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On the existence of uniformly consistent estimates

Author: Yannis G. Yatracos
Journal: Proc. Amer. Math. Soc. 94 (1985), 479-486
MSC: Primary 62G05; Secondary 62E20
MathSciNet review: 787899
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Abstract: Let $ \mathcal{M}$ be a family of probability measures on $ (\mathfrak{X},\mathcal{A})$ and $ U$ the uniform structure defined by vicinities of the form

$\displaystyle \left\{ (P, Q):\sup\limits_{1 \leqslant i \leqslant K} \vert P^n(A_{i,n}) - Q^n(A_{i,n}) \vert < \varepsilon \right\},$

where $ {P^n}$ is the product measure on $ ({\mathfrak{X}^n},{\mathcal{A}^n}),{A_{i,n}} \in {\mathcal{A}^n},\varepsilon > 0,n \wedge K \geqslant 1$. Let $ {\phi ^ * }:(\mathcal{M},U) \to ({\phi ^ * }(\mathcal{M}),d)$, where

$\displaystyle d\left(\phi^*(P), \phi^*(Q) \right) = \vert\vert P - Q\vert\vert _{L_1} = 2\sup\limits_{A \in \mathcal{A}} \vert P(A) - Q(A)\vert.$

We consider the case where the space of measures $ M$ is $ {L_1}$ separable and relate the existence of uniformly consistent estimates for $ {\phi ^ * }(P)$ with uniform continuity of $ {\phi ^ * }$ and $ {L_1}$-total boundedness of $ M$.

References [Enhancements On Off] (What's this?)

  • [G] Choquet [1969], Lectures on analysis, Vol. I, Chapter 2, §5, Benjamin, New York.
  • [E] Hille and R. Phillips [1957], Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I. MR 0089373 (19:664d)
  • [W] Hoeffding and J. Wolfowitz [1958], Distinguishability of sets of distributions, Ann. Statist. 29, 700-718. MR 0095555 (20:2057)
  • [C] Kraft [1955], Some conditions for consistency and uniform consistency of statistical procedures, Univ. of Calif. Publ. in Statist. 2, 125-142. MR 0073896 (17:505a)
  • [L] M. LeCam and L. Schwartz [1960], A necessary and sufficient condition for the existence of consistent estimates, Ann. Statist. 31, 140-150. MR 0142178 (25:5571)
  • [R] Moché [1977], Thèse de Doctorat d'Etat, Université de Lille, France.
  • [J] Pfanzagl [1968], On the existence of consistent estimates and tests, Z. Wahrsch. Verw. Gebiete 10, 43-62. MR 0233453 (38:1775)
  • [Y] G. Yatracos [1983], Uniformly consistent estimates and rates of convergence via minimum distance methods, Ph.D. Thesis, Univ. of California, Berkeley, Calif.

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Keywords: Uniformly consistent estimation of a functional of a measure, $ {L_1}$ total boundedness of the space of measures, minimum distance estimation
Article copyright: © Copyright 1985 American Mathematical Society

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