L-space-of-a-statistical-experiment

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

of distributions if all are dominated by a -finite measure on , there is no such possibility in the undominated case. Nevertheless, there is a substitute for the space generated by the which respects both the linear and the order structure of measures: the -space of the experiment, introduced in [a4]. This is a subspace of the Banach lattice of all signed measures on , and can be defined in three different ways, as follows [a1].

Denote by the vector lattice of all signed finite measures on , put and use as an abbreviation for . Equipped with the variational norm , is an order-complete Banach lattice. More precisely, is an abstract -space, which means that the norm is additive on . A solid linear subspace is called a band if whenever the satisfy for all .

If is a statistical experiment, then one defines

a) to be the smallest band (with respect to ) in containing ;

b) to be the -closure of , where

c) . Then . This space is called the -space of and is denoted by .

If there exists a such that for one has if and only if for all , then is dominated (and vice versa). In this case, the -space of is, as a Banach lattice, isomorphic to . The situation for undominated experiments is different. As an abstract -space, is always isomorphic to , with a Radon measure on a locally compact topological space [a3]. However, in general is not even semi-finite [a6] (i.e., lacks the finite subset property [a11]), and then there is no representation of the topological dual as . is called the -space of the experiment and generalizes the space of equivalence classes of bounded random variables in the following sense. Let denote the set of all real-valued functions defined on that are -measurable and bounded. For any , denote by the mapping assigning to every . Then coincides with the -closure of [a1], [a4], [a8]. For an alternative representation of , see [a9].

An experiment is called coherent if . Every dominated experiment is also coherent, due to the familiar isomorphism between and , 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 -finite, in general) and localizable [a11] measure on such that for one has if and only if for all . This result is an alternative interpretation of the fact that is isomorphic to if and only if is semi-finite and localizable [a11].

The experiment with , the Borel field, and is not coherent, since the counting measure is not localizable on because is countably generated but is not -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 -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)