Moments of von Mises and Fisher distributions and applications

The von Mises and Fisher distributions are spherical analogues to the Normal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth

Model Averaging in Risk Management with an Application to Futures Markets

This paper considers the problem of model uncertainty in the case of multi-asset volatility models and discusses the use of model averaging techniques as a way of dealing with the risk of inadvertently using false models in portfolio management. Evaluation of volatility models is then considered and a simple Value-at-Risk (VaR) diagnostic test is proposed for individual as well as ‘average ’ models. The asymptotic as well as the exact finite-sample distribution of the test statistic, dealing with the possibility of parameter uncertainty, are established. The model averaging idea and the VaR diagnostic tests are illustrated by an application to portfolios of daily returns on six currencies, four equity indices, four ten year government bonds and four commodities over the period 1991-2007. The empirical evidence supports the use of ‘thick’ model averaging strategies over single models or Bayesian type model averaging procedures

Multivariate Covariance Generalized Linear Models

We propose a general framework for non-normal multivariate data analysis called multivariate covariance generalized linear models (McGLMs), designed to handle multivariate response variables, along with a wide range of temporal and spatial correlation structures defined in terms of a covariance link function combined with a matrix linear predictor involving known matrices. The method is motivated by three data examples that are not easily handled by existing methods. The first example concerns multivariate count data, the second involves response variables of mixed types, combined with repeated measures and longitudinal structures, and the third involves a spatio-temporal analysis of rainfall data. The models take non-normality into account in the conventional way by means of a variance function, and the mean structure is modelled by means of a link function and a linear predictor. The models are fitted using an efficient Newton scoring algorithm based on quasi-likelihood and Pearson estimating functions, using only second-moment assumptions. This provides a unified approach to a wide variety of different types of response variables and covariance structures, including multivariate extensions of repeated measures, time series, longitudinal, spatial and spatio-temporal structures.Comment: 21 pages, 5 figure

Flexible modelling in statistics: past, present and future

In times where more and more data become available and where the data exhibit rather complex structures (significant departure from symmetry, heavy or light tails), flexible modelling has become an essential task for statisticians as well as researchers and practitioners from domains such as economics, finance or environmental sciences. This is reflected by the wealth of existing proposals for flexible distributions; well-known examples are Azzalini's skew-normal, Tukey's $g$-and-$h$, mixture and two-piece distributions, to cite but these. My aim in the present paper is to provide an introduction to this research field, intended to be useful both for novices and professionals of the domain. After a description of the research stream itself, I will narrate the gripping history of flexible modelling, starring emblematic heroes from the past such as Edgeworth and Pearson, then depict three of the most used flexible families of distributions, and finally provide an outlook on future flexible modelling research by posing challenging open questions.Comment: 27 pages, 4 figure

Generalized structured additive regression based on Bayesian P-splines

Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX