A complementary note on the supercritical birth,death and catastrophe process

A central limit theorem for the population size of the super-critical linear birth and death process with a linear catastrophe component has previously been obtained under a fourth order moment condition on the increment distribution. In this note we show that this result is valid under a second order moment condition, and that no lesser condition will suffice. This is accomplished by giving a new, self-contained and simple proof of the asymptotic normality of a certain tail sum of independent variates.     

A Note on Lancaster's Procedure for the Combination of Independent Events

Lancaster (1961) generalized Fisher's (1932) nonparametric procedure for combining independent p-values by transforming P_i from the i-th experiment to a chi-squared random variable with d_i degrees of freedom, with d_i not necessarily equal to 2. We explore the relationship between Lancaster's procedure and a weighted Lipták procedure (Koziol and Tuckwell, 1994) under which P_i is transformed to the standard normal scale. We investigate approximations to the null distribution of Lancaster's test procedure, chi-squared with d degrees of freedom. We find that the Cornish-Fisher (1960) expansions and the Lugannani-Rice (1980) saddlepoint approximations are quite accurate, for non-integral values of d, and for values of d as low as 20.     

A multi-dimensional scaling approach to shape analysis

We propose an alternative to Kendall's shape space for reflectionshapes of configurations in with k labelled vertices, where reflection shape consistsof all the geometric information that is invariant under compositionsof similarity and reflection transformations. The proposed approachembeds the space of such shapes into the space of (k – 1) x (k – 1) real symmetricpositive semidefinite matrices, which is the closure of an opensubset of a Euclidean space, and defines mean shape as the naturalprojection of Euclidean means in on to the embedded copy of the shape space. This approachhas strong connections with multi-dimensional scaling, and themean shape so defined gives good approximations to other commonlyused definitions of mean shape. We also use standard perturbationarguments for eigenvalues and eigenvectors to obtain a centrallimit theorem which then enables the application of standardstatistical techniques to shape analysis in two or more dimensions.     

A Note on Optimality of Certain Rank Tests for Ordered Alternatives in Randomized Blocks

Nonparametric tests for ordered alternatives in randomised block designs based on within block rankings have been proposed by many authors. This note is concerned with the optimality of the choice of so-called regression constants usually considered in such rank tests. Some examples are discussed.     

A note on Gauss--Hermite quadrature

For Gauss—Hermite quadrature, we consider a systematicmethod for transforming the variable of integration so thatthe integrand is sampled in an appropriate region. The effectivenessof the quadrature then depends on the ratio of the integrandto some Gaussian density being a smooth function, well approximatedby a low-order polynomial. It is pointed out that, in this approach,order one Gauss-Hermite quadrature becomes the Laplace approximationmxHermitequadrature becomes the Laplace approximationmxHermite quadraturebecomes the Laplace approximationmxHermite quadrature becomesthe Laplace approximation. Thus the quadrature as implementedhere can be thought of as a higher-order Laplace approximation.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.