On the use of simulation methods to compute probabilities: application to the first division Spanish soccer league

We consider the problem of using the points a given team has in the First Division Spanish Soccer League to estimate its probabilities of achieving a specific objective, such as, for example, staying in the first division or playing the European Champions League. We started thinking about this specific problem and how to approach it after reading that some soccer coaches indicate that a team in the first division guarantees its staying in that division if it has a total of 42 points at the end of the regular season. This problem differs from the typical probability estimation problem because we only know the actual cumulative score a given team has at some point during the regular season. Under this setting a series of different assumptions can be made to predict the probability of interest at the end of the season. We describe the specific theoretical probability model using the multinomial distribution and, then, introduce two approximations to compute the probability of interest, as well as the exact method. The different proposed methods are then evaluated and also applied to the example that motivated them. One interesting result is that the predicted probabilities can then be dynamically evaluated by using data from the current soccer competition.Peer Reviewe

On the use of simulation methods to compute probabilities: application to the first division Spanish soccer league

We consider the problem of using the points a given team has in the First Division Spanish Soccer League to estimate its probabilities of achieving a specific objective, such as, for example, staying in the first division or playing the European Champions League. We started thinking about this specific problem and how to approach it after reading that some soccer coaches indicate that a team in the first division guarantees its staying in that division if it has a total of 42 points at the end of the regular season. This problem differs from the typical probability estimation problem because we only know the actual cumulative score a given team has at some point during the regular season. Under this setting a series of different assumptions can be made to predict the probability of interest at the end of the season. We describe the specific theoretical probability model using the multinomial distribution and, then, introduce two approximations to compute the probability of interest, as well as the exact method. The different proposed methods are then evaluated and also applied to the example that motivated them. One interesting result is that the predicted probabilities can then be dynamically evaluated by using data from the current soccer competition.Peer Reviewe

Modelización de datos longitudinales con estructuras de covarianza no estacionarias: modelos de coeficientes aleatorios frente a modelos alternativos

Un tema que ha suscitado el interés de los investigadores en datos longitudinales durante las dos últimas décadas, ha sido el desarrollo y uso de modelos paramétricos explícitos para la estructura de covarianza de los datos. Sin embargo, el análisis de estructuras de covarianza no estacionarias en el contexto de datos longitudinales no se ha realizado de forma detallada principalmente debido a que las distintas aplicaciones no hacían necesario su uso. Muchos son los modelos propuestos recientemente, pero la mayoría son estacionarios de segundo orden. Algunos de éstos, sin embargo, no son estacionarios y suficientemente flexibles, de tal forma que es posible modelizar varianzas no constantes y/o correlaciones que no sean sólo función del tiempo que separa a dos observaciones dadas. Estudiaremos algunas de estas propuestas y las compararemos con los modelos de coeficientes aleatorios, evaluando sus ventajas y desventajas e indicando cuándo su uso no es apropiado o útil. Presentaremos dos ejemplos para ilustrar el ajuste de estos modelos y los compararemos entre sí, mostrando de esta forma cómo pueden modelizarse datos longitudinales de forma efectiva y simple. En estos ejemplos, los distintos modelos alternativos, especialmente los modelos antedependientes, fueron superiores a los modelos de coeficientes aleatorios

Comparing Bayesian Spatial Conditional Overdispersion and the Besag–York–Mollié Models: Application to Infant Mortality Rates

In this paper, we review overdispersed Bayesian generalized spatial conditional count data models. Their usefulness is illustrated with their application to infant mortality rates from Colombian regions and by comparing them with the widely used Besag–York–Mollié (BYM) models. These overdispersed models assume that excess of dispersion in the data may be partially caused from the possible spatial dependence existing among the different spatial units. Thus, specific regression structures are then proposed both for the conditional mean and for the dispersion parameter in the models, including covariates, as well as an assumed spatial neighborhood structure. We focus on the case of response variables following a Poisson distribution, specifically concentrating on the spatial generalized conditional normal overdispersion Poisson model. Models were fitted by making use of the Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximation (INLA) algorithms in the specific context of Bayesian estimation methods.This work was supported by Ministerio de Economía y Competitividad (Spain), Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF), under research grant MTM2016-74931-P (AEI/ERDF, EU), and by the Department of Education of the Basque Government (UPV/EHU Econometrics Research Group) under research grant IT-1359-19

Bayesian structured antedependence model proposals for longitudinal data

An important problem in Statistics is the study of longitudinal data taking into account the effect of other explanatory variables, such as treatments and time and, simultaneously, the incorporation into the model of the time dependence between observations on the same individual. The latter is specially relevant in the case of nonstationary correlations, and nonconstant variances for the different time point at which measurements are taken. Antedependence models constitute a well known commonly used set of models that can accommodate this behaviour. These covariance models can include too many parameters and estimation can be a complicated optimization problem requiring the use of complex algorithms and programming. In this paper, a new Bayesian approach to analyse longitudinal data within the context of antedependence models is proposed. This innovative approach takes into account the possibility of having nonstationary correlations and variances, and proposes a robust and computationally efficient estimation method for this type of data. We consider the joint modelling of the mean and covariance structures for the general antedependence model, estimating their parameters in a longitudinal data context. Our Bayesian approach is based on a generalization of the Gibbs sampling and Metropolis-Hastings by blocks algorithm, properly adapted to the antedependence models longitudinal data settings. Finally, we illustrate the proposed methodology by analysing several examples where antedependence models have been shown to be useful: the small mice, the speech recognition and the race data sets

Bayesian joint modelling of the mean and covariance structures for normal longitudinal data

We consider the joint modelling of the mean and covariance structures for the general antedependence model, estimating their parameters and the innovation variances in a longitudinal data context. We propose a new and computationally efficient classic estimation method based on the Fisher scoring algorithm to obtain the maximum likelihood estimates of the parameters. In addition, we also propose a new and innovative Bayesian methodology based on the Gibbs sampling, properly adapted for longitudinal data analysis, a methodology that considers linear mean structures and unrestrictedcovariance structures for normal longitudinal data. We illustrate the proposed methodology and study its strengths and weaknesses by analyzing two examples, the race and the cattle data sets

Two-stage nonparametric regression for longitudinal data

In the analysis of longitudinal data it is of main interest to investigate the existence of group and individual effects under correlated observations across time. In this paper, we develop a nonparametric two-step procedure that enables us to estimate group effects under a very general form of correlation across time. Moreover, we propose several methods to estimate the bandwidth and show their asymptotyc optimality. Since the asymptotic distribution is untractable, we develop a randomization test that is suitable for testing the group effects. Finally, we apply the estimation procedure, the bandwidth selection criteria and the randomization test to the data from the Iowa Cochlear Implant Project

Two-Stage Nonparametric Regression for Longitudinal Data

In the analysis of longitudinal data it is of main interest to investigate the existence of group and individual effects under correlated observations across time. In this paper, we develop a nonparametric two-step procedure that enables us to estimate group effects under a very general form of correlation across time. Moreover, we propose several methods to estimate the bandwidth and show their asymptotyc optimality. Since the asymptotic distribution is untractable, we develop a randomization test that is suitable for testing the group effects. Finally, we apply the estimation procedure, the bandwidth selection criteria and the randomization test to the data from the Iowa Cochlear Implant Project.This work was supported by Dirección General de Enseñanza Superior del Ministerio Español de Educación y Cultura and Universidad del País Vasco (UPV/EHU) under research grant PB95-0346

Survival Analysis Using a Censored Semiparametric Regression Model

In this work we study the effect of several covariates X on a censored response variable T with unknown probability distribution. A semiparametric model is proposed to consider situations where the functional form of the effect of one or more covariates is unknown. We provide its estimation procedure and, in addition, a bootstrap technique to make inference on the parameters. An application with a real dataset is presented, as well as some simulation results, to demonstrate the good behavior of the proposed estimation process and to analyze the effect of the censorship. This new model has an important application field in reliability, survival or lifetime data analysis.censorship, Kaplan-Meier, lifetime data models, bootstrap, nonparametric estimation