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In capture-recapture studies, the use of individual covariates has been recommended to get stable population estimates. However, some residual heterogeneity might still exist and ignoring such heterogeneity could lead to underestimating the population size ( N). In this work, we explore two new models with capture probabilities depending on both covariates and unobserved random effects, to estimate the size of a population. Inference techniques including Horvitz-Thompson estimate and confidence intervals for the population size, are derived. The selection of a particular model is carried out using the Akaike information criterion (AIC). First, we extend the random effect model of Darroch et al. (1993, Journal of American Statistical Association 88, 1137-1148) to handle unit level covariates and discuss its limitations. The second approach is a generalization of the traditional zero-truncated binomial model that includes a random effect to account for an unobserved heterogeneity. This approach provides useful tools for inference about N, since key quantities such as moments, likelihood functions and estimates of N and their standard errors have closed form expressions. Several models for the unobserved heterogeneity are available and the marginal capture probability is expressed using the Logit and the complementary Log-Log link functions. The sensitivity of the inference to the specification of a model is also investigated through simulations. A numerical example is presented. We compare the performance of the proposed estimator with that obtained under model [ABSTRACT FROM AUTHOR]

Random sums play an important role in insurance claim theory and risk models. This paper aims to explore the numerical characteristics of dependent random sums using copulas. We obtain theoretical formulae for the expectation and variance of random sums in the case of a general copula. The expectation is an extension of the Wald equality and can be used to estimate prospective value, while variance provides a theoretical guarantee for measuring the risk premium. As corollaries, the corresponding results for the compound Poisson process and random sums using Farlie-Gumbel-Morgenstern copulas have been obtained. We also conduct stochastic simulation to verify the accuracy of our theoretical results. [ABSTRACT FROM AUTHOR]

We propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location. We use copulas so that the marginal distributions and the dependence structure can be modeled independently. Dependence is modeled with a Gaussian or t-copula, so that there is an underlying latent Gaussian process. We model the marginal distributions using the skew t family. The mean, variance, and shape parameters are modeled nonparametrically as functions of location. A computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy-tailed marginal distributions is introduced. This framework provides a new set of tools for increasingly complex data collected in medical and public health studies. Our methods were motivated by and are illustrated with a state-of-the-art study of neuronal tracts in multiple sclerosis patients and healthy controls. Using the tools we have developed, we were able to find those locations along the tract most affected by the disease. However, our methods are general and highly relevant to many functional data sets. In addition to the application to one-dimensional tract profiles illustrated here, higher-dimensional extensions of the methodology could have direct applications to other biological data including functional and structural magnetic resonance imaging (MRI). [ABSTRACT FROM AUTHOR]

In life history studies, interest often lies in the analysis of the interevent, or gap times and the association between event times. Gap time analyses are challenging however, even when the length of follow-up is determined independently of the event process, because associations between gap times induce dependent censoring for second and subsequent gap times. This article discusses nonparametric estimation of the association between consecutive gap times based on Kendall's τ in the presence of this type of dependent censoring. A nonparametric estimator that uses inverse probability of censoring weights is provided. Estimates of conditional gap time distributions can be obtained following specification of a particular copula function. Simulation studies show the estimator performs well and compares favorably with an alternative estimator. Generalizations to a piecewise constant Clayton copula are given. Several simulation studies and illustrations with real data sets are also provided. [ABSTRACT FROM AUTHOR]

This paper discusses copula model selection procedures and goodness-of-fit tests under censoring. The proposed methodology is based on a comparison of nonparametric and model-based estimators of the probability integral transformation, K. New weighted estimators for K are introduced. The resulting tests are compared to an existing approach by simulation and illustrated with an example involving bleeding changes in a woman’s reproductive history. [ABSTRACT FROM PUBLISHER]

The product-limit estimator calculated from data subject to random left-truncation relies on the testable assumption of quasi-independence between the failure time and the truncation time. In this paper, we propose a model for a truncated sample of pairs (Xi,Yi) satisfying Yi > Xi. A possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas. The model also features a distribution function FX(.) and a survival distribution SY(.) associated with the marginal behaviours of X and Y in the observable region Y > X. Semiparametric estimators for these two functions are proposed; they do not make any parametric assumption about either FX(.) or SY(.). We derive an estimator for the copula parameter α based on the conditional Kendall's tau. We generalise the copula-graphic estimators of Zheng & Klein (1995) to truncated variables. The asymptotic distributions of all these estimators are then investigated. The methods are illustrated with a real dataset on HIV infection by transfusion and by simulations. [ABSTRACT FROM AUTHOR]

Assuming that the two failure times of interest with bivariate current status data follow a bivariate copula model, we propose a two‐stage estimation procedure to estimate the association parameter which is related to Kendall's tau. Asymptotic properties of the proposed semiparametric estimator show that, although the first‐stage marginal estimators have a convergence rate of only n1/3, the resulting parameter estimator still converges to a normal random variable with the usual n1/2 rate. The variance of the proposed estimator can be consistently estimated. Simulation results are presented, and a community‐based study of cardiovascular diseases in Taiwan provides an illustrative example. [ABSTRACT FROM PUBLISHER]

We consider estimation of the cross ratio in Clayton's (1978) copula in which covariates are incorporated into the marginal distributions via semiparametric accelerated life regression models. Generalisations of Oakes' (1982, 1986) concordance estimating equations yield a closed form for the association parameter under right censoring. Joint inferences about covariate effects in the marginal models and the cross ratio are obtained with U-statistics, martingales and a resampling technique for nonsmooth estimating equations. Simulating under an exponential model, we find that our procedure may provide a more precise estimate of association than other methods for proportional hazards models. A goodness-of-fit test for the assumed copula is presented and used in an analysis of a diabetic retinopathy dataset. [ABSTRACT FROM PUBLISHER]

We consider a bivariate regression model in which the error structure is arbitrary, except for equality of its marginal distributions. Two simple but inefficient approaches to analysis are (i) to work with the differences between the two components of each response and (ii) to derive working independence estimators as if the two component responses were independent. The former strategy does well if the explanatory variables are balanced over pairs and the correlation between the two components of the response is high; the latter strategy does well when the response correlation is low or the covariates are highly unbalanced. We show that combining the two estimators can give a procedure with high efficiency in all situations. We show that an adjusted marginal score can be calculated that is uncorrelated with the within-pair score, so that separate intrapair and interpair estimators can be calculated. For bivariate normal regression models our approach is identical to the usual procedure for combining intra- and interblock information. In general the approaches are not the same. Detailed calculations and a numerical example are given for a bivariate extreme value model due to Gumbel that is also important in survival analysis. Extensions from bivariate to general multivariate responses are indicated. [ABSTRACT FROM PUBLISHER]