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## Estimating functions

*Christopher G. Small and Jinfang Wang*

### in Numerical Methods for Nonlinear Estimating Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506881
- eISBN:
- 9780191709258
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506881.003.0002
- Subject:
- Mathematics, Probability / Statistics

This chapter gives a survey of the basic concepts of estimating functions, which are used in subsequent chapters. The concept of unbiasedness for estimating functions is introduced as a ... More

## Hertz on the Hamilton formalism

*JESPER LÜTZEN*

### in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0023
- Subject:
- Physics, History of Physics

In the introduction to his book Principles of Mechanics, Heinrich Hertz emphasised that one of the advantages of his geometric formulation of his mechanics is that it throws a bright light upon ... More

## Quantum Statistics of Fields

*STEPHEN M. BARNETT and PAUL M. RADMORE*

### in Methods in Theoretical Quantum Optics

- Published in print:
- 2002
- Published Online:
- February 2010
- ISBN:
- 9780198563617
- eISBN:
- 9780191714245
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198563617.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in ... More

## Characteristic Functions

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0011
- Subject:
- Economics and Finance, Econometrics

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its ... More

## Review of probability

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews probability theory, limiting the discussion to the study of random events as opposed to random processes, the latter being a sequence of random events extended over a period of ... More

## Weak Convergence of Distributions

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0022
- Subject:
- Economics and Finance, Econometrics

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved, and the chapter then considers the ... More

## PROBABILITY THEORY

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter begins with a review of elementary probability theory, conditional probability, and statistical independence. It explains the notion of a random variable and its distribution function or ... More

## Acts of Mathematical Creation

*Paul Erickson*

### in The World the Game theorists Made

- Published in print:
- 2015
- Published Online:
- May 2016
- ISBN:
- 9780226097039
- eISBN:
- 9780226097206
- Item type:
- chapter

- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226097206.003.0002
- Subject:
- History, History of Science, Technology, and Medicine

Despite the ex post facto identification of a number of “anticipations” of game-theoretic results through history, it is generally agreed that modern game theory’s founding work was mathematician ... More

## Blocking

*Debraj Ray*

### in A Game-Theoretic Perspective on Coalition Formation

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199207954
- eISBN:
- 9780191709104
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199207954.003.0011
- Subject:
- Economics and Finance, Financial Economics

This chapter begins a second approach to coalition formation, one firmly grounded in coalitions as fundamental behavioral units. A central notion is that of blocking and the attendant concepts that ... More

## Clarification and defence

*John Collins*

### in The Unity of Linguistic Meaning

- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199694846
- eISBN:
- 9780191732027
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199694846.003.0006
- Subject:
- Philosophy, Philosophy of Language, Philosophy of Mind

There are three strands of justification for the account of combinatorial unity presented in the last chapter. The first is negative: the other extant accounts of unity are, I have argued, either ... More

## Phase Space Distributions

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation ... More

## Characteristic functions

*J. Klafter and I.M. Sokolov*

### in First Steps in Random Walks: From Tools to Applications

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0001
- Subject:
- Physics, Soft Matter / Biological Physics

Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter ... More

## Langevin processes

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Just as Fokker–Planck processes are a generalisation of thermal noise, Langevin processes constitute a generalisation of shot noise. In general, Langevin processes permit a larger class of problems ... More

## The method of joint probability distribution functions, neighbourhoods, and representations

*Carmelo Giacovazzo*

### in Phasing in Crystallography: A Modern Perspective

- Published in print:
- 2013
- Published Online:
- November 2020
- ISBN:
- 9780199686995
- eISBN:
- 9780191918377
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199686995.003.0009
- Subject:
- Chemistry, Crystallography: Chemistry

Wilson statistics, described in Chapter 2, aims at calculating the distribution of the structure factor P(F) ≡ P(|F|, φ) when nothing is known about the structure; the ... More

## Canonical & Gauge Transformations

*Peter Mann*

### in Lagrangian and Hamiltonian Dynamics

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0018
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions ... More

## Estimation Bounds for Automated Object Recognition

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0014
- Subject:
- Computer Science, Programming Languages

Thus far we have only studied representations of the source. Now we add the channel, pushing us into the frameworks of estimate then examine estimation bounds for ... More

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