Contiguity (probability theory)
In probability theory, two sequences of probability measures 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 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.
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Definition
Let (Ωn, Fn) be a sequence of measurable spaces, each equipped with two absolutely continuous measures Pn and Qn. Then Qn is said to be contiguous with respect to Pn (denoted Qn ◁ Pn) if Pn(An) → 0 implies Qn(An) → 0 for every sequence of measurable sets An. The sequences Pn and Qn are said to be mutually contiguous or bi-contiguous (denoted Qn ◁▷ Pn) if both Qn is contiguous with respect to Pn and Pn is contiguous with respect to Qn. [2]
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 Q ≪ P) if P(A) = 0 implies Q(A) = 0 for any measurable set A. The definition of contiguity is exactly the same, only it involves the sequences of measures.
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 ƒ = dQ⁄dP, 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.
Applications
See also
External References
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- 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
Notes
- ^ Wolfowitz J.(1974) Review of the book: "Contiguity of Probability Measures: Some Applications in Statistics. by George G. Roussas", Journal of the American Statistical Association, 69, 278–279 jstor
- ^ van der Vaart (1998, p. 87)
- ^ http://www.samsi.info/200506/fmse/course-info/werker-updated-nov14.pdf
References
- Hájek, J. (1967). Theory of rank tests. New York: Academic Press.
{{cite book}}: Unknown parameter|coauthor=ignored (|author=suggested) (help) - Le Cam, Lucien (1960). "Locally asymptotically normal families of distributions". University of California Publications in Statistics. 3: 37–98.
- van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge University Press.
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