Contiguity (probability theory)

In mathematics, a contiguity space can be seen as a circular quality space. A qualitative field is described in terms of its topological and ordinal aspects, each boundary is partitioned into a contiguity space, where each contiguity segment describes an adjacency to another qualitative region. Contiguity is basic for the asymptotic methods of estimation theory. Contiguity of sequences is a property that may be used to derive limiting asymptotic distribution under the alternative of a statistical hypothesis.

Probability theory

Contiguity of sequences of probability measures is defined for a sequence of measurable spaces $(\Omega _{n},{\mathcal {F}}_{n})_{n=1}^{\infty }$ with two probability measure sequences, $(P_{n})_{n=1}^{\infty }$ and $(Q_{n})_{n=1}^{\infty }$.

where $(Q_{n})_{n=1}^{\infty }$ is contiguous to $(P_{n})_{n=1}^{\infty }$ if $\forall F_{n}\in {\mathcal {F}}_{n}:{\big (}P_{n}(F_{n})\to 0{\big )}\Rightarrow {\big (}Q_{n}(F_{n})\to 0{\big )}$ as $n\to \infty$,

and $(Q_{n})_{n=1}^{\infty }$ is bi-contiguous to $(P_{n})_{n=1}^{\infty }$ if $\forall F_{n}\in {\mathcal {F}}_{n}:{\big (}P_{n}(F_{n})\to 0{\big )}\Leftrightarrow {\big (}Q_{n}(F_{n})\to 0{\big )}$ as $n\to \infty$.

History

The concept was originally introduced by Lucien Le Cam in the 1960s 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.

Applications

• Econometrics

See also

• Probability space
• Probability theory

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