Alfred Kume

We show that a special case of method of moment estimator derived from the Stein class coincides with the class of generalized score matching estimator. Choosing a suitable weight function for generalized score matching is not straightforward. However, by placing it within the method of moment framework we can alleviate this problem by extending the Stein class to generalized method of moments. As a consequence we can work with a number of functions and hence derive generalized score matching estimators with optimal properties.

The paper considers the distribution of a general linear combination of central and non-central chi-square random variables by exploring the branch cut regions that appear in the standard Laplace inversion process. Due to the original interest from the directional statistics, the focus of this paper is on the density function of such distributions and not on their cumulative distribution function. In fact, our results confirm that the latter is a special case of the former. Our approach provides new insight by generating alternative characterizations of the probability density function in terms of a finite number of feasible univariate integrals. In particular, the central cases seem to allow an interesting representation in terms of the branch cuts, while general degrees of freedom and non-centrality can be easily adopted using recursive differentiation. Numerical results confirm that the proposed approach works well while more transparency and therefore easier control in the accuracy is ensured.

One of the main features of interest in analysing the light curves of stars is the underlying periodic behaviour. The corresponding observations are a complex type of time series with unequally spaced time points and are sometimes accompanied by varying measures of accuracy. The main tools for analysing these type of data rely on the periodogram-like functions, constructed with a desired feature so that the peaks indicate the presence of a potential period. In this paper, we explore a particular periodogram for the irregularly observed time series data, similar to Thieler et. al. (2013). We identify the potential periods at the appropriate peaks and more importantly with a quantifiable uncertainty. Our approach is shown to easily generalise to non-parametric methods including a weighted Gaussian process regression periodogram. We also extend this approach to correlated background noise. The proposed method for period detection relies on a test based on quadratic forms with normally distributed components. We implement the saddlepoint approximation, as a faster and more accurate alternative to the simulation-based methods that are currently used. The power analysis of the testing methodology is reported together with applications using light curves from the Hunting Outbursting Young Stars citizen science project.

In this paper we propose a regression model for size-and-shape response data. So far as we are aware, few such models have been explored in the literature to date. We assume a Gaussian model for labelled landmarks; these landmarks are used to represent the random objects under study. The regression structure, assumed in this paper to be linear in the ambient space, enters through the landmark means. Two approaches to parameter estimation are considered. The first approach is based directly on the marginal likelihood for the landmark-based shapes. In the second approach we treat the orientations of the landmarks as missing data, and we set up a model-consistent estimation procedure for the parameters using the EM algorithm. Both approaches raise challenging computational issues which we explain how to deal with. The usefulness of this regression modelling framework is demonstrated through real-data examples.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher–Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher–Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered.

We propose an alternative to Kendall's shape space for reflection shapes of configurations in with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space of (k − 1) × (k − 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in on to the embedded copy of the shape space. This approach has strong connections with multi-dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.