Pirámides Rectangulares con base un Polígono Regular de cantidad impar de

According to the above reasoning, the M-space structure of L∞(Ω,A) contains the essen- tial information about the sample space. So, we may “forget” any additional structure of L∞(Ω,A) which goes beyond its M-space structure. If we do this and consider ba(Ω,A) as the dual ofL∞(Ω,A), nothing remains left from ba(Ω,A) apart from its L-space struc- ture (remember that the dual space of an M-space is always an L-space; cf. Proposition 8.22 a) ). That is, the L-space structure contains the essential information about the elements of ba(Ω,A). In terms of L-spaces, the probability charges P ∈ ba(Ω,A) are precisely the normed, positive elements of ba(Ω,A) (cf. Subsection 2.2) – that is, the

probability charges can completely be identified by the L-space structure of ba(Ω,A) ! If we have a fixed precise model

E = (Pθ)θ∈Θ ⊂ ba+₁(Ω,A)

we usually do not have to consider the whole L-space ba(Ω,A) but it is enough to consider the smallest L-spaceL⊂ba(Ω,A) which contains our modelE = (Pθ)θ∈Θ. This L-space L is equal to the smallest band in the L-space ba(Ω,A) which contains E = (Pθ)θ∈Θ. This set L is calledL-space of E orL-space generated by E and is denoted by

L(E) in Le Cam (1986).

In classical statistics, E = (Pθ)θ∈Θ is a family of probability measures which is domi- nated12 by someσ-finite measure µ. In this case,

L(E) ⊂

ν ∈ba(Ω,A)

dν =f dµ , f ∈L₁(Ω,A, µ) In addition, assume that µ is also dominated13 _{byE. Then,}

L(E) = ν ∈ba(Ω,A) dν =f dµ , f ∈L₁(Ω,A, µ) (3.13) That is, L(E) can be identified with the set of densities L1(Ω,A, µ) then. Examples for this case are

• models (Pθ)θ∈Θ on (Z,2Z)

• the model (Pθ)θ∈Θ on (R,B) wherePθ =N(a, b) for every (a, b) = θ∈Θ

Note that

L(E) = ba(Ω,A) (3.14)

is also possible for a suitable chosen model E.14

So far, we have stated that the L-space structure ofL(E) contains all essential information about E – that is, it is not important that the elements Pθ of E are probability measures

on some specific sample space (Ω,A) and they may be defined as elements of any abstract L-space.

Accordingly, L. Le Cam proposes the following definition of experiments – called precise models in the present book. In order to avoid confusions with the definitions given in previous sections, the definitions given by L. Le Cam in his general setup carry the prefix “LC”. 12_{That is: µ(A) = 0, A∈ A ⇒ P} θ(A) = 0 ∀θ∈Θ 13_{That is: P} θ(A) = 0 ∀θ∈Θ, A∈ A ⇒ µ(A) = 0 14_{Choose Θ = ba}+ 1(Ω,A) andPθ=θ ∀θ∈Θ .

Definition 3.33 (Experiment / precise model)

An LC-experiment/precise LC-model indexed by the set Θ is a family (Pθ)θ∈Θ ⊂ L

where Lis an L-space and Pθ is a normed (kPθk= 1), positive (Pθ ≥0) element of L for

every θ∈Θ.

Cf. (Le Cam, 1986, p. 5).

It is clear that every ordinary precise model (according to Section 3.2) is a precise LC- model

Of course, the question arises if the above definition excessively generalizes the notion “precise model”. As described below, the answer is: no, essentially not! Indeed, it is less a generalization then an abstraction.

Since we have lost the sample space (Ω,A) now, we have also lost the random variables f ∈ L∞(Ω,A). In order to reintroduce them, we consider the dual space of Ldenoted by

L∗ =: M

The elements f∗ ∈M corresponds to the random variables f ∈ L∞(Ω,A).

For example, let the precise model E = (Pθ)θ consist of probability measures Pθ on a

sample space (Ω,A) such that Pθ is dominated by a σ-finite measure µ on (Ω,A) for

every θ ∈ Θ. Furthermore, assume that µ is also dominated by E. Put L = L(E); according to (3.13), we can identify L(E) with the set of all µ-densities. Next, (Dunford and Schwartz, 1958, Theorem IV.8.5) says that the dual space

L∗ =M

is equal to L∞(Ω,A, µ) . Here, corresponding elements f∗ ∈M and f ∈L∞(Ω,A, µ) are related by the identity

f∗(β) = Z

Ω

f(ω)β(ω)µ(dω) ∀β ∈L1(Ω,A, µ)

That is, in this special case which is quite common in classical statistics, L(E) is equal to the set of all µ-densities and M is equal to the set of all bounded random variables. The dual space of L=L(E) is called M-space of E or M-space generated by E and is denoted by

M = M(E)

in Le Cam (1986). This is indeed an M-space because L(E) is an L-space; cf. Section 8.1. The following schema illustrates the relations between the abstract setup and the traditional concepts:

bounded random variables L∞(Ω,A, µ) abstraction // M(E)

µ-densities L1(Ω,A, µ)

abstraction _//

L(E)

In order to describe a decision problem now, we need a set _D of possible decisions t∈ _D and a loss function

In Le Cam’s setup, we have

Wθ ∈ Γ ∀θ∈Θ

where Γ is a set of bounded functions

γ : _D → _R

which fulfills certain conditions (Γ is a so-called uniform lattice; cf. (Le Cam, 1986, 4 and 5)). As a special case, we may simply take

Γ = L∞(_D,D) where D is an algebra on _D.

Finally, Markov kernels play an important role in decision theory – especially as random- ized decision procedures. Since Le Cam dispenses with sample spaces in the definition of LC-experiments / precise LC-models, Markov kernels cannot be defined. Therefore, Markov kernels are replaced by transitions:

Definition 3.34 Let L1 and L2 be L-spaces. A transition from L1 toL2 is a map σ: L1 → L2

which is

• linear

• positive: T(µ)≥0 ∀µ≥0

• normalized: kT(µ)k=kµk ∀µ≥0 Now, let Θ be an index set, L an L-space and

E = (Pθ)θ∈Θ ⊂ L

a precise LC-model; L(E) denotes the L-space generated by E. Θ×_D → _R, (θ, t) 7→ Wθ(t)

is a loss function where

Wθ ∈ Γ :=L∞(D,D) ∀θ ∈Θ

According to Theorem 2.4, the dual space of Γ =L∞(_D,D) is equal to Γ∗ = ba(_D,D)

which is an L-space; cf. Theorem 2.6. With these predefinitions, decision procedures in the sense of (Le Cam, 1986, §1.3) can be defined:

Definition 3.35 (LC - decision procedures) A LC - decision procedure (based on E and taking values in _D) is a transition from L(E) to Γ∗

Next, the LC - risk function is defined to be

Θ → _R, θ 7→ σ(Pθ)[Wθ]

Since Γ∗ = ba(_D,D), we may also write σ(Pθ)[Wθ] =

Z

Wθ(t)Kθ(dt), where Kθ :=σ(Pθ) (3.15)

Instead of (3.15), Le Cam (1986) uses the notation WθσPθ := σ(Pθ)[Wθ]

How do these definitions fit into the usual setup based on sample spaces? In order to answer this question, let the precise model E = (Pθ)θ∈Θ consist of probability measures Pθ on a sample space (Ω,A) such thatPθ is dominated by a σ-finite measureµon (Ω,A)

for every θ ∈Θ. Furthermore, assume that µis also dominated by E. Put L=L(E) and M =M(E). As stated above, we can identify

L(E) = L1(Ω,A, µ) and

M(E) = L∞(Ω,A, µ)

As usual, (_D,D) is a decision space and the loss function is some (Wθ)θ∈Θ ⊂ L∞(D,D) =: Γ

Let τ be an ordinary (randomized) decision function, i.e. τ is a Markov kernel τ : Ω× D → _R, (ω, D) 7→ τω(D)

The Markov kernel τ defines a transition

σ : L(E) → ba(_D,D) = Γ∗ via σ(ν)[h] = Z Ω Z D h(t)τω(dt)ν(dω) = Z Ω Z D h(t)β(ω)τω(dt)µ(dω)

for every h ∈ L∞(_D,D) and every dν =β dµ,β ∈L1(Ω,A, µ) . Therefore, the risk function is equal to

θ 7→ WθσPθ = σ(Pθ)[Wθ] = Z Ω Z D Wθ(t)τω(dt)Pθ(dω)

In this way, LC - decision procedures generalize ordinary (randomized) decision functions. Again, the question arises if this concept is an excessive generalization. In the following, it is explained why the answer to this question is “no, essentially not”.