the performance of the proposed SEM algorithm. For this purpose, we assume the lifetimes due to each competing cause to follow a two-parameter generalized exponential distribution. We also compare the results obtained from the SEM algorithm with those obtained from the well-known expectation maximization (EM) algorithm. Furthermore, we investigate a simplified estimation procedure for both SEM and EM algorithms that allow the objective function to be maximized to split into simpler functions with lower dimensions with respect to model parameters. Moreover, we present examples where the EM algorithm fails to converge but the SEM algorithm still works. For illustrative purposes, we analyze a breast cancer survival data. Finally, we use a graphical method to assess the goodness-of-fit of the model with generalized exponential lifetimes.     

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.     

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.