Maximum Likelihood Estimation of Life-Span Based on Censored and Passively Registered Historical Data

We consider the estimation of life length of people who were born in the seventeenth or eighteenth century in England. The data consist of a sequence of times of life events that is either ended by a time of death or is right-censored by an unobserved time of migration. We propose a semi parametric model for the data and use a maximum likelihood method to estimate the unknown parameters in this model. We prove the consistency of the maximum likelihood estimators and describe an algorithm to obtain the estimates numerically. We have applied the algorithm to data and the estimates found are presented.     

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.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Comparison of Maximum Likelihood with Conditional Pairwise Likelihood Estimation of Person Parameters in the Rasch Model

This paper is concerned with person parameter estimation in the binary Rasch model. The loss of efficiency of a pseudo, quasi, or composite likelihood approach investigated. By means of a Monte Carlo study, two quasi likelihood estimators are compared to two well-established maximum likelihood approaches, one of which being a weighted likelihood procedure. The results show that the observed values of the root mean squared error are practically equivalent for the compared estimators in the case of a sufficiently large number of items.     

Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model

Multivariate failure time data arise when the sample consists of clusters and each cluster contains several possibly dependent failure times. The Clayton–Oakes model (Clayton, 1978; Oakes, 1982) for multivariate failure times characterizes the intracluster dependence parametrically but allows arbitrary specification of the marginal distributions. In this paper, we discuss estimation in the Clayton–Oakes model when the marginal distributions are modeled to follow the Cox (1972) proportional hazards regression model. Parameter estimation is based on an approximate generalized maximum likelihood estimator. We illustrate the model's application with example datasets.     

Maximum Likelihood Estimation in a Semiparametric Logistic/Proportional-Hazards Mixture Model

Abstract.  We consider large sample inference in a semiparametric logistic/proportional-hazards mixture model. This model has been proposed to model survival data where there exists a positive portion of subjects in the population who are not susceptible to the event under consideration. Previous studies of the logistic/proportional-hazards mixture model have focused on developing point estimation procedures for the unknown parameters. This paper studies large sample inferences based on the semiparametric maximum likelihood estimator. Specifically, we establish existence, consistency and asymptotic normality results for the semiparametric maximum likelihood estimator. We also derive consistent variance estimates for both the parametric and non-parametric components. The results provide a theoretical foundation for making large sample inference under the logistic/proportional-hazards mixture model.     

Maximum Likelihood Estimation of a Poisson Parameter in a Model of Equioverlapping Samples:A Simulation Study

In a model of equioverlapping samples maximum likelihood estimation of a Poisson parameter is examined and compared with two linear unbiased estimations by mean squared error. Since a likelihood estimator is not explicitly available in general, a simulation study has been performed and the results are illustrated     

A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

Prediction of Times to Failure of Censored Items for a Simple Step-Stress Model with Regular and Progressive Type I Censoring from the Exponential Distribution

The problem of predicting times to failure of units from the Exponential Distribution which are censored under a simple step-stress model is considered in this article. We discuss two types of censoring—regular and progressive Type I—and two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP) for each type of censoring. Numerical examples are used to illustrate the prediction methods. Using simulation studies, mean squared prediction error (MSPE) and prediction intervals are generated for these examples. MLP and the CMP are then compared with respect to MSPE and the prediction interval.     

Asymptotic Comparison Between Constant-stress Testing and Step-stress Testing for Type-I Censored Data from Exponential Distribution

By running the life tests at higher stress levels than normal operating conditions, accelerated life testing quickly yields information on the lifetime distribution of a test unit. The lifetime at the design stress is then estimated through extrapolation using a regression model. In constant-stress testing, a unit is tested at a fixed stress level until failure or the termination time point of the test, while step-stress testing allows the experimenter to gradually increase the stress levels at some pre-fixed time points during the test. In this article, the optimal k-level constant-stress and step-stress accelerated life tests are compared for the exponential failure data under Type-I censoring. The objective is to quantify the advantage of using the step-stress testing relative to the constant-stress one. A log-linear relationship between the mean lifetime parameter and stress level is assumed and the cumulative exposure model holds for the effect of changing stress in step-stress testing. The optimal design point is then determined under C-optimality, D-optimality, and A-optimality criteria. The efficiency of step-stress testing compared to constant-stress testing is discussed in terms of the ratio of optimal objective functions based on the information matrix.     

Inference for a Simple Step-Stress Model with Type-I Censoring and Lognormally Distributed Lifetimes

Accelerated life-testing (ALT) is a very useful technique for examining the reliability of highly reliable products. It allows the experimenter to obtain failure data more quickly at increased stress levels than under normal operating conditions. A step-stress model is one special class of ALT, and in this article we consider a simple step-stress model under the cumulative exposure model with lognormally distributed lifetimes in the presence of Type-I censoring. We then discuss inferential methods for the unknown parameters of the model by the maximum likelihood estimation method. Some numerical methods, such as the Newton–Raphson and quasi-Newton methods, are discussed for solving the corresponding non-linear likelihood equations. Next, we discuss the construction of confidence intervals for the unknown parameters based on (i) the asymptotic normality of the maximum likelihood estimators (MLEs), and (ii) parametric bootstrap resampling technique. A Monte Carlo simulation study is carried out to examine the performance of these methods of inference. Finally, a numerical example is presented in order to illustrate all the methods of inference developed here.     

On Parameter Inference for Step-Stress Accelerated Life Test with Geometric Distribution

In this article, we present the parameter inference in step-stress accelerated life tests under the tampered failure rate model with geometric distribution. We deal with Type-II censoring scheme involved in experimental data, and provide the maximum likelihood estimate and confidence interval of the parameters of interest. With the help of the Monte-Carlo simulation technique, a comparison of precision of the confidence limits is demonstrated for our method, the Bootstrap method, and the large-sample based procedure. The application of two industrial real datasets shows the proposed method efficiency and feasibility.