Difference between revisions of "L-space-of-a-statistical-experiment"
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− | + | An order-complete [[Banach lattice|Banach lattice]] (cf. also [[Riesz space|Riesz space]]) of measures on a [[Measurable space|measurable space]] ( \Omega, {\mathcal F} ) , | |
+ | defined in the context of [[Statistical decision theory|statistical decision theory]] [[#References|[a2]]], [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[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|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 [[#References|[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 [[#References|[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|Radon measure]] on a locally compact topological space [[#References|[a3]]]. However, in general m | ||
+ | is not even semi-finite [[#References|[a6]]] (i.e., lacks the finite subset property [[#References|[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} } [[#References|[a1]]], [[#References|[a4]]], [[#References|[a8]]]. For an alternative representation of M ( {\mathcal E} ) , | ||
+ | see [[#References|[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., [[#References|[a6]]]). However, every coherent experiment is weakly dominated (and vice versa) in the following sense [[#References|[a7]]]: there exists a semi-finite (not \sigma - | ||
+ | finite, in general) and localizable [[#References|[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 [[#References|[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 [[#References|[a6]]] (this argument needs the assumption that each uncountable metric space contains a non-Borel set). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.M. Bomze, "A functional analytic approach to statistical experiments" , Longman (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Heyer, "Theory of statistical experiments" , Springer (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Kakutani, "Concrete representation of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030104.png" />-spaces and the mean ergodic theorem" ''Ann. of Math.'' , '''42''' (1941) pp. 523–537</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Le Cam, "Sufficiency and approximate sufficiency" ''Ann. Math. Stat.'' , '''35''' (1964) pp. 1419–1455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Le Cam, "Asymptotic methods in statistical decision theory" , Springer (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Luschgy, D. Mussmann, "Products of majorized experiments" ''Statistics and Decision'' , '''4''' (1986) pp. 321–335</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E. Siebert, "Pairwise sufficiency" ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''46''' (1979) pp. 237–246</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> H. Strasser, "Mathematical theory of statistics" , de Gruyter (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> E.N. Torgersen, "On complete sufficient statistics and uniformly minimum variance unbiased estimators" ''Teoria statistica delle decisioni. Symp. Math.'' , '''25''' (1980) pp. 137–153</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> E.N. Torgersen, "Comparison of statistical experiments" , Cambridge Univ. Press (1991)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A.C. Zaanen, "Integration" , North-Holland (1967)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.M. Bomze, "A functional analytic approach to statistical experiments" , Longman (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Heyer, "Theory of statistical experiments" , Springer (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Kakutani, "Concrete representation of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030104.png" />-spaces and the mean ergodic theorem" ''Ann. of Math.'' , '''42''' (1941) pp. 523–537</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Le Cam, "Sufficiency and approximate sufficiency" ''Ann. Math. Stat.'' , '''35''' (1964) pp. 1419–1455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Le Cam, "Asymptotic methods in statistical decision theory" , Springer (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Luschgy, D. Mussmann, "Products of majorized experiments" ''Statistics and Decision'' , '''4''' (1986) pp. 321–335</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E. Siebert, "Pairwise sufficiency" ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''46''' (1979) pp. 237–246</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> H. Strasser, "Mathematical theory of statistics" , de Gruyter (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> E.N. Torgersen, "On complete sufficient statistics and uniformly minimum variance unbiased estimators" ''Teoria statistica delle decisioni. Symp. Math.'' , '''25''' (1980) pp. 137–153</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> E.N. Torgersen, "Comparison of statistical experiments" , Cambridge Univ. Press (1991)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A.C. Zaanen, "Integration" , North-Holland (1967)</TD></TR></table> |
Revision as of 22:15, 5 June 2020
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 ![]() |
[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) |
L-space-of-a-statistical-experiment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-space-of-a-statistical-experiment&oldid=47546