An estimator for the scale parameter of the two parameter weibull distribution for type i singly right-censored data

For type I censoring, in addition to the failure times, the number failures is also observed as part of the data. Using this feature of type I singly right-censored data a simple estimator is obtained for the scale parameter of the two parameter Weibull distribution. The exact mean and variance of the estimator are derived and computed for finite sample sizes. Its limiting properties such as asymptotic normality and asymptotic relative efficiency are obtained. The estimator has high efficiency for moderate and heavy censoring. Its use is illustrated by means of an example.     

Self-consistent estimators of survival functions with doubly-censored data

The consistency and asymptotic normality of self-consistent estimators (SCE) of survival functions with doubly-censored data have been studied by many authors. However, to the best of our knowledge, expressions of the asymptotic variance of the SCE have not been derived in the literature. In this paper, under the assumption that the survival time and censoring time distributions are discrete with finitely many jump points, an expression and a consistent estimator of the asymptotic variance of the SCE are presented. A proof of the strong consistency of the SCE is also presented. Our simu¬lation studies indicate that the estimate of the asymptotic variance is very close to the true value even with moderate sample sizes and high censoring rates     

Comparative Study of Blu and ML Estimators of Location and Scale Parameters of an Extreme Value Distribution for Small Samples and Type I Censorship

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators.     

An Iterative Weighted Least Squares Algorithm and Simulation Study for Censored Data M-Estimates

A popular linear regression estimator for censored data is the one proposed by Buckley and James (1979). However, this estimator is not robust to outliers, which is not surprising since it is a modified version of the uncensored data least squares estimator. Lai and Ying (1994) have proposed an M-estimator for censored data that is a generalization of the Buckley- James estimator. In this paper we discuss a weighted least squares algorithm for computing these M-estimates and compare the performance of two Huber M-estimators with the Buckley-James estimator in a simulation study. We find that the Huber M-estimators perform more robustly for a broad range of censoring and error distributions.     

Inference in the Pareto distribution based on progressive Type II censoring with random removals

This study considers the estimation problem for the Pareto distribution based on progressive Type II censoring with random removals. The number of units removed at each failure time has a discrete uniform distribution. We are going to use the maximum likelihood method to obtain the estimator of parameter. The expectation and variance of the maximum likelihood estimator will be derived. The expected time required to complete such an experiment will be computed. Some numerical results of expected test times are carried out for this type of progressive censoring and other sampling schemes.     

Renovating inverval-censored responses

In this note we provide a general framework for describing interval-censored samples including estimation of the magnitude and rank positions of data that have been interval-censored so as to counteract the effect of censoring. This process of sample adjustment, or renovation, allows samples to be compared graphically, using diagrams (such as boxplots) which are based on ranks. The renovation process is based on Buckley-James regression estimators for linear regression with censored data.     

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.     

Tobit regression for modeling mean survival time using data subject to multiple sources of censoring

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re‐weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large‐sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite‐sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait‐listed for kidney transplantation.     

Some results on maximum likelihood estimators of parameters of exponential distribution under type I progressive censoring with changing failure rates

In this paper the study of relative bias (RB), exact variance and mean square error (MSE) of the maximum likelihood estimators of the exponential distribution under type I progressive censoring with changing failure rates is considered. A minimum mean square error (MMSE) estimator for the parameter at each stage is proposed. The numerical evalution of their relative performance is made for selected values of n and p. Further results concerning group-censoring, total expected waiting time and optimal spacings of the times of censoring are derived and results obtained by Kendell and Anderson (1971) are deduced as special cases.     

Dependence Estimation Over a Finite Bivariate Failure Time Region

This articleconcerns nonparametric estimation of association between bivariatefailure times. In the presence of independent right censoring,the support for failure time variates may be restricted and measuresof dependence over a finite failure time region may be of particularinterest. To this end, the reciprocal cross ratio function, weightedby the bivariate failure time density, is proposed as a summarymeasure of dependence over a failure time region. This `relativerisk' estimator is shown to be consistent and asymptoticallynormally distributed, with consistent bootstrap variance estimator.A finite-region version of Kendall's tau, which is suitable forcensored failure time data, is also proposed, and correspondingasymptotic distribution theory is noted. The accuracy of theseasymptotic approximations is studied in simulations and an illustrationis provided.     

Nonparametric Survival Estimation when Death is Reported with Delay

In disease registries there can be a delay between death of asubject and the reporting of this death to the data analyst.If researchers use the Kaplan-Meier estimator and implicitlyassumed that subjects who have yet to have death reported arestill alive, i.e. are censored at the time of analysis, the Kaplan-Meierestimator is typically inconsistent. Assuming censoring is independentof failure, we provide a simple estimator that is consistentand asymptotically efficient. We also provide estimates of theasymptotic variance of our estimator and simulations that demonstratethe favorable performance of these estimators. Finally, we demonstrateour methods by analyzing AIDS survival data. This analysis underscoresthe pitfalls of not accounting for delay when estimating thesurvival distribution and suggests a significant reduction inbias by using our estimator.     

On progressive hybrid censored exponential distribution

In this paper, we introduce a new adaptive Type-I progressive hybrid censoring scheme, which has some advantages over the progressive hybrid censoring schemes already discussed in the literature. Based on an adaptive Type-I progressively hybrid censored sample, we derive the exact distribution of the maximum-likelihood estimator (MLE) of the mean lifetime of an exponential distribution as well as confidence intervals for the failure rate using exact distribution, asymptotic distribution, and three parametric bootstrap resampling methods. Furthermore, we provide computational formula for the expected number of failures and investigate the performance of the point and interval estimation for the failure rate in this case. An alternative simple form for the distribution of the MLE under adaptive Type-II progressive hybrid censoring scheme proposed by Ng et al. [Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme, Naval Res. Logist. 56 (2009), pp. 687–698] is obtained. Finally, from the exact distribution of the MLE, we establish the explicit expression for the Bayes risk of a sampling plan under adaptive Type-II progressive hybrid censoring scheme when a general loss function is used, and present some optimal Bayes solutions under four different progressive hybrid censoring schemes to illustrate the effectiveness of the proposed method.     

Blinded sample size re‐estimation in superiority and noninferiority trials: bias versus variance in variance estimation

The internal pilot study design allows for modifying the sample size during an ongoing study based on a blinded estimate of the variance thus maintaining the trial integrity. Various blinded sample size re‐estimation procedures have been proposed in the literature. We compare the blinded sample size re‐estimation procedures based on the one‐sample variance of the pooled data with a blinded procedure using the randomization block information with respect to bias and variance of the variance estimators, and the distribution of the resulting sample sizes, power, and actual type I error rate. For reference, sample size re‐estimation based on the unblinded variance is also included in the comparison. It is shown that using an unbiased variance estimator (such as the one using the randomization block information) for sample size re‐estimation does not guarantee that the desired power is achieved. Moreover, in situations that are common in clinical trials, the variance estimator that employs the randomization block length shows a higher variability than the simple one‐sample estimator and in turn the sample size resulting from the related re‐estimation procedure. This higher variability can lead to a lower power as was demonstrated in the setting of noninferiority trials. In summary, the one‐sample estimator obtained from the pooled data is extremely simple to apply, shows good performance, and is therefore recommended for application. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Tests of fit for the Gumbel distribution: EDF-based tests against entropy-based tests

In this article, we propose some tests of fit based on sample entropy for the composite Gumbel (Extreme Value) hypothesis. The proposed test statistics are constructed using different entropy estimates. Through a Monte Carlo simulation, critical values of the test statistics for various sample sizes are obtained. Since the tests based on the empirical distribution function (EDF) are commonly used in practice, the power values of the entropy-based tests with those of the EDF tests are compared against various alternatives and different sample sizes. Finally, two real data sets are modeled by the Gumbel distribution.KEYWORDS: Entropy estimator, Gumbel distribution, Monte Carlo simulation, test power     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

On Nonparametric Maximum Likelihood Estimations of Multivariate Distribution Function Based on Interval-Censored Data

This article considers the nonparametric maximum likelihood estimator (NPMLE) of a joint distribution function when the multivariate failure times of interest are interval-censored. With different types of interval censoring mechanism, the NPMLE's of the multivariate distribution function are studied and the strong consistency for the NPMLEs is obtained in terms of a self-consistency equation. Furthermore, the convergence rate of the estimator is given, which depends on the types of interval censoring mechanism.     

The effect of different data structures,sample sizes on model fit measures

In this study, our aim was to investigate the changes of different data structures and different sample sizes on the structural equation modeling and the influence of these factors on the model fit measures. Examining the created structural equation modeling under different data structures and sample sizes, the evaluation of model fit measures were performed with a simulation study. As a result of the simulation study, optimization and negative variance estimation problems have been encountered depending on the sample size and changing correlations. It was observed that these problems disappeared either by increasing the sample size or the correlations between the variables in factor. For upcoming studies, the choice of RMSEA and IFI model fit measures can be suggested in all sample sizes and the correlation values for data sets are ensured the multivariate normal distribution assumption.     

Estimation from Censored Data with Incomplete Information

Phillips and Sweeting [J. R. Statist. Soc. B 58 (1996) 775–783.] considered estimation of the parameter of the exponential distribution with censored failure time data when there is incomplete knowledge of the censoring times. It was shown that, under particular models for the censoring mechanism and censoring errors, it will usually be safe to ignore such errors provided they are not expected to be too large. A flexible model is introduced which includes the extreme cases of no censoring errors and no information on the censoring values. The effect of alternative assumptions about knowledge of the censoring values on the estimation of failure rate is investigated.