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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$.

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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