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Escarela, G. y Hernández, A. (2009). Modelado de Parejas Aleatorias Usando Cópulas (Modelling Random Couples Using Copulas). Revista Colombiana de Estadística, 32:(1) 33-58. ISSN: 0120-1751.  |
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Las cópulas se han convertido en una herramienta útil para el modelado
multivariado tanto estocástico como estadístico. En este artículo se revisan
propiedades fundamentales de las cópulas que permitan caracterizar la estructura
de dependencia de familias de distribución bivariadas definidas por
la cópula. También se describen algunas clases de cópulas, enfatizando en la
importancia de la cópula Gaussiana y la familia Arquimediana. Se resalta la
utilidad de las cópulas para el modelado de parejas de variables aleatorias
continuas y el de las discretas. La aplicación de la cópula se ilustra con la
construcción de modelos de regresión de Markov de primer orden para respuestas
no Gaussianas. |
| Abstract: |
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Copulas have become a useful tool for the multivariate modelling in both
stochastics and statistics. In this article, fundamental properties that allow
the characterization of the dependence structure of families of the bivariate
distributions defined by the copula are reviewed. Also, the importance of
both the Gaussian copula and the Archimedean family is emphasized while
some classes of copulas are described. The usefulness for modelling either
discrete or continuous random couples is highlighted. The construction of
first-order Markov regression models for non-Gaussian responses illustrates
the application of the copula. |
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Dependencia; Cópula; Medida de asociación; [Estadística
Aplicada];  de Kendall;  de Spearman; Correlación serial |
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Dependence; Copula; Measure of association; [Applied Statistics];
Kendall ; Spearman ; Serial correlation |
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Hernandez-Quintero, A.; Dupuy, J.-F. and Escarela, G. (2009). Analysis of a Semi-Parametric Mixture Model for Competing Risks. Annals of the Institute of Statistical Mathematics. DOI: 10.1007/s10463-009-0229-1, ISSN: 0020-3157, eISSN: 1572-9052.  |
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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. |
| Keywords: |
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Censored failure time data; Competing risks; Large-sample properties; Maximum likelihood estimation; Mixture model; Multinomial logistic; Proportional hazards model |
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Escarela, G.; Pérez-Ruíz, L.C. and Bowater, R. (2009). A Copula-Based Markov Chain Model for the Analysis of Binary Longitudinal Data. Journal of Applied Statistics, 36:(6) 647-657. DOI: 10.1080/02664760802499287, ISSN: 0266-4763, eISSN: 1360-0532. |
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A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods. |
| Keywords: |
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copula; discrete time series; Markov regression models; maximum likelihood; probit regression model; serial correlation |
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Escarela,
G. and Bowater, R. (2008). Fitting a Semi-Parametric Mixture
Model for Competing Risks in Survival Data. Communications in
Statistics: Theory and Methods, 37:(2) 277-293. DOI: 10.1080/03610920701649134, ISSN: 0361-0926, eISSN: 1532-415X. |
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A model for survival analysis is studied that is relevant for samples which are subject to multiple types of failure. In comparison with a more standard approach, through the appropriate use of hazard functions and transition probabilities, the model allows for a more accurate study of cause-specific failure with regard to both the timing and type of failure. A semiparametric specification of a mixture model is employed that is able to adjust for concomitant variables and allows for the assessment of their effects on the probabilities of eventual causes of failure through a generalized logistic model, and their effects on the corresponding conditional hazard functions by employing the Cox proportional hazards model. A carefully formulated estimation procedure is presented that uses an EM algorithm based on a profile likelihood construction. The methods discussed, which could also be used for reliability analysis, are applied to a prostate cancer data set. |
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EM algorithm; Event-specific hazard; Multiple decrement data; Product-limit estimate; Split-population models |
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Dupuy,
J.-F. and Escarela, G. (2007). Modélisation
de Risques Concurrents par un Modèle de Mélange Semi-Paramétrique
(Modeling Competing Risks with a Semi-Parametric Mixture Model). Comptes
Rendus Mathematique, 344:(10) 641-644. DOI: 10.1016/j.crma.2007.03.031, ISSN: 1631-073X. |
| Résumé: |
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Nous nous intéressons à un modèle de mélange semi-paramétrique proposé dans la littérature pour modéliser des risques concurrents dans un contexte de durées de vie censurées à droite et observées avec des variables explicatives. Nous montrons la convergence forte des estimateurs qui ont été proposés pour estimer les paramètres de ce modèle. |
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We consider a semi-parametric mixture model for competing risks in right-censored survival data with explanatory variables. We prove strong consistency of estimators of the parameters in this model. |
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Escarela,
G., Mena, R. and Castillo-Morales, A. (2006). A Flexible Class
of Parametric Transition Regression Models Based on Copulas: Application
to Pollomyelitis Incidence. Statistical Methods in Medical Research, 15:(6)
593-609. DOI: 10.1177/0962280206070645, ISSN: 0962-2802, eISSN: 1477-0334. |
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This paper presents an extension of a general parametric class of transitional models of order p. In these models, the conditional distribution of the current observation, given the present and past history, is a mixture of conditional distributions, each of them corresponding to the current observation, given each one of the p-lagged observations. Such conditional distributions are constructed using bivariate copula models which allow for a rich range of dependence suitable to model non-Gaussian time series. Fixed and time varying covariates can be included in the models. These models have the advantage of straightforward construction and estimation for the analysis of time series and more general longitudinal data. A poliomyelitis incidence data set is used to illustrate the proposed methods, contrary to other researches’ conclusions whose methods are mainly based on linear models, we find significant evidence of a decreasing trend in polio infection after accounting for seasonality. |
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Escarela,
G. and Carrière, J. (2006). A Bivariate Model of Claim
Frequencies and Severities. Journal of Applied Statistics, 12:(8) 867-883. DOI: 10.1080/02664760600743969, ISSN: 0266-4763, eISSN: 1360-0532. |
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Bivariate claim data come from a population that consists of insureds who may claim either one, both or none of the two types of benefits covered by a policy. In the present paper, we develop a statistical procedure to fit bivariate distributions of claims in presence of covariates. This allows for a more accurate study of insureds' choice and size in the frequency and severity of the two types of claims. A generalised logistic model is employed to examine the frequency probabilities, whilst the three parameter Burr distribution is suggested to model the underlying severity distributions. The bivariate copula model is exploited in such a way that it allows us to adjust for a range of frequency dependence structures; a method for assessing the adequacy of the fitted severity model is outlined. A health claims dataset illustrates the methods; we describe the use of orthogonal polynomials for characterising the relationship between age and the frequency and severity models. |
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Bivariate loss distribution; Frank's copula; Survival copula; Burr regression; Diagnostics |
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Escarela,
G. and Carrière, J. (2003). Fitting Competing Risks
with an Assumed Copula. Statistical Methods in Medical Research,
12:(2) 333-349. DOI: 10.1191/0962280203sm335ra, ISSN: 0962-2802, eISSN: 1477-0334. |
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We propose a fully parametric model for the analysis of competing risks data where the types of failure may not be independent. We show how the dependence between the cause-specific survival times can be modelled with a copula function. Features include: identifiability of the problem; accessible understanding of the dependence structures; and flexibility in choosing marginal survival functions. The model is constructed in such a way that it allows us to adjust for concomitant variables and for a dependence parameter to assess the effects of these on each marginal survival model and on the relationship between the causes of death. The methods are applied to a prostate cancer data set. We find that, with the copula model, more accurate inferences are obtained than with the use of a simpler model such as the independent competing risks approach. |
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Escarela,
G.; Francis, B. and Soothill, K. (2000). Competing Risks,
Persistence and Desistance in Analyzing Recidivism. Journal of
Quantitative Criminology, 16:(4) 385-414. DOI: 10.1023/A:1007586031274, ISSN: 0748-4518, eISSN: 1573-7799. |
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A statistical procedure is developed to analyze recidivism in samples whichare subject to the presence of desisters and to multiple modes ofreconviction. This allows for a more accurate study of individuals'transition and hazard in the type and timing of offenses following aspecific type of conviction. The use of a nonparametric approach forinvestigating failure in the presence of other acting causes is shown;initial estimators of the probabilities of reconviction for different typesof offenses are obtained, and the method can be used both to display thedata and to choose an appropriate parametric family for the survivaltimes. An exponential mixture model for competing risks is presented insuch a way that it allows us to adjust for concomitant variables and toassess their effects on the probabilities both of reconviction forpredetermined types of offenses and desistance and of the hazards ofreconviction; a method for assessing calibration of predicted survivalprobabilities is suggested. A 21-year follow-up of persons convicted ofindecent assault on a female in 1973 illustrates the methods; we find ahigh probability of sexual reconviction for individuals with previoussexual convictions and evidence of diversity and a raised hazard ofreconviction for young chronic offenders. |
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survival analysis; event-specific hazard; split population models; sex offenders; indecent assault |
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Soothill, K.; Francis, B. and Escarela, G. (1999). White-Collars and Black Sheep: A Twenty-Year Criminological Follow-Up
of White-Collar Ex-Offenders. The Australian and New Zealand Journal
of Criminology, 32:(3) 303-314. ISSN: 0004-8658, eISSN: 0157-1532. |
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This paper describes a further analysis of the outcome o a twenty-year criminological follow-up of a consecutive series of 348 male ex-offenders seeking white-collar employment who were offered the services of a specialist employment agency (APEX) in the early 1970s. This analysis shows the value of a more sophisticated statistical analysis which combines survival analysis with smoothing models: this provides new insights into the theoretical understanding of the data. There is evidence that remaining in contact with the organisation, irrespective of whether a suitable job is found, benefits those with around four to twelve previous convictions. |
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Abstracts (Articles) 