# 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).

How to Cite This Entry:
L-space-of-a-statistical-experiment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-space-of-a-statistical-experiment&oldid=50135
This article was adapted from an original article by I.M. Bomze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article