Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Quasi ex-ante inflation forecast uncertainty

We propose a measure of the effects of monetary policy based on an analysis of the distribution of the ex-post inflation forecast uncertainty. We argue that the difference between the distributions of the ex-ante and ex-post uncertainties reflects the impact of monetary policy decisions. Using the theoretical background of the New Keynesian model with imperfect information and a monetary policy rule, we derive a proxy for ex-ante inflation uncertainty called quasi ex-ante forecast uncertainty, which is free to a certain extent of the effects of monetary policy decisions. Furthermore, we introduce the compound strength measure of monetary policy, as well as the uncertainty ratio, which approximates the impact of monetary policy on the reduction of the inflation forecast uncertainty. Our empirical results show that the greatest policy effect in reducing the inflation forecast uncertainty occurs for countries which conduct either a well-established or a relatively pure inflation targeting policy.     

Akaike's Information Criterion,Cp and Estimators of Loss for Elliptically Symmetric Distributions

In this article, we develop a modern perspective on Akaike's information criterion and Mallows's C_p for model selection, and propose generalisations to spherically and elliptically symmetric distributions. Despite the differences in their respective motivation, C_p and Akaike's information criterion are equivalent in the special case of Gaussian linear regression. In this case, they are also equivalent to a third criterion, an unbiased estimator of the quadratic prediction loss, derived from loss estimation theory. We then show that the form of the unbiased estimator of the quadratic prediction loss under a Gaussian assumption still holds under a more general distributional assumption, the family of spherically symmetric distributions. One of the features of our results is that our criterion does not rely on the specificity of the distribution, but only on its spherical symmetry. The same kind of criterion can be derived for a family of elliptically contoured distribution, which allows correlations, when considering the invariant loss. More specifically, the unbiasedness property is relative to a distribution associated to the original density.     

Ruin probabilities in multivariate risk models with periodic common shock

We propose a multidimensional risk model where the common shock affecting all classes of insurance business is arriving according to a non-homogeneous periodic Poisson process. In this multivariate setting, we derive upper bounds of Lundberg-type for the probability that ruin occurs in all classes simultaneously using the martingale approach via piecewise deterministic Markov processes theory. These results are numerically illustrated in a bivariate risk model, where the beta-shape periodic claim intensity function is considered. Under the assumption of dependent heavy-tailed claims, asymptotic bounds for the finite-time ruin probabilities associated to three types of ruin in this multivariate framework are investigated.     

Moment-based density approximations for aggregate losses

The determination of the distribution of aggregate losses is of crucial importance for an insurer. In this paper, we propose a technique for approximating the distribution of univariate and bivariate aggregate losses, which is solely based on their moments. Accordingly, this methodology can be implemented without any specific knowledge of the claim number or size distributions. The numerical examples presented herein indicate that the proposed approach constitutes a viable alternative to the commonly used recursive and FFT methods.     

Non‐parametric Analysis of Gap Times for Multiple Event Data: An Overview

Multiple event data are frequently encountered in medical follow‐up, engineering and other applications when the multiple events are considered as the major outcomes. They may be repetitions of the same event (recurrent events) or may be events of different nature. Times between successive events (gap times) are often of direct interest in these applications. The stochastic‐ordering structure and within‐subject dependence of multiple events generate statistical challenges for analysing such data, including induced dependent censoring and non‐identifiability of marginal distributions. This paper provides an overview of a class of existing non‐parametric estimation methods for gap time distributions for various types of multiple event data, where sampling bias from induced dependent censoring is effectively adjusted. We discuss the statistical issues in gap time analysis, describe the estimation procedures and illustrate the methods with a comparative simulation study and a real application to an AIDS clinical trial. A comprehensive understanding of challenges and available methods for non‐parametric analysis can be useful because there is no existing standard approach to identifying an appropriate gap time method that can be used to address research question of interest. The methods discussed in this review would allow practitioners to effectively handle a variety of real‐world multiple event data.     

Errors of misclassification in discrimination of dimensional coherent elliptic random field observations

The problem of classification of dimensional coherent elliptic random field observations into one of two populations specified by different regression mean models and common stationary scale matrix is considered, under the further assumption that the observations to be classified are dependent on the training samples. In this statistical frame, the behaviour of linear discriminant function is studied and an asymptotic expression for the distribution function of the probabilities of misclassification is derived.     

Power laws, discontinuities and regional city size distributions

Urban systems are manifestations of human adaptation to the natural environment. City size distributions are the expression of hierarchical processes acting upon urban systems. In this paper, we test the entire city size distributions for the southeastern and southwestern United States (1990), as well as the size classes in these regions for power law behavior. We interpret the differences in the size of the regional city size distributions as the manifestation of variable growth dynamics dependent upon city size. Size classes in the city size distributions are snapshots of stable states within urban systems in flux.     

Explanations of the inconsistencies in survey respondents’ forecasts

A comparison of the point forecasts and the probability distributions of inflation and output growth made by individual respondents to the US Survey of Professional Forecasters indicates that the two sets of forecasts are sometimes inconsistent. We evaluate a number of possible explanations, and find that not all forecasters update their histogram forecasts as new information arrives. This is supported by the finding that the point forecasts are more accurate than the histograms in terms of first-moment prediction.