Ole E Barndorff-Nielsen - Böcker

This book is a slightly revised and expanded version of a set I I I of notes used for a lecture series given at the Ecole dlEte de I Probabilites at st. Flour in August 1986. In view of the statistical nature of the material discussed herein it was agreed to publish the material as a separate volume in the statistics series rather than, as is the tradition, in a joint volume in the Lecture Notes in Mathematics Series. It is a genuine pleasure to have this opportunity to thank I I I the organizers of Les Ecoles dlEte, and in particular Professor P. -L. Hennequin, for the excellent arrangements of these Summer Schools which form a very significant forum for the exchange of scientific ideas relating to probability. The efficient, careful and patient preparation of the typescript by Oddbj~rg Wethelund is also gratefully acknowledged. Aarhus, June 1988 O. E. Barndorff-Nielsen Parametric statistical Models and Likelihood O. E. Barndorff-Nielsen o. Introduction 0. 1. Outline of contents 1 0. 2. A few preliminaries 2 1. Likelihood and auxiliary statistics 1. 1. Likelihood 4 1. 2. Moments and cumulants of log likelihood derivatives 10 1. 3. Parametrization invariance 13 1. 4. Marginal and conditional likelihood 15 * 1. 5. Combinants, auxiliaries, and the p -model 19 1. 6. Orthogonal parameters 27 1. 7. Pseudo likelihood, profile likelihood and modified 30 profile likelihood 1. 8. Ancillarity and conditionality 33 41 1. 9. Partial sufficiency and partial ancillarity 1. 10.

The present set of notes grew out of our interest in the study of statistical transformation models, in particular exponential transfor- mation models. The latter class comprises as special cases all fully tractable models for mUltivariate normal observations. The theory of decomposition and invariance of measures provides essential tools for the study of transformation models. While the major aspects of that theory are treated in a number of mathematical monographs, mostly as part of much broader contexts, we have found no single account in the literature which is sufficiently comprehensive for statistical pur- poses. This volume aims to fill the gap and to indicate the usefulness of measure decomposition and invariance theory for the methodology of statistical transformation models. In the course of the work with these notes we have benefitted much from discussions with steen Arne Andersson, J0rgen Hoffmann-J0rgensen and J0rgen Granfeldt Petersen. We are also very indebted to Jette Ham- borg and Oddbj0rg Wethelund for their eminent secretarial assistance.

A Lévy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lévy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Lévy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lévy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.

A Lévy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lévy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Lévy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lévy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.

Drawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development.Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context.Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling.

These two volumes contain a collection of selected papers on a wide range of aspects of natural transport and re-sorting of sand and soil. Apart from one paper on avalanche and a small number in which sea waves are also an agent, the papers all deal exclusively with wind erosion. One paper reviews in depth the contributions to the literature of aeolian transport which have been published during the last five years. The purpose of the workshop from which these volumes derive was to discuss work carried out during these five years but not yet published. The collection of papers was reviewed in light of these discussions and can therefore be taken to be a particularly well informed statement about the subject as it stood in mid 1990. Several important field problem classes were examined as were several new observational techniques. This activity, together with widespread concern about soil degradation and desertification in wind transported soils, will probably attract many new workers to the field during the next decade. This summary of its present status will be extremely helpful to them.

The first of two volumes, this book contains a collection of selected papers on a wide range of aspects of the natural transport and resorting of sand and soil. Apart from one paper on avalanches and a small number in which sea waves are also an agent, the papers deal exclusively with wind erosion. This second volume reviews the contributions to the literature of aeolian transport which have been published during the last five years, concentrating on soil degradation and desertification. The purpose of the workshop from which these volumes are drawn was to discuss work carried out during these five years but which has not, as yet, been published. The books examine several important field problem classes and a number of new observational techniques. This book of proceedings on geography (geomorphology), agriculture, sedimentology, civil engineering and statistics is intended for research teams, graduate students, senior undergraduates in some subjects, design teams and researchers.

Drawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development.Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context.Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling.

This is a readily accessible introduction to the theory of stochastic processes with emphasis on processes with independent increments and Markov processes. After preliminaries on infinitely divisible distributions and martingales, Chapter 1 gives a thorough treatment of the decomposition of paths of processes with independent increments, today called the Levy-Ito decomposition, in a form close to Ito's original paper from 1942. Chapter 2 contains a detailed treatment of time-homogeneous Markov processes from the viewpoint of probability measures on path space. Two separate Sections present about 70 exercises and their complete solutions. The text and exercises are carefully edited and footnoted, while retaining the style of the original lecture notes from Aarhus University.

This is the second volume of two volumes containing the revised and completed notes lectures given at the school "Quantum Independent Increment Processes: Structure and Applications to Physics". This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald during the period March 9 - 22, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics.The present second volume contains the following lectures: "Random Walks on Finite Quantum Groups" by Uwe Franz and Rolf Gohm, "Quantum Markov Processes and Applications in Physics" by Burkhard Kummerer, Classical and Free Infinite Divisibility and Levy Processes" by Ole E. Barndorff-Nielsen, Steen Thorbjornsen, and "Levy Processes on Quantum Groups andDual Groups" by Uwe Franz

The volume Stochastic Processes by K. Itö was published as No. 16 of Lecture Notes Series from Mathematics Institute, Aarhus University in August, 1969, based on Lectures given at that Institute during the academie year 1968­ 1969. The volume was as thick as 3.5 cm., mimeographed from typewritten manuscript and has been out of print for many years. Since its appearance, it has served, for those abIe to obtain one of the relatively few copies available, as a highly readable introduetion to basic parts of the theories of additive processes (processes with independent increments) and of Markov processes. It contains, in particular, a clear and detailed exposition of the Lévy-It ö decomposition of additive processes. Encouraged by Professor It ó we have edited the volume in the present book form, amending the text in a number of places and attaching many footnotes. We have also prepared an index. Chapter 0 is for preliminaries. Here centralized sums of independent ran­ dom variables are treated using the dispersion as a main tooI. Lévy's form of characteristic functions of infinitely divisible distributions and basic proper­ ties of martingales are given. Chapter 1 is analysis of additive processes. A fundamental structure the­ orem describes the decomposition of sample functions of additive processes, known today as the Lévy-Itó decomposition. This is thoroughly treated, as­ suming no continuity property in time, in a form close to the original 1942 paper of Itó, which gave rigorous expression to Lévy's intuitive understanding of path behavior.

Over the past 10-15 years, we have seen a revival of general Levy ' processes theory as well as a burst of new applications. In the past, Brownian motion or the Poisson process have been considered as appropriate models for most applications. Nowadays, the need for more realistic modelling of irregular behaviour of phen- ena in nature and society like jumps, bursts, and extremeshas led to a renaissance of the theory of general Levy ' processes. Theoretical and applied researchers in elds asdiverseas quantumtheory,statistical physics,meteorology,seismology,statistics, insurance, nance, and telecommunication have realised the enormous exibility of Lev ' y models in modelling jumps, tails, dependence and sample path behaviour. L' evy processes or Levy ' driven processes feature slow or rapid structural breaks, extremal behaviour, clustering, and clumping of points. Toolsandtechniquesfromrelatedbut disctinct mathematical elds, such as point processes, stochastic integration,probability theory in abstract spaces, and differ- tial geometry, have contributed to a better understanding of Le 'vy jump processes.As in many other elds, the enormous power of modern computers has also changed the view of Levy ' processes. Simulation methods for paths of Levy ' p- cesses and realisations of their functionals have been developed. Monte Carlo simulation makes it possible to determine the distribution of functionals of sample paths of Levy ' processes to a high level of accuracy.

This book communicates some contemporary mathematical and statistical developments in river basin hydrology as they pertain to space-time rainfall, spatial landform and network structures and their role in understanding averages and fluctuations in the hydrologic water balance of river basins. While many of the mathematical and statistical nations have quite classical mathematical roots, the river basin data structure has led to many variations on the problems and theory.

This proceedings volume focusses on the applications of geometry in present day science. It contains contributions from a variety of fields, including biology, computer science, mathematics, medicine, physics and stochastics.

Change of Time and Change of Measure provides a comprehensive account of two topics that are of particular significance in both theoretical and applied stochastics: random change of time and change of probability law.Random change of time is key to understanding the nature of various stochastic processes, and gives rise to interesting mathematical results and insights of importance for the modeling and interpretation of empirically observed dynamic processes. Change of probability law is a technique for solving central questions in mathematical finance, and also has a considerable role in insurance mathematics, large deviation theory, and other fields.The book comprehensively collects and integrates results from a number of scattered sources in the literature and discusses the importance of the results relative to the existing literature, particularly with regard to mathematical finance. It is invaluable as a textbook for graduate-level courses and students or a handy reference for researchers and practitioners in financial mathematics and econometrics.