Exact inference for joint Type-I hybrid censoring model with exponential competing risks data

Assuming that the failure time under different risk factors follows the independent exponential distribution, a joint model under Type-I hybrid censoring is addressed in detail. Based on the Maximum likelihood estimates (MLEs) of unknown parameters, we obtain exact distributions of MLEs by using the moment generating function (MGF). Confidence intervals (CIs) of parameters are constructed through both the exact method and the parametric bootstrap method. Then we compare the performances of different methods by Monte Carlo simulations. Finally, the validity of the proposed models and methods are demonstrated by a numerical example.     

A non-parametric competing risks model for manpower planning

One of the most important variables for manpower planners is duration until a specified event occurs. This is frequently the completed length of service until leaving a job, but may also include such variables as length of service in a grade until promotion, or length of a spell of withdrawal from the labour force. In this paper we develop non-parametric maximum likelihood estimators for the survivor functions of length of stay in a grade until leaving for a number of different possible destinations. Since the data are statistically incomplete, including right censored and left truncated durations, as well as complete durations, we must modify the competing risks theory in the biostatistical literature to take such incompleteness into account. Right censored durations arise when the individual is still in the grade when data collection ceases and left truncated durations when the individual is already in service when data collection commences. The competing risks model is fitted to data for Northern Ireland nursing service and used to predict staff flows between grades. We may thus estimate future movements within the system and predict the future manpower stocks.     

Analysis of a semiparametric mixture model for competing risks

Semiparametric mixture regression models have recently been proposed to model competing risks data in survival analysis. In particular, Ng and McLachlan (Stat Med 22:1097–1111, 2003) and Escarela and Bowater (Commun Stat Theory Methods 37:277–293, 2008) have investigated the computational issues associated with the nonparametric maximum likelihood estimation method in a multinomial logistic/proportional hazards mixture model. In this work, we rigorously establish the existence, consistency, and asymptotic normality of the resulting nonparametric maximum likelihood estimators. We also propose consistent variance estimators for both the finite and infinite dimensional parameters in this model.     

The signal model: A model for competing risks of opportunistic maintenance

This paper presents a competing risks reliability model for a system that releases signals each time its condition deteriorates. The released signals are used to inform opportunistic maintenance. The model provides a framework for the determination of the underlying system lifetime from right-censored data, without requiring explicit assumptions about the type of censoring to be made. The parameters of the model are estimated from observational data by using maximum likelihood estimation. We illustrate the estimation process through a simulation study. The proposed signal model can be used to support decision-making in optimising preventive maintenance: at a component level, estimates of the underlying failure distribution can be used to identify the critical signal that would trigger maintenance of the individual component; at a multi-component system level, accurate estimates of the component underlying lifetimes are important when making general maintenance decisions. The benefit of good estimation from censored data, when adequate knowledge about the dependence structure is not available, may justify the additional data collection cost in cases where full signal data is not available.     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

Objective Bayesian analysis for competing risks model with Wiener degradation phenomena and catastrophic failures

In this paper, the objective Bayesian method is applied to investigate the competing risks model involving both catastrophic and degradation failures. By modeling soft failure as the Wiener degradation process, and hard failures as a Weibull distribution, we obtain the noninformative priors (Jefferys prior and two reference priors) for the parameters. Moreover, we show that their posterior distributions have good properties and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two reference priors. Some simulation studies are conducted to illustrate the superiority of objective Bayesian method. Finally, we apply our methods to two real data examples and compare the objective Bayesian estimates with the other estimates.     

Asymptotic properties for the semiparametric regression model with randomly censored data

Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + ε, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ². It is assumed that (X,T) is independent of E. The estimators andg _n (*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that is asymptotically normal andg _n (*) is weak consistence with rateO _p(n-1/3). Project supported by China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

Determining cutoff values of prognostic factors in survival data with competing risks

In clinical studies, patients are often classified into high or low risk groups based on prognostic factors related to survival outcomes. Using maximally selected linear rank statistics, several methods have been developed to determine a cutoff value of the prognostic factor. We propose an extension of these methods for the circumstances that competing risks are encountered in conjunction with an event outcome of interest. A simulation study is carried out to demonstrate the performance of the proposed method using some commonly used measures such as bias, precision, and power. We also apply our method to two real datasets involving lung cancer and hepatocellular carcinoma, illustrating optimal determinations of cutoff values for binary decisions on prognosis.     

Approximations of some hazard rate estimators in a competing risks model

A general model involving k competing risks is studied and the hazard rates of these risks are simultaneously estimated. The estimators are strongly approximated by Gaussian processes and the limiting distribution of certain statistics are obtained.     

Nonparametric estimation of cumulative incidence functions for competing risks data with missing cause of failure

In this paper, we develop a fully nonparametric approach for the estimation of the cumulative incidence function with Missing At Random right-censored competing risks data. We obtain results on the pointwise asymptotic normality as well as the uniform convergence rate of the proposed nonparametric estimator. A simulation study that serves two purposes is provided. First, it illustrates in detail how to implement our proposed nonparametric estimator. Second, it facilitates a comparison of the nonparametric estimator to a parametric counterpart based on the estimator of Lu and Liang (2008). The simulation results are generally very encouraging.     

Concentration curve for truncated and censored data

Concentration curve is the inverse Lorenz curve. Together, they form the basis for most measures of distributional inequality. In this paper, we consider the empirical estimator of the concentration curve when the data are subjected to random left truncation and/or right censorship. Simultaneous strong Gaussian approximations for the associated Lorenz and normed concentration processes are established under appropriate assumptions. Functional laws of the iterated logarithm for the two processes are established as easy consequences. The construction provides a solid foundation for the study of functional statistics based on the two processes.     

Non-parametric estimation from simultaneous renewal–failure–degradation data with competing risks

A joint model for linear degradation and competing failure data with partial renewals is proposed. Non-parametric estimation procedures for failure intensities and failure probabilities as functions of degradation level are given. Asymptotic properties of the estimators are investigated. To cite this article: V. Bagdonavi?ius et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Histogram selection for possibly censored data

We propose in this article a unified approach to functional estimation problems based on possibly censored data. The general framework that we define allows, for instance, to handle density and hazard rate estimation based on randomly right-censored data, or regression. Given a collection of histograms, our estimation procedure consists in selecting the best histogram among that collection from the data, by minimizing a penalized least-squares type criterion. For a general collection of histograms, we obtain nonasymptotic oracle-type inequalities. Then, we consider the collection of histograms built on partitions into dyadic intervals, a choice inspired by an approximation result due to DeVore and Yu. In that case, our estimator is also adaptive in the minimax sense over a wide range of smoothness classes that contain functions of inhomogeneous smoothness. Besides, its computational complexity is only linear in the size of the sample.     

Empirical entropy for right censored data

The maximum entropy method has been widely used in many fields, such as statistical mechanics,economics, etc. Its crucial idea is that when we make inference based on partial information, we must use the distribution with maximum entropy subject to whatever is known. In this paper, we investigate the empirical entropy method for right censored data and use simulation to compare the empirical entropy method with the empirical likelihood method. Simulations indicate that the empirical entropy method gives better coverage probability than that of the empirical likelihood method for contaminated and censored lifetime data.     

Model testing for multivariate censored data

Summary. In the fields like Astronomy and Ecology, the need for proper statistical analysis of data that are censored is being increasingly recognized. Such data occur when, due to noise or other factors, instruments fail to detect low luminosities of celestial objects, or low concentrations of certain pollutants. For multivariate censored data sets there are very few distribution free methods available and researchers in the various fields often impose an assumption on the joint distribution, such as multivariate normality, and carry out parametric inferences. Under censoring, however, such parametric inferences are asymptotically wrong if the imposed assumption is incorrect. In this paper we propose a class of goodness-of-fit procedures for testing assumptions about the multivariate distribution under random censoring. The test procedures generalize Pearson's goodness-of-fit test in the sense that they are based on the concept of observed-minus-expected frequencies. The theory of the test statistic, however, differs from that for the classical Pearson test due to the accommodation of censored data. Received: 24 May 1994 / In revised form: 3 March 1996     

Exact goodness-of-fit tests for censored data

The statistic introduced in Fortiana and Grané (J R Stat Soc B 65(1):115–126, 2003) is modified so that it can be used to test the goodness-of-fit of a censored sample, when the distribution function is fully specified. Exact and asymptotic distributions of three modified versions of this statistic are obtained and exact critical values are given for different sample sizes. Empirical power studies show the good performance of these statistics in detecting symmetrical alternatives.     

Analysis of two-sample censored data using a semiparametric mixture model

In this article we study a semiparametric mixture model for the two-sample problem with right censored data. The model implies that the densities for the continuous outcomes are related by a parametric tilt but otherwise unspecified. It provides a useful alternative to the Cox （1972） proportional hazards model for the comparison of treatments based on right censored survival data. We propose an iterative algorithm for the semiparametric maximum likelihood estimates of the parametric and nonparametric components of the model. The performance of the proposed method is studied using simulation. We illustrate our method in an application to melanoma.     

Nonparametric inference under competing risks and selection-biased sampling

The aim of this paper is to carry out statistical inference in a competing risks setup when only selection-biased observation of the data of interest is available. We introduce estimators of the cumulative incidence functions and study their joint large sample behavior.