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February, 1979 A Functional Relationship between the Different $r$-means for Indicator Functions
D. Landers, L. Rogge
Ann. Probab. 7(1): 166-169 (February, 1979). DOI: 10.1214/aop/1176995159


Let $P$ be a probability measure defined on a $\sigma$-field $\mathscr{F}$ over $\Omega$. Let $\mathfrak{L}\subset \mathscr{F}$ be a $\sigma$-lattice and $r > 1$. For each $A \in \mathscr{F}$ denote by $P_r(A/\mathfrak{L})$ the unique nearest point projection of $1_A$ onto the closed convex subspace of all "$\mathfrak{L}$-measurable" equivalence-classes of $L_r(\Omega, \mathscr{F}, P)$. It is shown that there exists a functional relationship between $P_r(A/\mathfrak{L})$ and $P_2(A/\mathfrak{L})$ of the form $$P_r(A/\mathfrak{L}) = \varphi(P_2(A/\mathfrak{L}))$$ where the function $\varphi$ depends only on $r$ but not on $A, P$ or $\mathfrak{L}$. This relationship is applied to the theory of sufficiency.


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D. Landers. L. Rogge. "A Functional Relationship between the Different $r$-means for Indicator Functions." Ann. Probab. 7 (1) 166 - 169, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.46020
MathSciNet: MR515824
Digital Object Identifier: 10.1214/aop/1176995159

Primary: 46E30
Secondary: 62B05

Keywords: $\sigma$-lattice , conditional expectation , Projection in $L_r$ , sufficiency

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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