SOME COMMENTS ON CONCENTRATION AND EXPANSION FUNCTIONS AS APPLIED TO BIVARIATE DEPENDENCE

Potential utilization of concentration and expansion functions in the detection of dependence of two random variables is investigated. Also, a brief literature survey is explored. Pitfalls and drawbacks of such applications are emphasized.


INTRODUCTION
The exposition presented in the remainder of this work shall frequently refer to the immediate following definition and remark.The definition is based essentially on Raoult [5] (c.f., also, the monograph of Hengartner and Theodorescu [2]).(1.2) Remark 1.
(i) The above definition can also be extended to a family of bi-measure spaces such as ) ), , , , (( where J is an index set. (ii) Existence of concentration function (the upper bound for (1.1)) is provided in Raoult [5] in terms of Neymann-Pearson test procedure (c.f., also, [2]).As a matter of fact, a typical example for application of concentration to a statistical area is Neymann-Hypothesis testing procedure, where A is rejection region, μ represents the measure under null hypothesis and ϑ (and hence a ϑ ) denotes the measure under alternative hypothesis.
(iii) If U ℑ = or if the relevant algebra is clear from the context, the index U of u f and u g can be ignored.
(iv) Note that μ ϑ << ℑ Ω is a bi-probability space, i.e., if ϑ and μ are probability measures on (vi) Also, for the special case in such a bi-probability space, we have  The presentation below is connected to the case where ) , , , (

{ }
is a biprobability space.In the two-dimensional case involving families of distributions like Fréchet class, we shall also use a family of spaces ). ), , , , (( Throughout the remainder of discussions, it is assumed that concentration and expansion functions exist.Within this setup, we let,

{
}, Thus, for t A defined as The last integral is based on the existence of a distribution function Note that when both ϑ and μ are dominated by a common measure ν with ν ϑ the last quotient being so-called likelihood ratio.
will otherwise be referred as generalized likelihood ratio below.Integrating out by parts, the last term of (1.6) becomes , and, by substituting (1.8) in the relation given in Remark 1 above, the expansion in (1.7) on the other hand boils down to The quantity t c in (1.6)-(1.7)and (1.8)-(1.9) is the so-called t-quantile for the population of In a concluding remark, Cifarelli and Regazzini [ ] 1 point potentials of (1.3) for application to such probabilistic issues as homogeneity, association, etc. Upon the suggestion, Scarsini [ ] 6 attempts to extend the concept of concentration to the twodimensional case and investigate its potential uses for ordering Fréchet class of bivariate distributions in terms of the degree of dependence they display.The conclusion reached by the author appears however to be hardly optimistic.As will be clear in the following pages, these nonpromissing results stem from the facts that (i), without (1.2) and/or (1.4), (1.1) and/or (1.5) alone provides only a partial, and often misleading, picture for dependence, especially in the presence of positive quadrant dependence (c.f., Lehmann [ ], 3 , for the concept); (ii) as is also posed by Scarsini [ ] 6 , the question that whether the concept of concentration and expansion does really coincide with the concept of dependence of random variables needs further investigations.These issues will be taken up next in a sequence of sections below.First, however, we summarize the known properties of these functions (Lemma 1 and Corollary to this lemma).We also prove some properties of expansion function (Lemma 2), which, in fact, do not seem to exist in literature.In order to be able to detect bivariate dependence in terms of concentration and expansion functions, we then set up a two-dimensional framework.The final two sections investigate the relationship of dependence to the concept of concentrationexpansion functions.

ANALYTICAL PROPERTIES OF CONCENTRATION AND EXPANSION FUNCTIONS
As explicitly given by Raoult [ ], 5 the concentration and expansion functions is also an increasing (nondecreasing) and continuous convex function in I; (iv) the functions 2) for a bi-probability space.
Proof: Noting that the functions are probability measures, the nondecreasing and continuity properties are easy to see.As for the concavity (convexity), this will be discussed below.See, also, [ ].In addition to the properties mentioned in Lemma 1, the expansion function u ϕ displays some further features -the index U of u ϕ will henceforth be ignored: is singular with respect to μ , i.e. , .
. Hence, from (1.7), we obtain The left-hand side inequality follows from (1.4).To show this let Now these two measures must satisfy This however is against the initial assumption that there is no Thence, the result follows.
(iii) When ϑ is singular with respect to , μ , so will be the measure a ϑ and hence the integral in (1.7) will be zero.QED Remark 3. The case where ( ], is interesting to note: In fact, for this case, we have This is tantamount to stating that This case will be resumed in connection with the discussions on twodimensional case (c.f., end of Section 3 below).

Corollary:
In view of the properties of ) 1 ( t − ϕ mentioned in the foregoing lemma and remark, the concentration function ( ]; Remark 4. Through proved differently, excepting for (vii), the results of this corollary can also be found in Theorem 2.3 of [ ].
1 Part of the proof of (vii) is in Remark 3. The remainder of the proof can be found at the end of Section Three where we dwell on Fréchet bounds.
When concentration and expansion functions both exist, their convex and concave natures seem to render them dual to each other for potential applications in Statistics.For easy reference, this duality is emphasized below.In fact, let for the respective concentration and expansion functions corresponding to (2.3), we have, (2.4) On the other hand, if, so that, for this case, expansion functions will be more appropriate.Conversely, if and thus concentration functions will be more appropriate for this latter case.Remark 5. When ϑ and μ are indexed and identified by some real-valued parameter θ with the respective values ϑ θ and μ θ concentration and expansion functions can meaningfully be associated with the well-known information integral of Kulbak: For

{ }
), 1 , 0 ( , ) ( : provided that they are all defined.When multiplied by (-1) and taken over the entire space { } , ) ( : ω ω l the integral on the left hand side is so-called Kulback's mean information or Kulback's information integral on .
represent H-functions of Boltzmann, conditional on .B, Accordingly, a bi-probability space (3.5)By substituting 0 μ for μ in Definition 1 above, + μ for , ϑ X for , Ω M for Q and the Borel algebra of subsets of M for U, a setup parallel to the one in Definition 1 is obtained.Provided that the relevant distributions exist, we can set When no confusion is expected to arise, the probability measures in (3.3) and the distributions in (3.6) will interchangeably be used below.
As in the univariate case, the distribution gives the concentration function in (1.5), where, for some yields the expansion function in (1.4) for the two-dimensional case.
An important point to note for the bivariate case is the fact that .In practice, this means that In other words, the superscript a of a + μ is unrequired in the bivariate case.
On the other hand, for the discussion to follow and for future reference, it should be recalled that independence is defined in terms of , ) , such that negation of independence refers to dependence; negative complete dependence on the other hand corresponds to Fréchet lower bound, i.e., ) ( whereas positive complete dependence applies to Fréchet upper bound, i.e., ) ( which, in terms of distributions, can alternatively be re-expressed as As will be noted below, the Fréchet bounds for bivariate measures (distributions) are not necessarily identical with the respective lower and upper bounds for concentration and expansion functions.For practical reasons, this constitutes a drawback in detecting the phenomenon of complete dependence through these functions.
Returning to the comment made in Remark 3 above, for some The constant value + t l of the generalized likelihood ratio . On the other hand, the Fréchet upper bound is The lower bound for concentration functions is For this to hold, we must have .Now the Fréchet lower bound for That is equal to zero everywhere in M, when

IMPLICATIONS FOR BIVARIATE DEPENDENCE
As is noted in earlier, concentration functions in (1.4) are appropriate for the case { } .These two respective cases can be matched with negative and positive dependencies in the bivariate setup: For two distinct random variables such as X and Y defined in (3.1) above, the concepts of negative and positive quadrant dependence are defined respectively as (c.f., [3]), Obviously, the relations in (4.1)-(4.2) follow from the following respective relations of their derivatives: Therefore, the corresponding generalized likelihood ratios become standing for the respective ratios for (4.1) and (4.2).Thus, for { }B and for the respective two cases (4.3) and (4.4), we have Application of concentration functions to positive quadrant dependence will thus result in an analytical inconsistency like .Such inconsistencies are often come across in literature dealing with positive dependence ordering.
To sum up the foregoing, we have:   which actually is the optimal way to screen out such a detection.Since the larger t is the smaller the interval ) 1 , ( t t − will (i) If they are to be used for statistical applications on bivariate dependence, concentration and expansion functions must be both be used, because concentration is seemingly appropriate for negative dependence, and expansion appears on the contrary to be suitable for positive dependence.These functions are nonetheless incapable of detecting complete dependences, especially positive complete dependence.
(ii) Clearly, the concepts of concentration and expansion are intrinsically not identical with the concept of dependence bearing on two random variables.The former relate to detection of continuity or singularity of two probability measures.The latter bears on the question whether joint probabilities (distributions) are formed by the product of their marginal measures (distributions) or not.Therefore, care should be taken in applications of concentration and expansion to dependence.
(iii) Since both the marginals and the joint measures are defined on a common measurable space, joint measures (distributions) cannot be singular with respect to the product of their marginals (marginal distributions) everywhere in the relevant space, unless we are confronted with an empty probability space.
(iv) The foregoing exposition comprises some initial results of an ongoing research on the topic.The authors intend to present full results in a separate paper in near future.

Definition 1 .
For a bi-measure space defined on it), the measure ϑ is assumed to have Lebesgue decomposition with respect to the measure .Furthermore, if Q c is a subset of Ω , such that concentration (resp., expansion) of the measure ϑ with respect to the measure μ on the subalgebra U, if,

5
discusses the concepts of concentration and expansion (étalement) in connection with Lebesgue decompositions covering a general bi-measure space on a unidimensional basis.The concentration in (1.1) is also noted [ ]. 2 An alternate better-known notion of concentration is due to Levy [ ], 4 who uses the concept for sums of random variables.To discriminate it from the former, the latter type of concentration is generally labelled as Lévy concentration functions.Raoult [ ] 5 shows the relation between these two types of concentration -Further developments concerning Lévy concentration functions can be found in [ ] 2 -The expansion function in (1.2) is however less known and hence has apparently no statistical and probabilistic applications.Within a different setting and in a relatively recent attempt to obtain a general formulation for the so-called Gini-type concentration indices, on the other hand, Gifarelli and Regazzini [ ] 1 re-dwell upon concentration, and to this end, the authors specify a function (c.f., Theorem 2.2 in [ ].1 ).

∈
∀t I.The relevant measure space for (1.3) is again a bi-probability space such as expansion function in the sense of(1.2), and yet, it is called concentration function by the authors.In fact, except for (1.3), the remaining results of Cifarelli and Regazzini [ ] 1 pertain to concentration functions.It is thus possible to classify Cifarelli-Regazzini type of concentration functions within the general category of concentration functions given in Definition1.
increasing (nondecreasing) and continuous concave function in I;(ii) given two sub-algebras * U and 0

5 Remark 2 .
Cifarelli and Regazzini (Theorem 2.3 in [ ] 1 ) maintains without proof that concentration functions are convex.However, in connection with decomposition concentrations, a proof for the concavity of concentration functions is provided in Theorem 4.2.2 of [ ].

2
The discussions in both Raoult [ ] 5 and Cifarelli and Regazzini [ ] 1 run in a unidimensional setup.To provide a framework for applications of concentration and expansion functions to bivariate dependence, a two-dimensional setup must hence be introduced: Given a probability space distinct measurable functions (random variables) on Ω

Z
be obtained.The measure + μ cannot be singular with respect to the measure 0 μ everywhere in B, simply for the reason that marginal probabilities X μ and Y μ are obtained from + μ and that f., the final paragraph of Section 4 for further discussions).Therefore, + μ is assumed to have the Lebesgue decomposition

1 , 0 .
concentration and expansion functions at the same time, the interval thus be reduced by its half, i.e., [ ] 2 The inequalities in (4.5) and (4.6) indicate on the other hand that, given some 5)stand for the corresponding concentration function.If no confusion is expected to arise, the index U of u