Random Measures Do Produce Generalized Likelihoods

In this article, we randomize the notion of the likelihood function by using random variables which may come from random measures. This randomization may be applied all over the positions of a sample and it is related to the notion of the base of a cone. Mathematics Subject Classification: 60G57; 97K50


Random measures and some comments on their properties
We remind the definition given in [3] (this is the Deinition [3, Def.1.1]slightly altered) of the random measure.We consider a probability space (Ω, F , μ) and a measurable space (E, E).
Definition 1.1 A random measure ξ on (E, E) over the probability space (Ω, F , μ) is a map ξ : E × Ω → R + , such that 1. the map ω → ξ(A, ω) is a random variable for any A ∈ E

the map A → ξ(A, ω) is a (probability) measure on E, μalmost surely in Ω.
What kind of random variables are the ξ A : Ω → R + , A ∈ E which are defined above?First of all, we may suppose that they belong to the linear space of the measurable R-valued random variables L 0 (Ω, F , μ) with respect to the probability space (Ω, F , μ).We also may suppose that these random variables ξ A , A ∈ E also belong to L 1 (Ω, F , μ).Since by definition the realizationsvalues of these variables are positive real numbers, they actually belong to Without loss of generality we may suppose that ξ ω is a probability measure on E for any ω ∈ Ω.If we suppose that E is a compact metric space, we may suppose that E is the Borel σ-algebra generated by it and we may consider the space of the σ-additive measures ca(E) over the σ-algebra E. This space is endowed with the total variation norm, according to which the set of the probability measures on E is the following set {ν ∈ ca . This set is the base of the cone ca + (E) defined by 1.But the space ca(E) is the dual space of the space C(E) of the continuous real-valued functions on E. This implies the definition of a random variable We also may supose that (E, E) = (Ω, F ), while Ω is a compact metric space and F its Borel σ-algebra.In this case, ξ ω is an element of the base {ν ∈ ca + (F )||ν|(Ω) = ν(Ω) = 1}.Hence by definition, Ω dξ ω (t) = 1, ω ∈ Ω.Also, we may define the random variable X f,ξ which may be taken to belong either to L 0 (Ω, F , μ) or to L 1 (Ω, F , μ) and specifically to L 1 (Ω, F , μ) for any f ∈ C(Ω).
belong to L 1 (Ω, F , μ), we may define measures on (Ω, F ) which are absolutely continuous with respect to μ.
Proof: If we take a certain A ∈ F and we consider the map In the same way, we may take a certain f ∈ C(Ω) and we may also consider the map As it is well-known the above maps define measures of bounded variation, or else Q ξ A , Q X f,ξ ∈ ca(F ), which are absolutely continuous with respect to μ.This implies the existence of the Radon -Nikodym derivatives Finally, we may also suppose that the measures Q ξ A , Q X f,ξ ∈ ca(F ) are actually probability measures, but this is not always necessary.

Generalized likelihood functions
As it is well-known from elementary statistics, a random sample of length n is a vector (X 1 , X 2 , ..., X n ) of independent, identically distributed random variables X i , i = 1, 2, ..., n whose common cumulative distribution function is F .In terms of the probability space (Ω, F , μ), the distribution probability measure Hence, we introduce the notion of (I, ξ I )-randomization, where , where each ξ i j , j = 1, 2, ..., k is a random measure on (Ω, F ) over (Ω, F , μ).Specifically, (I, ξ I )-randomization obtains two expressions.The first expression is the (I, ξ I , A I )-randomization and the second expression is the (I, ξ I , f I )-randomization.
Remark 2.1 By the first expression we mean that the Q X i j distribution measure is replaced by the probability measure beause they are approximated by them.By the second expression we mean that the Q X i j distribution measure is replaced by the probability measure Q X ξ i j ,f i j , f i j ∈ C(Ω), j = 1, 2, ..., k, for the same reason.

Remark 2.2
The results concerning the (I, ξ I , A I ) or the (I, ξ I , f I ) randomization cope with the approximation either of the Theorem 2.3 Consider a locally compact topological space Ω and let F be its Borel σ-algebra.Then for any X ∈ L 1 (Ω, F , μ), where μ is a probability measure on (Ω, F ), there is at least one random measure ξ : Proof: It is well-known that C c (Ω) (being the space of the continuous functions with compact support) is . 1 -dense in L 1 (Ω, F , μ), see [1, Th.12.9].Thus, for any n, we deduce the existence of some f n ∈ C c (Ω) such that Hence, we have to find fn ∈ L 1 and a random measure ξ such that f n (ω) = fn , ξ ω , μ-a.e.If we consider some ξ for which ξ ω = μ, μ-a.e., then we may put fn = f n .Also, if we suppose that ξ(A) ∈ L 1 , A ∈ F, then for the second case we select A n , n ∈ N to be the sets Corollary 2.4 Consider a locally compact topological space Ω and let F be its Borel σ-algebra.Then for any X ∈ L 2 (Ω, F , μ), where μ is a probability measure on (Ω, F ), there is at least one random measure ξ : Proof: [1, Th.12.9] also holds for L 2 (Ω, F , μ).
According to Remark 2.1, we may re-state more exact a definition of (I, ξ I , A I )-randomization and (I, ξ I , f I )-randomization.
Definition 2.8 The (I, ξ I , A I )-generalized likehihood with respect to the sample (X 1 , X 2 , ..., X n ) and the index set I = {1, 2, ..., k} which concerns the positions of randomization, is the sequence of joint densities In the next Propositions, we suppose that Ω is locally compact topological space.
Proposition 2.9 Proof: It arises from the Corollary 2.4 and the Hölder Inequality.

Proposition 2.10 ξ(I, A)
if the k factors of the product belong to L ∞ .
Proof: It arises from the Corollary 2.4 and the Hölder Inequality.

Appendix
In this Section, we give some essential notions and results from the theory of partially ordered linear spaces which are used in this paper.For these notions and definitions, see [2, Ch.
where E is a normed linear space, is called uniformly monotonic functional of C if there is some real number a > 0 such that f (x) ≥ a x for any x ∈ C. In case where a uniformly monotonic functional of C exists, C is a cone.C 0 = {f ∈ E * |f (x) ≥ 0 for any x ∈ C} is the dual wedge of C in E * .Also, by C 00 we denote the subset (C 0 ) 0 of E * * .It can be easily proved that if C is a closed wedge of a reflexive space, then C 00 = C.If C is a wedge of E * , then the set C 0 = {x ∈ E|x(f ) ≥ 0 for any f ∈ C} is the dual wedge of C in E, whereˆ: E → E * * denotes the natural embedding map from E to the second dual space E * * of E. Note that if for two wedges K, C of E, K ⊆ C holds, then C 0 ⊆ K 0 .
If C is a cone, then a set B ⊆ C is called base of C if for any x ∈ C\{0} there exists a unique λ x > 0 such that λ x x ∈ B. The set B f = {x ∈ C|f (x) = 1} where f is a strictly positive functional of C is the base of C defined by f .B f is bounded if and only if f is uniformly monotonic.If B is a bounded base of C such that 0 / ∈ B then C is called well-based.If C is well-based, then a bounded base of C defined by a g ∈ E * exists.If E = C − C then the wedge C is called generating, while if E = C − C it is called almost generating.If C is generating, then C 0 is a cone of E * in case where E is a normed linear space.Also, f ∈ E * is a uniformly monotonic functional of C if and only if f ∈ intC 0 , where intC 0 denotes the norm-interior of C 0 .

Definition 2 . 5 Definition 2 . 6 Definition 2 . 7
We obtain a (I, ξ I , A I )-randomization with respect to the set I and for the sample (X 1 , X 2 , ..., X n ), if we replace the density dQ j = 1, 2, ..., k, n ∈ N. We obtain a (I, ξ I , f I )-randomization with respect to the set I and for the sample (X 1 , X 2 , ..., X n ), if we replace the densitydQ X i j dμ by dQ X ξ i j ,f i j ,n dμ for any j = 1, 2, ..., k, n ∈ N. The (I, ξ I , f I )-generalized likehihood with respect to the sample (X 1 , X 2 , ..., X n )and the index set I = {1, 2, ..., k} which concerns the positions of randomization, is the sequence of joint densities 1, Ch.2, Ch.3].Let E be a (normed) linear space.A set C ⊆ E satisfying C + C ⊆ C and λC ⊆ C for any λ ∈ R + is called wedge.A wedge for which C ∩ (−C) = {0} is called cone.A pair (E, ≥) where If x ≥ y then λx ≥ λy for any λ ∈ R + and x + z ≥ y + z for any z ∈ E, where x, y ∈ E (compatible with the linear structure of E), is called partially ordered linear space.The binary relation ≥ in this case is a partial ordering on E. The set P = {x ∈ E|x ≥ 0} is called (positive) wedge of the partial ordering ≥ of E. Given a wedge C in E, the binary relation ≥ C defined as follows: x ≥ C y ⇐⇒ x − y ∈ C, is a partial ordering on E, called partial ordering induced by C on E. If the partial ordering ≥ of the space E is antisymmetric, namely if x ≥ y and y ≥ x implies x = y, where x, y ∈ E, then P is a cone.E denotes the linear space of all linear functionals of E, called algebraic dual while E * is the norm dual of E, in case where E is a normed linear space.Suppose that C is a wedge of (ii) If x ≥ y and y ≥ z then x ≥ z, where x, y, z ∈ E (transitive) (iii)