Contiguity (probability theory)
Encyclopedia
In probability theory
, two sequences of probability measure
s are said to be contiguous if asymptotically they share the same support
. Thus the notion of contiguity extends the concept of absolute continuity
to the sequences of measures.
The concept was originally introduced by as part of his contribution to the development of abstract general asymptotic theory
in mathematical statistics
. Le Cam was instrumental during the period in the development of abstract general asymptotic theory in mathematical statistics. He is best known for the general concepts of local asymptotic normality and contiguity.
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The notion of contiguity is closely related to that of absolute continuity. We say that a measure Q is absolutely continuous with respect to P (denoted ) if for any measurable set A, implies . That is, Q is absolutely continuous with respect to P if the support
of Q is a subset of the support of P. The contiguity property replaces this requirement with an asymptotic one: Qn is contiguous with respect to Pn if the “limiting support” of Qn is a subset of the limiting support of Pn.
It is possible however that each of the measures Qn be absolutely continuous with respect to Pn, while the sequence Qn not being contiguous with respect to Pn.
The fundamental Radon–Nikodym theorem
for absolutely continuous measures states that if Q is absolutely continuous with respect to P, then Q has density with respect to P, denoted as , such that for any measurable set A
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, two sequences of probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s are said to be contiguous if asymptotically they share the same support
Support (measure theory)
In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...
. Thus the notion of contiguity extends the concept of absolute continuity
Absolute continuity
In mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity...
to the sequences of measures.
The concept was originally introduced by as part of his contribution to the development of abstract general asymptotic theory
Asymptotic theory
Asymptotic theory or large sample theory is the branch of mathematics which studies properties of asymptotic expansions.The most known result of this field is the prime number theorem:...
in mathematical statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. Le Cam was instrumental during the period in the development of abstract general asymptotic theory in mathematical statistics. He is best known for the general concepts of local asymptotic normality and contiguity.
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Definition
Let be a sequence of measurable spaces, each equipped with two measures Pn and Qn.- We say that Qn is contiguous with respect to Pn (denoted ) if for every sequence An of measurable sets, implies .
- The sequences Pn and Qn are said to be mutually contiguous or bi-contiguous (denoted ) if both Qn is contiguous with respect to Pn and Pn is contiguous with respect to Qn.
The notion of contiguity is closely related to that of absolute continuity. We say that a measure Q is absolutely continuous with respect to P (denoted ) if for any measurable set A, implies . That is, Q is absolutely continuous with respect to P if the support
Support (measure theory)
In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...
of Q is a subset of the support of P. The contiguity property replaces this requirement with an asymptotic one: Qn is contiguous with respect to Pn if the “limiting support” of Qn is a subset of the limiting support of Pn.
It is possible however that each of the measures Qn be absolutely continuous with respect to Pn, while the sequence Qn not being contiguous with respect to Pn.
The fundamental Radon–Nikodym theorem
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
for absolutely continuous measures states that if Q is absolutely continuous with respect to P, then Q has density with respect to P, denoted as , such that for any measurable set A
-
which is interpreted as being able to “reconstruct” the measure Q from knowing the measure P and the derivative ƒ. A similar result exists for contiguous sequences of measures, and is given by the Le Cam’s third lemma.
Additional literature
- Roussas, George G. (1972), Contiguity of Probability Measures: Some Applications in Statistics, CUP, ISBN 9780521090957.
- Scott, D.J. (1982) Contiguity of Probability Measures, Australian & New Zealand Journal of Statistics, 24 (1), 80–88.
External references
- Contiguity Asymptopia: 17 October 2000, David Pollard
- Asymptotic normality under contiguity in a dependence case
- A Central Limit Theorem under Contiguous Alternatives
- Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems
- Testing statistical hypotheses
- Necessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser et al 1982 Russ. Math. Surv. 37 107–136
- The unconscious as infinite sets By Ignacio Matte Blanco, Eric (FRW) Rayner
- "Contiguity of Probability Measures", David J. Scott, La Trobe University
- "On the Concept of Contiguity", Hall, Loynes