Christopher G. Small

Hilbert space methods in probability and statistical inference

Hilbert Spaces. Probability Theory. Estimating Functions. Orthogonality and Nuisance Parameters. Martingale Estimating Functions and Projected Likelihood. Stochastic Integration and Product Integrals. Estimating Functions and the Product Integral Likelihood for Continuous Time Stochastic Processes. Hilbert Spaces and Spline Density Estimation. Bibliography. Index.

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X1),...,f(Xn). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension.

Numerical methods for nonlinear estimating equations

Introduction Estimating functions Numerical algorithms Working with roots Methodologies for root selection Artificial likelihoods and estimating functions Root selection and dynamical systems Bayesian estimating functions Bibliography Index

Multidimensional scaling of simplex shapes

Abstract Bookstein (Statist. Sci. 1 (1986) 181–242; Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991) has proposed a method for the representation of triangle shape as points in the Poincare half plane – a space of constant negative curvature. Small (The Statistical Theory of Shape, Springer, New York, 1996) provided an extension of the Bookstein representation by representing the shapes of n-simplexes on manifolds. These manifolds are quite distinct from those proposed by D.G. Kendall based upon Procrustes arguments. In this paper, we examine the geometrical properties of these simplex shape spaces in greater detail. In particular, explicit formulas are given for the geodesic distance between any two points in these spaces. Such formulas permit the implementation of multidimensional scaling methods for the statistical shape analysis of two- and three-dimensional objects. In addition, the curvatures of the simplex shape spaces are examined. It is shown that the spaces of n-simplex shapes are not of constant curvature unless n =2.

The statistical analysis of dynamic curves and sections

Abstract By a curve, we shall understand a one-dimensional smooth path lying in R 2 or R 3 which can be naturally parametrized by a real coordinate. The coordinate could represent physical time or any other variable which can be interpreted dynamically. In some cases, the curve will arise as the linear section of a higher-dimensional structure such as the planar section of a surface in R 3 . In this paper, we develop a model for the shape of planar curves, based on their curvatures, that is reasonably robust to the location of landmarks or knots used to approximate the contours of the curve. The measurement for the shape difference between two curves that we propose is also based on the curvatures of the curves and directly inherits the simple Euclidean property for averages.

Multiple roots of estimating functions

Estimating functions can have multiple roots. In such cases, the statistician must choose among the roots to estimate the parameter. Standard asymptotic theory shows that in a wide variety of cases, there exists a unique consistent root, and that this root will lie asymptotically close to other consistent (possibly inefficient) estimators for the parameter. For this reason, attention has largely focused on the problem of selecting this root and determining its approximate asymptotic distribution. In this paper, however, we concentrate on the exact distribution of the roots as a random set. In particular, we propose the use of higher-order root intensity functions as a tool for examining the properties of the roots and determining their most problematic features. The use of root intensity functions of first and second order is illustrated by application to the score function for the Cauchy location model.

Form representions and means for landmarks: A survey and comparative study

The form of an object may be briefly defined as the totality of all geometrical features in the object. Thus different images (as two-dimensional objects) can be said to have the same form provided they convey identical geometrical information, regardless of the coordinates used to express this information. In this paper, we survey some of the principal approaches to the representation and statistical analysis of forms of landmark data which can be extracted from images directly or by means of computer tomography. In particular, we survey the various ways in which forms can be pooled by form averaging. We propose a set of criteria for an ideal mean form, and study the available methods in light of these criteria. An annotated bibliography is provided as a guide to various theoretical and applied papers on the subject.

Looking for circular structures in post hole distributions: quantitative analysis of two settlements from Bronze Age England

Abstract A statistical procedure is suggested for the analyses of hypothetical round-house plans among post hole patterns. The data are compared with an analogous random scattering of points and differences are noted. Plausible interpretations are contrasted with chance configurations to determine their strength. Single-link cluster analysis and statistical geometry are used. The emphasis throughout the paper is on data analysis rather than hypothesis testing.

Reconstructing convex bodies from random projected images

We consider the problem of reconstructing a convex body from samples of lower-dimensional projections, or shadows. Such problems have been studied using mean cross-sectional measures of convex sets. We develop new methods based upon inner and outer estimates of the convex body, for which rates of convergence are obtained. These asymptotic results for the inner and outer estimates are compared with each other as well as those based upon Cauchys formula.

Calculating the simplex median

While much attention has recently focussed on the use of multivariate medians for estimating the centre of a data set, a particular median based upon random simplexes proposed by Liu has been overshadowed by progress on other multivariate medians. The purpose of this paper is to redress the balance, and to show that Lius simplex median is tractable, and has distinctive desirable properties that recommend it for use in data analysis.