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 Le Cam (1960) 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.^[1]

Definition

Let $(\Omega _{n},{\mathcal {F}}_{n})$ be a sequence of measurable spaces, each equipped with two measures P_n and Q_n.

We say that Q_n is contiguous with respect to P_n (denoted Q_n ◁ P_n) if for every sequence A_n of measurable sets, P_n(A_n) → 0 implies Q_n(A_n) → 0.
The sequences P_n and Q_n are said to be mutually contiguous or bi-contiguous (denoted Q_n ◁▷ P_n) if both Q_n is contiguous with respect to P_n and P_n is contiguous with respect to Q_n.^[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 for any measurable set A, P(A) = 0 implies Q(A) = 0. 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: 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.

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 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

Q(A)=\int _{A}f\,{\mathrm {d}}P,\,

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

Econometrics ^[3]

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.; Šidák, Z. (1967). Theory of rank tests. New York: Academic Press.
Le Cam, Lucien (1960). "Locally asymptotically normal families of distributions". University of California Publications in Statistics. 3: 37–98.
Roussas, George G. (2001), "Contiguity of probability measures", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge University Press.

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

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