On the identifiability of copulas in bivariate competing risks models

In competing risks models, the joint distribution of the event times is not identifiable even when the margins are fully known, which has been referred to as the “identifiability crisis in competing risks analysis” (Crowder, 1991). We model the dependence between the event times by an unknown copula and show that identification is actually possible within many frequently used families of copulas. The result is then extended to the case where one margin is unknown. The Canadian Journal of Statistics 41: 291–303; 2013 © 2013 Statistical Society of Canada     

Cox regression for current status data with mismeasured covariates

Covariate measurement error problems have been extensively studied in the context of right‐censored data but less so for current status data. Motivated by the zebrafish basal cell carcinoma (BCC) study, where the occurrence time of BCC was only known to lie before or after a sacrifice time and where the covariate (Sonic hedgehog expression) was measured with error, the authors describe a semiparametric maximum likelihood method for analyzing current status data with mismeasured covariates under the proportional hazards model. They show that the estimator of the regression coefficient is asymptotically normal and efficient and that the profile likelihood ratio test is asymptotically Chi‐squared. They also provide an easily implemented algorithm for computing the estimators. They evaluate their method through simulation studies, and illustrate it with a real data example. The Canadian Journal of Statistics 39: 73–88; 2011 © 2011 Statistical Society of Canada     

The CLT under right censorship and reporting delays

We extend the central limit theorem (CLT) under right censorship to the case when at the time of analysis we may have reporting delays. Under weak moment assumptions we derive an i.i.d. representation of the estimator, from which asymptotic normality easily follows.     

A linear transformation model for multivariate interval‐censored failure time data

This paper discusses multivariate interval‐censored failure time data observed when several correlated survival times of interest exist and only interval censoring is available for each survival time. Such data occur in many fields, for instance, studies of the development of physical symptoms or diseases in several organ systems. A marginal inference approach was used to create a linear transformation model and applied to bivariate interval‐censored data arising from a diabetic retinopathy study and an AIDS study. The results of simulation studies that were conducted to evaluate the performance of the presented approach suggest that it performs well. The Canadian Journal of Statistics 41: 275–290; 2013 © 2013 Statistical Society of Canada     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Relative risk regression for current status data in case‐cohort studies

We propose using the weighted likelihood method to fit a general relative risk regression model for the current status data with missing data as arise, for example, in case‐cohort studies. The missingness probability is either known or can be reasonably estimated. Asymptotic properties of the weighted likelihood estimators are established. For the case of using estimated weights, we construct a general theorem that guarantees the asymptotic normality of the M‐estimator of a finite dimensional parameter in a class of semiparametric models, where the infinite dimensional parameter is allowed to converge at a slower than parametric rate, and some other parameters in the objective function are estimated a priori. The weighted bootstrap method is employed to estimate the variances. Simulations show that the proposed method works well for finite sample sizes. A motivating example of the case‐cohort study from an HIV vaccine trial is used to demonstrate the proposed method. The Canadian Journal of Statistics 39: 557–577; 2011. © 2011 Statistical Society of Canada     

Combined multiple testing by censored empirical likelihood

We propose a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. We prove a version of Wilks’ theorem for the multiple constrained censored empirical likelihood ratio, which provides a simple reference distribution for the test statistic of our proposed method. A useful application of the proposed method is, for example, examining the survival experience of different populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. In a simulation study of two sample survival data, we compare the proposed method of combining tests to previously developed procedures. The results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures. Statistical software is available in the R package ‘emplik’.     

Representations of moments using lower-order conditional moments

Moments and central moments of a random variable X   are expressed as integrals of functions of lower-order conditional moments and the cumulative distribution of X$X$. In particular, sample central moments of order 2k$2k$ are expressed as the sum of between groups variations, providing an analogue to the analysis of variance. Similar expressions are obtained for the expectations of real-valued and measurable functions of X$X$.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

The Harris Extended Exponential Distribution

A new lifetime distribution is proposed and studied. The Harris extended exponential is obtained from a mixture of the exponential and Harris distributions, which arises from a branching process. Several structural properties of the new distribution are discussed, including moments, generating function and order statistics. The new distribution can model data with increasing or decreasing failure rate. The shape of the hazard rate function is controlled by one of the added parameters in an uncomplicated manner. An application to a real dataset illustrates the usefulness of the new distribution.     

Additive hazards regression with censoring indicators missing at random

In this article, the authors consider a semiparametric additive hazards regression model for right‐censored data that allows some censoring indicators to be missing at random. They develop a class of estimating equations and use an inverse probability weighted approach to estimate the regression parameters. Nonparametric smoothing techniques are employed to estimate the probability of non‐missingness and the conditional probability of an uncensored observation. The asymptotic properties of the resulting estimators are derived. Simulation studies show that the proposed estimators perform well. They motivate and illustrate their methods with data from a brain cancer clinical trial. The Canadian Journal of Statistics 38: 333–351; 2010 © 2010 Statistical Society of Canada     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

On the incidence–prevalence relation and length‐biased sampling

For many diseases, logistic constraints render large incidence studies difficult to carry out. This becomes a drawback, particularly when a new study is needed each time the incidence rate is investigated in a new population. By carrying out a prevalent cohort study with follow‐up it is possible to estimate the incidence rate if it is constant. The authors derive the maximum likelihood estimator (MLE) of the overall incidence rate, λ, as well as age‐specific incidence rates, by exploiting the epidemiologic relationship, (prevalence odds) = (incidence rate) × (mean duration) (P/[1 ? P] = λ × µ). The authors establish the asymptotic distributions of the MLEs and provide approximate confidence intervals for the parameters. Moreover, the MLE of λ is asymptotically most efficient and is the natural estimator obtained by substituting the marginal maximum likelihood estimators for P and µ into P/[1 ? P] = λ × µ. Following‐up the subjects allows the authors to develop these widely applicable procedures. The authors apply their methods to data collected as part of the Canadian Study of Health and Ageing to estimate the incidence rate of dementia amongst elderly Canadians. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

Moderate deviations for parameter estimation in some time inhomogeneous diffusions

We study moderate deviations for the maximum likelihood estimation of some inhomogeneous diffusions. The moderate deviation principle with explicit rate functions is obtained. Moreover, we apply our result to the parameter estimation in α$\alpha$-Wiener bridges.     

Confidence intervals for the scale parameter of exponential distribution based on Type II doubly censored samples

This paper deals with the problem of interval estimation of the scale parameter in the two-parameter exponential distribution subject to Type II double censoring. Base on a Type II doubly censored sample, we construct a class of interval estimators of the scale parameter which are better than the shortest length affine equivariant interval both in coverage probability and in length. The procedure can be repeated to make further improvement. The extension of the method leads to a smoothly improved confidence interval which improves the interval length with probability one. All improved intervals belong to the class of scale equivariant intervals.