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Olkin & Vaeth (1981) consider a two-way classification model with k rows and p columns in which the residuals (ei1, ..., eip) for row i are independently distributed with a multivariate normal distribution with zero means and common covariance matrix ((ij)). They concentrate their attention particularly on the situation where the row classification corresponds to k subjects, each treated in one of two or more ways; the column classification corresponds to p repetitions, perhaps in time, from each subject, probably resulting in correlated errors. Replication of subjects within treatments enables the elements of the covariance matrix to be estimated from the data, which in turn enables maximum likelihood estimates of the row and column parameters to be computed and likelihood ratio tests to be carried out without having to make assumptions about the form of the covariance matrix. The careful analysis of data of this kind has been the subject of numerous papers, Wishart (1938) perhaps being the first to concentrate on this topic. Despite this, however, the use of erroneous methods in published material is widespread and this provided the stimulus for the publication of our earlier paper (Rowell & Walters, 1976), and also for the present communication. We note here that there is confusion in Olkin & Vaeth's row/column terminology in their worked example: the rows in their tables are referred to as columns in the text, and vice versa. In what follows, when referring to their numerical example, the rows classification will refer to the high/low factor, and the columns classification to the time factor.

Summary Exposure mappings facilitate investigations of complex causal effects when units interact in experiments. Current methods require experimenters to use the same exposure mappings to define the effect of interest and to impose assumptions on the interference structure. However, the two roles rarely coincide in practice, and experimenters are forced to make the often questionable assumption that their exposures are correctly specified. This paper argues that the two roles exposure mappings currently serve can, and typically should, be separated, so that exposures are used to define effects without necessarily assuming that they are capturing the complete causal structure in the experiment. The paper shows that this approach is practically viable by providing conditions under which exposure effects can be precisely estimated when the exposures are misspecified. Some important questions remain open.

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights "minimal weights" and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance., 41 pages

The envelope model allows efficient estimation in multivariate linear regression. In this paper, we propose the sparse envelope model, which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients. The envelope estimator is consistent but not sparse, and in many situations it is important to identify the response variables for which the regression coefficients are zero. The sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model. Response variable selection arises naturally in many applications, but has not been studied as thoroughly as predictor variable selection. In this paper, we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts. In response variable selection, even if a response has zero coefficients, it should still be retained to improve the estimation efficiency of the nonzero coefficients. This is different from the practice in predictor variable selection. We establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator.

In a recent paper Dette et al. (2014) introduced optimal design problems for dose finding studies with an active control. These authors concentrated on regression models with normal distributed errors (with known variance) and the problem of determining optimal designs for estimating the smallest dose, which achieves the same treatment effect as the active control. This paper discusses the problem of designing active-controlled dose finding studies from a broader perspective. In particular, we consider a general class of optimality criteria and models arising from an exponential family, which are frequently used analyzing count data. We investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control. Optimal designs are constructed for several situations and the differences arising from different distributional assumptions are investigated in detail. In particular, our results are applicable for constructing optimal experimental designs to analyze active-controlled dose finding studies with discrete data, and we illustrate the efficiency of the new optimal designs with two recent examples from our consulting projects.

This paper proposes a unified framework to characterize the rate function of a recurrent event process through shape and size parameters. In contrast to the intensity function, which is the event occurrence rate conditional on the event history, the rate function is the occurrence rate unconditional on the event history, and thus it can be interpreted as a population-averaged count of events in unit time. In this paper, shape and size parameters are introduced and used to characterize the association between the rate function λ(⋅) and a random variable X. Measures of association between X and λ(⋅) are defined via shape- and size-based coefficients. Rate-independence of X and λ(⋅) is studied through tests of shape-independence and size-independence, where the shape- and size-based test statistics can be used separately or in combination. These tests can be applied when X is a covariable possibly correlated with the recurrent event process through λ(⋅) or, in the one-sample setting, when X is the censoring time at which the observation of N(⋅) is terminated. The proposed tests are shape- and size-based, so when a null hypothesis is rejected, the test results can serve to distinguish the source of violation.

This paper deals with the design of the likelihood continual reassessment method, which is an increasingly widely used model-based method for dose-finding studies. It is common to implement the method in a two-stage approach, whereby the model-based stage is activated after an initial sequence of patients has been treated. While this two-stage approach is practically appealing, it lacks a theoretical framework, and it is often unclear how the design components should be specified. This paper develops a general framework based on the coherence principle, from which we derive a design calibration process. A real clinical-trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner, while offering competitive operating characteristics. We explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose-toxicity model.

Summary For the case with a single causal variable, Dawid et al. (2014) defined the probability of causation, and Pearl (2000) defined the probability of necessity to assess the causes of effects. For a case with multiple causes that could affect each other, this paper defines the posterior total and direct causal effects based on the evidence observed for post-treatment variables, which could be viewed as measurements of causes of effects. Posterior causal effects involve the probabilities of counterfactual variables. Thus, as with the probability of causation, the probability of necessity and direct causal effects, the identifiability of posterior total and direct causal effects requires more assumptions than the identifiability of traditional causal effects conditional on pre-treatment variables. We present assumptions required for the identifiability of posterior causal effects and provide identification equations. Further, when the causal relationships between multiple causes and an endpoint can be depicted by causal networks, we can simplify both the required assumptions and the identification equations of the posterior total and direct causal effects. Finally, using numerical examples, we compare the posterior total and direct causal effects with other measures for evaluating the causes of effects and the population attributable risks.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics.

This paper considers one-step targeted maximum likelihood estimation methodology for multi-dimensional causal parameters in general survival and competing risks settings where event times take place on the positive real line ℝ+ and are subject to right-censoring. We focus on effects of baseline treatment decisions possibly confounded by pre-treatment covariates, but remark that our work generalizes to settings with time-varying treatment regimes and time-dependent confounding. We point out two overall contributions of our work. First, our methods can be used to obtain simultaneous inference for treatment effects on multiple absolute risks in competing risks settings. Second, our methods can be used to achieve inference for the full survival curve, or a full absolute risk curve, across time. The one-step targeted maximum likelihood procedure is based on a one-dimensional universal least favourable submodel for each cause-specific hazard that we implement in recursive steps along a corresponding non-universal multivariate least favourable submodel. Our empirical study demonstrates the practical use of the methods.

SUMMARY Rooted and ranked phylogenetic trees are mathematical objects that are useful in modelling hierarchical data and evolutionary relationships with applications to many fields such as evolutionary biology and genetic epidemiology. Bayesian phylogenetic inference usually explores the posterior distribution of trees via Markov Chain Monte Carlo methods. However, assessing uncertainty and summarizing distributions remains challenging for these type of structures. While labelled phylogenetic trees have been extensively studied, relatively less literature exists for unlabelled trees which are increasingly useful, for example when % Similarly, in many instances, one seeks to summarize samples of trees obtained with different methods, or from different samples and environments, and wishes to assess the stability and generalizability of these summaries. In our paper, we exploit recently proposed distance metrics of unlabelled ranked binary trees and unlabelled ranked genealogies, or trees equipped with branch lengths, to define the Fréchet mean, variance, and interquartile sets as summaries of these tree distributions. We provide an efficient combinatorial optimization algorithm for computing the Fréchet mean of a sample or of distributions on unlabelled ranked tree shapes and unlabelled ranked genealogies. We show the applicability of our summary statistics for studying popular tree distributions and for comparing the SARS-CoV-2 evolutionary trees across different locations during the COVID-19 epidemic in 2020. Our current implementations are publicly available at github.com/RSamyak/fmatrix

This paper considers benchmarking issues in the context of small area estimation. We find optimal estimators within the class of benchmarked linear estimators under linear constraints. This extends existing results for external and internal benchmarking, and also links the two. Necessary and sufficient conditions for self-benchmarking are found for an augmented model. Most results of this paper are found using ideas of orthogonal projection Copyright 2013, Oxford University Press.

This paper derives the asymptotic null distribution of a quasilikelihood ratio test statistic for an autoregressive moving average model against its threshold extension. The null hypothesis is that of no threshold, and the error term could be dependent. The asymptotic distribution is rather complicated, and all existing methods for approximating a distribution in the related literature fail to work. Hence, a novel bootstrap approximation based on stochastic permutation is proposed in this paper. Besides being robust to the assumptions on the error term, our method enjoys more flexibility and needs less computation when compared with methods currently used in the literature. Monte Carlo experiments give further support to the new approach, and an illustration is reported. Copyright 2011, Oxford University Press.

Summary We consider the problem of estimating and inferring treatment effects in randomized experiments. In practice, stratified randomization, or more generally, covariate-adaptive randomization, is routinely used in the design stage to balance treatment allocations with respect to a few variables that are most relevant to the outcomes. Then, regression is performed in the analysis stage to adjust the remaining imbalances to yield more efficient treatment effect estimators. Building upon and unifying recent results obtained for ordinary-least-squares adjusted estimators under covariate-adaptive randomization, this paper presents a general theory of regression adjustment that allows for model mis-specification and the presence of a large number of baseline covariates. We exemplify the theory on two lasso-adjusted treatment effect estimators, both of which are optimal in their respective classes. In addition, nonparametric consistent variance estimators are proposed to facilitate valid inferences, which work irrespective of the specific randomization methods used. The robustness and improved efficiency of the proposed estimators are demonstrated through numerical studies.

In recent years, several authors have investigated response-adaptive allocation rules for comparative clinical trials, in order to favour, at each stage of the trial, the treatment that appears to be best. In this paper, we define admissible allocations, namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion, reflecting the inferential properties of the experiment, and the proportion of patients assigned to the best treatment or treatments; we survey existing designs from this viewpoint. We also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria, with weights that depend on the actual treatment effects. This compound criterion leads to a locally optimal allocation that can be targeted by some response-adaptive randomization rule. The paper mainly deals with the case of two treatments, but the suggested methodology is shown to extend to more than two. Copyright 2010, Oxford University Press.

SummaryIn this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U-type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings.

A Gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non-Gaussian. Our paper presents a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous marginals, and generalises Gaussian graphical models to the Gaussian copula framework. Posterior inference is carried out using a novel and efficient simulation method. The methods in the paper are applied to simulated and real data.

This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias.

SUMMARY Properties of the periodogram are seldom studied in the setting of nonparametric regression, although that is the context in which the periodogram is widely applied in astronomy. There it is a competitor with more recent least-squares methods. The periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem. However, as we show in the present paper, it also has drawbacks. The estimator that it produces has somewhat higher variance than its least-squares counterpart, and a periodogram-based approach is more prone to suffer difficulties caused by periodicity of the observation schedule. While the periodogram remains a very attractive tool, the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting. This aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset.

This paper is concerned with the problem of comparing the population means of two groups of independent observations. An approximate randomization test procedure based on the test statistic of Chen and Qin (2010) is proposed. The asymptotic behavior of the test statistic as well as the randomized statistic is studied under weak conditions. In our theoretical framework, observations are not assumed to be identically distributed even within groups. No condition on the eigenstructure of the covariance matrices is imposed. And the sample sizes of the two groups are allowed to be unbalanced. Under general conditions, all possible asymptotic distributions of the test statistic are obtained. We derive the asymptotic level and local power of the approximate randomization test procedure. Our theoretical results show that the proposed test procedure can adapt to all possible asymptotic distributions of the test statistic and always has correct test level asymptotically. Also, the proposed test procedure has good power behavior. Our numerical experiments show that the proposed test procedure has favorable performance compared with several alternative test procedures., Comment: 52 pages. To appear in Biometrika

Summary This paper develops a functional hybrid factor regression modelling framework to handle the heterogeneity of many large-scale imaging studies, such as the Alzheimer’s disease neuroimaging initiative study. Despite the numerous successes of those imaging studies, such heterogeneity may be caused by the differences in study environment, population, design, protocols or other hidden factors, and it has posed major challenges in integrative analysis of imaging data collected from multicentres or multistudies. We propose both estimation and inference procedures for estimating unknown parameters and detecting unknown factors under our new model. The asymptotic properties of both estimation and inference procedures are systematically investigated. The finite-sample performance of our proposed procedures is assessed by using Monte Carlo simulations and a real data example on hippocampal surface data from the Alzheimer’s disease study.

Summary Relational arrays represent measures of association between pairs of actors, often in varied contexts or over time. Trade flows between countries, financial transactions between individuals, contact frequencies between school children in classrooms and dynamic protein-protein interactions are all examples of relational arrays. Elements of a relational array are often modelled as a linear function of observable covariates. Uncertainty estimates for regression coefficient estimators, and ideally the coefficient estimators themselves, must account for dependence between elements of the array, e.g., relations involving the same actor. Existing estimators of standard errors that recognize such relational dependence rely on estimating extremely complex, heterogeneous structure across actors. This paper develops a new class of parsimonious coefficient and standard error estimators for regressions of relational arrays. We leverage an exchangeability assumption to derive standard error estimators that pool information across actors, and are substantially more accurate than existing estimators in a variety of settings. This exchangeability assumption is pervasive in network and array models in the statistics literature, but not previously considered when adjusting for dependence in a regression setting with relational data. We demonstrate improvements in inference theoretically, via a simulation study, and by analysis of a dataset involving international trade.

Our paper proposes adaptive Monte Carlo sampling schemes for Bayesian variable selection in linear regression that improve on standard Markov chain methods. We do so by considering Metropolis-Hastings proposals that make use of accumulated information about the posterior distribution obtained during sampling. Adaptation needs to be done carefully to ensure that sampling is from the correct ergodic distribution. We give conditions for the validity of an adaptive sampling scheme in this problem, and for simulating from a distribution on a finite state space in general, and suggest a class of adaptive proposal densities which uses best linear prediction to approximate the Gibbs sampler. Our sampling scheme is computationally much faster per iteration than the Gibbs sampler, and when this is taken into account the efficiency gains when using our sampling scheme compared to alternative approaches are substantial in terms of precision of estimation of posterior quantities of interest for a given amount of computation time. We compare our method with other sampling schemes for examples involving both real and simulated data. The methodology developed in the paper can be extended to variable selection in more general problems.

The recent paper by Peng & Yao (2003) gave an interesting extension of least absolute deviation estimation to generalised autoregressive conditional heteroscedasticity, GARCH, time series models. The asymptotic distributions of absolute residual autocorrelations and squared residual autocorrelations from the GARCH model estimated by the least absolute deviation method are derived in this paper. These results lead to two useful diagnostic tools which can be used to check whether or not a GARCH model fitted by using the least absolute deviation method is adequate. Some simulation experiments give further support to the asymptotic theory and a real data example is also reported.

SUMMARY This paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters. In a statistical model with unknown nuisance parameters, the efficiency of an estimator of a parameter usually increases when the nuisance para meters are known. However the opposite phenomenon can sometimes occur. In this paper, we elucidate the occurrence of this paradox by examining estimating functions. In particular, we focus on the projected estimating function, which is defined by the projection of the score function on to a given estimating function. A sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters. In addition, a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators.

SUMMARY In this paper we consider optimal design of experiments in the case of correlated obser- vations. We use and further develop the concept of design measures introduced by Paizman & Mtiller (1998) for the construction of a simple, quick and elegant design algorithm. We support the construction of this algorithm for a general correlation structure by an interpretation in terms of norms. Examples demonstrate that our results are useful for generating exact designs by sampling from the obtained design measures. Most of the literature on experimental design operates under the assumption of uncorre- lated errors and by exploiting the concept of a design measure introduced by Kiefer (1959). In this setting an optimal design measure is usually computed by an iterative algorithm and then an exact design with independent replications approximately proportional to that measure is employed. In the correlated case when replications are not allowed the implementation of design measures is not so straightforward. Recently, however, by adding virtual design-dependent noise to the process, Paizman & Muiller (1998) have introduced a way of using this popular concept, albeit requiring a very different interpretation. Furthermore, Paizman & Muller (2000) showed that solving minimisation problems in terms of these measures corresponds to computation of exact designs. These measures serve as building blocks for the methods presented in this paper, which provides a general numerical tool for screening designs. By a different type of smoothing of some important nondifferentiable terms in the algorithm, and without imposing restric- tions on the number of support points, we obtain a design measure. The relative magnitude of this measure for each support point reflects the importance of its inclusion in any exact design. This algorithm can also be interpreted in terms of certain 'information' norms as pre

SUMMARY This paper studies simultaneous feature selection and extraction in supervised and unsupervised learning. We propose and investigate selective reduced rank regression for constructing optimal explanatory factors from a parsimonious subset of input features. The proposed estimators enjoy sharp oracle inequalities, and with a predictive information criterion for model selection, they adapt to unknown sparsity by controlling both rank and row support of the coefficient matrix. A class of algorithms is developed that can accommodate various convex and nonconvex sparsity-inducing penalties, and can be used for rank-constrained variable screening in high-dimensional multivariate data. The paper also showcases applications in macroeconomics and computer vision to demonstrate how low-dimensional data structures can be effectively captured by joint variable selection and projection.

Summary Network latent space models assume that each node is associated with an unobserved latent position in a Euclidean , and such latent variables determine the probability of two nodes connecting with each other. In many applications, nodes in the network are often observed along with high-dimensional node variables, and these node variables provide important information for understanding the network structure. However, classical network latent space models have several limitations in incorporating node variables. In this paper, we propose a joint latent space model where we assume that the latent variables not only explain the network structure, but are also informative for the multivariate node variables. We develop a projected gradient descent algorithm that estimates the latent positions using a criterion incorporating both network structure and node variables. We establish theoretical properties of the estimators and provide insights into how incorporating high-dimensional node variables could improve the estimation accuracy of the latent positions. We demonstrate the improvement in latent variable estimation and the improvements in associated downstream tasks, such as missing value imputation for node variables, by simulation studies and an application to a Facebook data example.

Summary Robins (1998) introduced marginal structural models, a general class of counterfactual models for the joint effects of time-varying treatments in complex longitudinal studies subject to time-varying confounding. Robins (1998) established the identification of marginal structural model parameters under a sequential randomization assumption, which rules out unmeasured confounding of treatment assignment over time. The marginal structural Cox model is one of the most popular marginal structural models for evaluating the causal effect of time-varying treatments on a censored failure time outcome. In this paper, we establish sufficient conditions for identification of marginal structural Cox model parameters with the aid of a time-varying instrumental variable, in the case where sequential randomization fails to hold due to unmeasured confounding. Our instrumental variable identification condition rules out any interaction between an unmeasured confounder and the instrumental variable in its additive effects on the treatment process, the longitudinal generalization of the identifying condition of Wang & Tchetgen Tchetgen (2018). We describe a large class of weighted estimating equations that give rise to consistent and asymptotically normal estimators of the marginal structural Cox model, thereby extending the standard inverse probability of treatment weighted estimation of marginal structural models to the instrumental variable setting. Our approach is illustrated via extensive simulation studies and an application to estimating the effect of community antiretroviral therapy coverage on HIV incidence.

Summary In this article we develop a method based on model-X knockoffs to find conditional associations that are consistent across environments, while controlling the false discovery rate. The motivation for this problem is that large datasets may contain numerous associations that are statistically significant and yet misleading, as they are induced by confounders or sampling imperfections. However, associations replicated under different conditions may be more interesting. In fact, sometimes consistency provably leads to valid causal inferences even if conditional associations do not. Although the proposed method is widely applicable, in this paper we highlight its relevance to genome-wide association studies, in which robustness across populations with diverse ancestries mitigates confounding due to unmeasured variants. The effectiveness of this approach is demonstrated by simulations and applications to UK Biobank data.

Summary Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much attention recently. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjee’s new correlation coefficient with three established rank correlations that also facilitate consistent tests of independence, namely Hoeffding’s $D$, Blum–Kiefer–Rosenblatt’s $R$, and Bergsma–Dassios–Yanagimoto’s $\tau^*$. We compare the computational efficiency of these rank correlation coefficients in light of recent advances, and investigate their power against local rotation and mixture alternatives. Our main results show that Chatterjee’s coefficient is unfortunately rate-suboptimal compared to $D$, $R$ and $\tau^*$. The situation is more subtle for a related earlier estimator of Dette et al. (2013). These results favour $D$, $R$ and $\tau^*$ over Chatterjee’s new correlation coefficient for the purpose of testing independence.

Summary Consider a survival time $T$ that is subject to random right censoring, and suppose that $T$ is stochastically dependent on the censoring time $C$. We are interested in the marginal distribution of $T$. This situation is often encountered in practice. Consider, for example, the case where $T$ is a patient’s time to death from a certain disease. Then the censoring time $C$ could be the time until the patient leaves the study or the time until death from another cause. If the reason for leaving the study is related to the health condition of the patient, or if the patient dies from a disease that has similar risk factors to the disease of interest, then $T$ and $C$ are likely to be dependent. In this paper we propose a new model that takes such dependence into account. The model is based on a parametric copula for the relationship between $T$ and $C$, and on parametric marginal distributions for $T$ and $C$. Unlike most other authors, we do not assume that the parameter defining the copula is known. We give sufficient conditions on these parametric copulas and marginals under which the bivariate distribution of $(T,C)$ is identified. These sufficient conditions are then checked for a wide range of common copulas and marginals. We also study the estimation of the model, and carry out extensive simulations and analysis on a pancreatic cancer dataset to illustrate the proposed model and estimation procedure.