Paul Marriott

An introduction to differential geometry in econometrics

In this introductory chapter we seek to cover sufficient differential geometry in order to understand its application to Econometrics. It is not intended to be a comprehensive review of either differential geometric theory, nor of all the applications which geometry has found in statistics. Rather it is aimed as a rapid tutorial covering the material needed in the rest of this volume and the general literature. The full abstract power of a modern geometric treatment is not always necessary and such a development can often hide in its abstract constructions as much as it illuminates. In Section 2 we show how econometric models can take the form of geometrical objects known as manifolds, in particular concentrating on classes of models which are full or curved exponential families. This development of the underlying mathematical structure leads into Section 3 where the tangent space is introduced. It is very helpful, to be able view the tangent space in a number of different, but mathematically equivalent ways and we exploit this throughout the chapter. Section 4 introduces the idea of a metric and more general tensors illustrated with statistically based examples. Section 5 considers the most important tool that a differential geometric approach offers, the affine connection. We look at applications of this idea to asymptotic analysis, the relationship between geometry and information theory and the problem of the choice of parameterisation. The last two sections look at direct applications of this geometric framework. In particular at the problem of inference in curved families and at the issue of information loss and recovery. Note that while this chapter aims to give a reasonably precise mathematical development of the required theory an alternative and perhaps more intuitive approach can be found in the chapter by Critchley, Marriott and Salmon later in this volume. For a more exhaustive and detailed review of current geometrical statistical theory see Kass and Vos (1997) or from a more purely mathematical background, see Murray and Rice (1993).

An elementary account of Amari's expected geometry

Differential geometry has found fruitful application in statistical inference. In particular, Amari’s (1990) expected geometry is used in higher order asymptotic analysis, and in the study of sufficiency and ancillarity. However, we can see three drawbacks to the use of a differential geometric approach in econometrics and statistics more generally. Firstly, the mathematics is unfamiliar and the terms involved can be difficult for the econometrician to fully appreciate. Secondly, their statistical meaning can be less than completely clear, and finally the fact that, at its core, geometry is a visual subject can be obscured by the mathematical formalism required for a rigorous analysis, thereby hindering intuition. All three drawbacks apply particularly to the differential geometric concept of a non metric affine connection. The primary objective of this paper is to attempt to mitigate these drawbacks in the case of Amari’s expected geometric structure on a full exponential family. We aim to do this by providing an elementary account of this structure which is clearly based statistically, accessible geometrically and visually presented.

Bayesian error analysis of Rutherford backscattering spectra

Rutherford backscattering spectrometry has been used to obtain quantitative and traceable information about homogeneous films. However, inhomogeneous films cannot be analysed with the same manual calculation methods. We have previously demonstrated that a machine algorithm using simulated annealing is available to extract depth profiles from spectra obtained from such inhomogeneous samples. In this work we show how a Bayesian error analysis can be used to construct error bounds on these depth profiles reflecting the uncertainty introduced by Poisson noise on the data. Hence we present, for the first time, RBS analysis which not only gives fully quantitative depth profiles for complex samples, but also reliable estimates of the errors due to collection statistics on these profiles.

On the Differential Geometry of the Wald Test with Nonlinear Restrictions

In this paper we exploit the tools of differential geometry to provide a clear explanation for the finite sample lack of invariance of the Wald statistic to algebraically equivalent reformulations of the null hypothesis

Seasonality of birth in Epilepsy: A Southern Hemisphere study

A consistent pattern has emerged from research in Northern Hemisphere populations indicating differences in the seasonality of birth between patients with epilepsy and the general population. This is the first study using similar methodology to look at Southern Hemisphere data. The population studied is composed of patients discharged from Australian hospitals with a diagnosis of epilepsy, in the period 1998-1999. The results show a significantly higher incidence of epilepsy in the patients born during the Australian winter and summer and a deficit of patients born during the spring and fall. This pattern is consistent with the Northern Hemisphere findings. This study provides further evidence of the existence of a seasonal aetiological agent(s) for epilepsy acting in the perinatal period.

Season of birth: aetiological implications for epilepsy

In most cases of epilepsy it is not possible to reach an aetiological diagnosis. Recent research points to a pre-perinatal disruption of the neurodevelopment as being the cause of at least some of these epilepsies of unknown aetiology. The object of this study was to corroborate this hypothesis from an epidemiological perspective and identify the most likely candidates for causes of this damage. The approach used was an analysis of the seasonal pattern of births in a large sample of epileptic patients discharged from NHS hospitals in England and Wales. The results illustrated that the seasonality of the births in the epileptic sample was significantly different from that of the general population, with an excess of patients born in December and January and a deficit of those born in September. This seasonality" was present only in the patients born before the late 1950s. These results are suggestive of the existence of an aetiological factor for epilepsy with a seasonal presence in the environment and which is epileptogenic when acting in the pre-perinatal period. Prenatal infections, obstetric complications and nutrititional deficiencies are amongst the hypotheses developed on the nature of this agent(s).

The hormonal environment in utero as a potential aetiological agent for schizophrenia.

AbstractThere is consistent evidence in the literature that the foetal neurodevelopmental period is crucial for the genesis of schizophrenia later in adult life. There are also strong indications that the schizophrenic illness has sexually dimorphic features. A hypothesis consistent with both findings is that sexual hormones may act as aetiological agents for schizophrenia during the foetal period influencing the neurodevelopment in a differential way in males and females. The aim of this study is to verify this hypothesis exploiting the correlation between fingers’ length in adults and hormonal concentrations in utero, which has been demonstrated in previous studies. More specifically, the literature shows that the lengths of the second and fourth finger in adults are proportional to the foetal concentrations of respectively oestrogens and androgens. When the sample of patients suffering from schizophrenia analysed in this study was compared with healthy subjects, it was observed that the average length of the second digit in the female schizophrenic sample resulted significantly shorter than in the female controls. There was no significant difference when the male schizophrenic sample was compared with male controls. The result of the study is, therefore, compatible with the hypothesis that oestrogenic hormones protect female foetuses from damage during the neurodevelopment in utero and ultimately give more benign characteristics to the schizophrenic illness in women.

The RBS data furnace: Simulated annealing

Abstract A computer program was written which carries out an automatic analysis of Rutherford Backscattering (RBS) data with minimal human involvement. The inputs which are required are the system parameters (e.g. experimental geometry, energy calibration), and the elements present in the sample. Parameters such as the number of layers, layer thickness and layer composition, are determined automatically during the procedure. The global optimisation simulated annealing (SA) algorithm was used, due to its two main features: First, the solution is independent of the initial guess chosen, and therefore a human-input initial layer structure is not needed. Second, it tends asymptotically to the absolute global minimum rather than a local minimum as in conventional minimisation algorithms, and hence high quality solutions can be achieved.

Local mixture models of exponential families

Exponential families are the workhorses of parametric modelling theory. One reason for their popularity is their associated inference theory, which is very clean, both from a theoretical and a computational point of view. One way in which this set of tools can be enriched in a natural and interpretable way is through mixing. This paper develops and applies the idea of local mixture modelling to exponential families. It shows that the highly interpretable and flexible models which result have enough structure to retain the attractive inferential properties of exponential families. In particular, results on identification, parameter orthogonality and log-concavity of the likelihood are proved.

Computational Information Geometry in Statistics: Theory and Practice

A broad view of the nature and potential of computational information geometry in statistics is offered. This new area suitably extends the manifold-based approach of classical information geometry to a simplicial setting, in order to obtain an operational universal model space. Additional underlying theory and illustrative real examples are presented. In the infinite-dimensional case, challenges inherent in this ambitious overall agenda are highlighted and promising new methodologies indicated.