Volume 24 - Issue 2 - December 1999

Geometry, algebra and data analysis

John Gower


This is an edited version of an inaugural lecture given at the Open University, UK. The paper is mainly concerned with different types of visualisation, and especially in the complementary roles of geometry and algebra in developing visual displays of multivariate data.

Boundary value problems with Dirac delta function

Behzad Salimi


A simple procedure is shown for the derivation of the formal solution of two-point boundary value problems with Dirac delta function as the inhomogeneous or the source term. The problem consists of a linear second-order ordinary differential equation and the general inhomogeneous boundary conditions of the third kind. The method of solution presented here involves concepts from rudimentary calculus only. A simple example to demonstrate the technique, and the extension of the method to problems with the most general form of the self-adjoint differential operator are discussed.

An application of cross products in probability

Y. H. Wang, Lingqi Tang and You Shi Lou


In this paper we extend Hamilton's cross product in R3 to the general finite dimensional vector space Rn, n >= 2. We then give an application of the cross product in probability theory.

Spectral representation of four finite birth and death processes

W. Van Assche, P. R. Parthasarathy and R. B. Lenin


We obtain the spectral representation of the transition probabilities of four particular finite birth and death processes in terms of finite systems of classical orthogonal polynomials. The explicit knowledge of these orthogonal polynomials and their roots allows us to study the asymptotic behaviour of the processes as time tends to infinity, including their (exponential) rate of convergence.

Stable Paretian models in econometrics: part II

Svetlozar T. Rachev, Jeong-Ryeol Kim and Stefan Mittnik


In this, the second part of our survey paper, we summarize the theoretical behavior of Dickey-Fuller unit root tests when disturbances follow a stable Paretian distribution with infinite variance. The results reveal that both finite- and large-sample critical values are heavily dependent on the index of stability alpha, which determines the tail thickness of such a distribution. We also summarize the asymptotic results of statistical inference for a cointegrating parameter in the multiple regression framework with heavy-tailed integrated variables, introduced by Phillips (1991).

Epidemic models and social networks

Håkan Andersson


This survey paper discusses a class of stochastic continuous time models for the spread of an epidemic across a static or dynamic social network. Various simple graphs are considered: Bernoulli random graphs, graphs with prescribed degrees, graphs with a certain number of short loops, overlapping subgraphs representing the superposition of independent networks, and dynamically changing graphs. For each of these, expressions for important epidemiological quantities such as the basic reproduction number, the final size of the epidemic and the time dynamics of the proportion of susceptible and infectious individuals, are derived. The modelling assumptions are meaningful for finite populations, but the results obtained are only valid asymptotically as the population size tends to infinity. The theoretical work is illustrated by computer simulations and numerical calculations.

On 2x2 Contingency Tables

W. F. Scott


2x2 contingency tables arise in connection with (i) tests of equality of proportions, and (ii) tests of lack of association between two attributes. We show that if the minimum expected cell frequency exceeds a certain value, which varies from 1.5 to 6 depending on the level of significance of the test, then the Type I error of the c2-test is reasonably accurate.