An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

The analysis of sports data, in particular football match outcomes, has always produced an immense interest among the statisticians. In this paper, we adopt the generalized Poisson difference distribution (GPDD) to model the goal difference of football matches. We discuss the advantages of the proposed model over the Poisson difference (PD) model, which was also used for the same purpose. The GPDD model, like the PD model, is based on the goal difference in each game that allows us to account for the correlation without explicitly modelling it. The main advantage of the GPDD model is its flexibility in the tails by considering shorter as well as longer tails than the PD distribution. We carry out the analysis in a Bayesian framework in order to incorporate external information, such as historical knowledge or data, through the prior distributions. We model both the mean and the variance of the goal difference and show that such a model performs considerably better than a model with a fixed variance. Finally, the proposed model is fitted to the 2012–2013 Italian Serie A football data, and various model diagnostics are carried out to evaluate the performance of the model.     

Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log-concave densities

For a multivariate random vector X = (X ₁,...,X _n ) with a log-concave density function, it is shown that the minimum min{X ₁,...,X _n } has an increasing failure rate, and the maximum max{X ₁,...,X _n } has a decreasing reversed hazard rate. As an immediate consequence, the result of Gupta and Gupta (in Metrika 53:39–49, 2001) on the multivariate normal distribution is obtained. One error in Gupta and Gupta method is also pointed out.      

A general framework for approximate sampling with an application to generating points on the boundary of bounded convex regions

We consider the problem of generating a sample of points according to some given probability distribution over some region. We give a general framework for constructing approximate sampling algorithms based on the theory of Markov chains. In particular, we show how it can be proven that a Markov chain has a limiting distribution. We apply these results to prove convergence for a class of so-called Shake-and-Bake algorithms, which can be used to approximate any absolutely continuous distribution over the boundary of a full-dimensional convex body.     

Limited information Bayesian analysis of a simultaneous equation with an autocorrelated error term and its application to the U.S. gasoline market

Using Markov Chain Monte Carlo algorithms within the limited information Bayesian framework, we estimate the parameters of the structural equation of interest and test weak exogeneity in a simultaneous equation model with white noise as well as autocorrelated error terms. A numerical example and an estimation of the supply and demand equations of the U.S. gasoline market show that if we ignore autocorrelation we obtain unreasonable posterior distributions of the parameters of interest. Also we find that the hypothesis of the asymmetric effect of the changes in oil price on the changes in gasoline price is rejected. Oil inventory has a significant negative effect on the gasoline price.