L-space-of-a-statistical-experiment

An order-complete Banach lattice (cf. also Riesz space) of measures on a measurable space $ ( \Omega, {\mathcal F} ) $, defined in the context of statistical decision theory [a2], [a5], [a7], [a8], [a10]. Prime object of this theory is the statistical experiment $ {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) $ where $ {\mathcal P} $ is a set of probability measures on $ ( \Omega, {\mathcal F} ) $. A statistical decision problem is to determine which of the distributions in $ {\mathcal P} $ are most likely to generate the observations (or data) collected. While the Radon–Nikodým theorem guarantees that one can operate with densities

$$ { \frac{dP }{d \mu } } \in L _ {1} ( \mu ) $$

of distributions if all $ P \in {\mathcal P} $ are dominated by a $ \sigma $- finite measure $ \mu $ on $ ( \Omega, {\mathcal F} ) $, there is no such possibility in the undominated case. Nevertheless, there is a substitute for the space generated by the $ { {dP } / {d \mu } } $ which respects both the linear and the order structure of measures: the $ L $- space $ L ( {\mathcal E} ) $ of the experiment, introduced in [a4]. This is a subspace of the Banach lattice of all signed measures on $ ( \Omega, {\mathcal F} ) $, and can be defined in three different ways, as follows [a1].

Denote by $ { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ the vector lattice of all signed finite measures on $ ( \Omega, {\mathcal F} ) $, put $ | \mu | = \sup ( \mu, - \mu ) \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ and use $ \mu \perp \nu $ as an abbreviation for $ \inf ( | \mu | , | \nu | ) = 0 $. Equipped with the variational norm $ \| \mu \| = | \mu | ( \Omega ) $, $ { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ is an order-complete Banach lattice. More precisely, $ { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ is an abstract $ L $- space, which means that the norm $ \| \cdot \| $ is additive on $ { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) _ {+} $. A solid linear subspace $ {\mathcal D} \subseteq { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ is called a band if $ \sup _ {i \in I } \mu _ {i} \in {\mathcal D} $ whenever the $ \mu _ {i} \in {\mathcal D} $ satisfy $ \mu _ {i} \leq \mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ for all $ i \in I $.

If $ {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) $ is a statistical experiment, then one defines

a) $ L _ {1} ( {\mathcal E} ) $ to be the smallest band (with respect to $ \subseteq $) in $ { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ containing $ {\mathcal P} $;

b) $ L _ {2} ( {\mathcal E} ) $ to be the $ \| \cdot \| $- closure of $ L ^ \prime ( {\mathcal E} ) $, where

$$ L ^ \prime ( {\mathcal E} ) = \left \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : \left | \mu \right | \leq \sum _ {i = 1 } ^ { n } \alpha _ {i} P _ {i} \right . $$

$$ \left . {\textrm{ for some } P _ {i} \in {\mathcal P}, \alpha _ {i} \geq 0, \textrm{ all } i ; n \in \mathbf N } \right \} ; $$

c) $ L _ {3} ( {\mathcal E} ) = \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : {\mu \perp \sigma \textrm{ for all } \sigma \perp {\mathcal P} } \} $. Then $ L _ {1} ( {\mathcal E} ) = L _ {2} ( {\mathcal E} ) = L _ {3} ( {\mathcal E} ) $. This space is called the $ L $- space of $ {\mathcal E} $ and is denoted by $ L ( {\mathcal E} ) $.

If there exists a $ Q \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $ such that for $ A \in {\mathcal F} $ one has $ Q ( A ) = 0 $ if and only if $ P ( A ) = 0 $ for all $ P \in {\mathcal P} $, then $ {\mathcal E} $ is dominated (and vice versa). In this case, the $ L $- space $ L ( {\mathcal E} ) $ of $ {\mathcal E} $ is, as a Banach lattice, isomorphic to $ L ^ {1} ( Q ) $. The situation for undominated experiments is different. As an abstract $ L $- space, $ L ( {\mathcal E} ) $ is always isomorphic to $ L ^ {1} ( m ) $, with $ m $ a Radon measure on a locally compact topological space [a3]. However, in general $ m $ is not even semi-finite [a6] (i.e., lacks the finite subset property [a11]), and then there is no representation of the topological dual $ M ( {\mathcal E} ) = L ( {\mathcal E} ) ^ {*} $ as $ L ^ \infty ( m ) $. $ M ( {\mathcal E} ) $ is called the $ M $- space of the experiment $ {\mathcal E} $ and generalizes the space of equivalence classes of bounded random variables in the following sense. Let $ X $ denote the set of all real-valued functions defined on $ \Omega $ that are $ {\mathcal F} $- measurable and bounded. For any $ \varphi \in X $, denote by $ {\dot \varphi } $ the mapping assigning $ \int _ \Omega \varphi {d \mu } $ to every $ \mu \in L ( {\mathcal E} ) $. Then $ M ( {\mathcal E} ) $ coincides with the $ \sigma ( M ( {\mathcal E} ) ,L ( {\mathcal E} ) ) $- closure of $ {\dot{X} } $[a1], [a4], [a8]. For an alternative representation of $ M ( {\mathcal E} ) $, see [a9].

An experiment $ {\mathcal E} $ is called coherent if $ M ( {\mathcal E} ) = {\dot{X} } $. Every dominated experiment is also coherent, due to the familiar isomorphism between $ [ L ^ {1} ( Q ) ] ^ {*} $ and $ L ^ \infty ( Q ) $, the reverse implication being false in general (for even larger classes of statistical experiments, see, e.g., [a6]). However, every coherent experiment is weakly dominated (and vice versa) in the following sense [a7]: there exists a semi-finite (not $ \sigma $- finite, in general) and localizable [a11] measure $ \mu $ on $ ( \Omega, {\mathcal F} ) $ such that for $ A \in {\mathcal F} $ one has $ \mu ( A ) = 0 $ if and only if $ P ( A ) = 0 $ for all $ P \in {\mathcal P} $. This result is an alternative interpretation of the fact that $ L ^ \infty ( \mu ) $ is isomorphic to $ [ L ^ {1} ( \mu ) ] ^ {*} $ if and only if $ \mu $ is semi-finite and localizable [a11].

The experiment $ {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) $ with $ \Omega = [ 0,1 ] $, $ {\mathcal F} $ the Borel field, and $ {\mathcal P} = \{ {\delta _ {x} } : {x \in [ 0,1 ] } \} $ is not coherent, since the counting measure $ \mu $ is not localizable on $ {\mathcal F} $ because $ {\mathcal F} $ is countably generated but $ \mu $ is not $ \sigma $- finite [a6] (this argument needs the assumption that each uncountable metric space contains a non-Borel set).

References

[a1]	I.M. Bomze, "A functional analytic approach to statistical experiments" , Longman (1990)
[a2]	H. Heyer, "Theory of statistical experiments" , Springer (1982)
[a3]	S. Kakutani, "Concrete representation of abstract $L$-spaces and the mean ergodic theorem" Ann. of Math. , 42 (1941) pp. 523–537
[a4]	L. Le Cam, "Sufficiency and approximate sufficiency" Ann. Math. Stat. , 35 (1964) pp. 1419–1455
[a5]	L. Le Cam, "Asymptotic methods in statistical decision theory" , Springer (1986)
[a6]	H. Luschgy, D. Mussmann, "Products of majorized experiments" Statistics and Decision , 4 (1986) pp. 321–335
[a7]	E. Siebert, "Pairwise sufficiency" Z. Wahrscheinlichkeitsth. verw. Gebiete , 46 (1979) pp. 237–246
[a8]	H. Strasser, "Mathematical theory of statistics" , de Gruyter (1985)
[a9]	E.N. Torgersen, "On complete sufficient statistics and uniformly minimum variance unbiased estimators" Teoria statistica delle decisioni. Symp. Math. , 25 (1980) pp. 137–153
[a10]	E.N. Torgersen, "Comparison of statistical experiments" , Cambridge Univ. Press (1991)
[a11]	A.C. Zaanen, "Integration" , North-Holland (1967)