In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, two sequences of
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''. The difference between a probability measure and the more gener ...
s are said to be contiguous if asymptotically they share the same
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
. Thus the notion of contiguity extends the concept of
absolute continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
to the sequences of measures.
The concept was originally introduced by as part of his foundational contribution to the development of
asymptotic theory
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
in mathematical
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
. He is best known for the general concepts of
local asymptotic normality In statistics, local asymptotic normality is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after a rescaling of the parameter. An important example when t ...
and contiguity.
Definition
Let
be a sequence of
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
s, each equipped with two measures ''P
n'' and ''Q
n''.
* We say that ''Q
n'' is contiguous with respect to ''P
n'' (denoted ) if for every sequence ''A
n'' of
measurable set
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
s, implies .
* The sequences ''P
n'' and ''Q
n'' are said to be mutually contiguous or bi-contiguous (denoted ) if both ''Q
n'' is contiguous with respect to ''P
n'' and ''P
n'' is contiguous with respect to ''Q
n''.
The notion of contiguity is closely related to that of
absolute continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
. 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 may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
of ''Q'' is a subset of the support of ''P'', except in cases where this is false, including, e.g., a measure that concentrates on an open set, because its support is a closed set and it assigns measure zero to the boundary, and so another measure may concentrate on the boundary and thus have support contained within the support of the first measure, but they will be mutually singular. In summary, this previous sentence's statement of absolute continuity is false. The ''contiguity'' property replaces this requirement with an asymptotic one: ''Q
n'' is contiguous with respect to ''P
n'' if the "limiting support" of ''Q
n'' is a subset of the limiting support of ''P
n''. By the aforementioned logic, this statement is also false.
It is possible however that each of the measures ''Q
n'' be absolutely continuous with respect to ''P
n'', while the sequence ''Q
n'' not being contiguous with respect to ''P
n''.
The fundamental
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
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''.
Properties
* For the case
for all ''n'' it applies
.
* It is possible that
is true for all ''n'' without
.
Le Cam's first lemma
For two sequences of measures
on measurable spaces
the following statements are equivalent:
*
*
*
*
for any statistics
.
where
and
are random variables on
.
Applications
*
Econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
See also
*
Asymptotic theory (statistics) In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size may grow indefinitely; the properties of estimato ...
*
Contiguity (disambiguation)
Contiguity or contiguous may refer to:
*Contiguous data storage, in computer science
*Contiguity (probability theory)
*Contiguity (psychology)
*Contiguous distribution of species, in biogeography
*Geographic contiguity
Geographic contiguity is th ...
*
Probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
Notes
References
*
*
*
*
Additional literature
:*Roussas, George G. (1972), ''Contiguity of Probability Measures: Some Applications in Statistics'', CUP, {{ISBN, 978-0-521-09095-7.
:*Scott, D.J. (1982) Contiguity of Probability Measures, ''Australian & New Zealand Journal of Statistics'', 24 (1), 80–88.
External links
Contiguity Asymptopia: 17 October 2000, David PollardAsymptotic normality under contiguity in a dependence case A Central Limit Theorem under Contiguous AlternativesSuperefficiency, Contiguity, LAN, Regularity, Convolution TheoremsTesting statistical hypothesesNecessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser et al 1982 Russ. Math. Surv. 37 107–136The 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
Probability theory