Baire category results for stochastic orders

In the sense of Baire categories, we prove that the elements of a typical pair of univariate distribution functions (defined on a bounded subset of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}) cannot be compared in the sense of the usual stochastic order, the increasing convex order and the mean residual lifetime order. A similar result is also proved in the class of copulas, i.e. multivariate distribution functions with standard uniform marginals, equipped with the orthant order.


Introduction
A large literature has appeared in recent decades about stocastic orders, i.e. partial orders introduced in the class of distribution functions (see, for instance, Müller and Stoyan [7] and Shaked and Shanthikumar [14] for an excellent overview of the state-of-art). Given a partial order ≤ * and two distribution functions F and G, one may wonder whether either F ≤ * G or G ≤ * F. Such a question has relevance, for instance, when F and G represent risk distributions (see [7] However, roughly speaking, it is not clear how many pairs of distribution functions can be compared according to a given partial order ≤ * . Here, by using the topological description of a set given by Baire categories (see [9]), we are going to show that the set of all pairs of univariate distributions that can be compared by means of some popular stochastic orders is topologically small in the set of all possible pairs of distributions.
Moreover, we discuss an extension of this problem in the class of copulas (see, e.g., Durante and Sempi [4]; Joe [6]; Nelsen [8]) equipped with the lower orthant order by showing that two pairs of copulas are typically not comparable.

Preliminaries
Let I denote the interval [0, 1]. Let F be the class of (right-continuous) distribution functions F with support on I and such that F(0 + ) ≥ 0 and F(1) = 1. Given F ∈ F , we denote by F the associated survival function given by F(x) = 1 − F(x) for every x ∈ I. Moreover, the associated mean residual life function is defined as for every x ∈ I such that F(x) > 0. On F , we consider the weak convergence so that, given F ∈ F and a sequence (F n ) n∈N ⊆ F , F n tends to F weakly (write F n w − → F) whenever F n (x) → F(x), as n → ∞, for all the continuity points x of F. Thanks to Helly-Bray Theorem (see, e.g., Dudley [3] for all continuous and bounded functions φ : R → R. The Lévy metric d L on F is a metrization of the weak convergence, that is F n w − → F, as n tends to ∞, if and only if, d L (F n , F) → 0. The metric d L is defined, for every F, G ∈ F , as and it makes (F , d L ) a complete space (see, e.g., Schweizer and Sklar [11]; Sempi [12]). Given F ∈ F and ε > 0, we shall denote by B d L (F, ε) (respectively, B d L (F, ε)) the closed (respectively, open) ball of radius ε centered on F. Thus The following definitions can be found, e.g., in Müller and Stoyan [7]; Shaked and Shanthikumar [14].
for every increasing convex function φ : R → R such that the integrals in (2) exist. (c) F ≤ mrl G in the mean residual life order if for every x ∈ I such that F(x) > 0 and G(x) > 0.
The three partial orders on F introduced above are linked together as follows.
Proposition 1 Let F, G ∈ F . Then: We recall that part (a) of Proposition 1 follows from the fact that F ≤ st G is equivalent to for every increasing function φ : R → R such that the expectations in (4)  The following multivariate stochastic order will be also considered (see [7,14]).
Notice that ≤ lo is closed under weak convergence and, it is implied by the multivariate version of the order ≤ st (see [7,Sect. 3.3]).
Finally, we recall some terminology from Baire category (see [9]). Given a topological space (X , d), a subset A ⊆ X is called nowhere dense if its closure has empty interior, i.e., A = ∅. If A can be expressed as countable union of nowhere dense subsets of X , then A is called meager, or of first category. Subsets of X that are not of first category are called of second category. If A c = X \ A is meager, then A is called co-meager. Loosely speaking, we will also refer to the elements of a co-meager set as typical.
An important result that we need to recall is Baire Category Theorem, which asserts that a complete metric space (X , d) is a Baire space (we refer to Charalambos and Border [2] for a proof of this result). Baire spaces are, by definition, metric spaces in which every nonempty open set is of second category. Obviously, if X is a Baire space, X itself is of second category. A relevant observation is that, in Baire spaces, e.g. complete metric spaces, a co-meager set A ⊆ X is necessarily of second category, otherwise we would express X = A ∪ A c as a countable union of nowhere dense sets, hence X would be of first category, a contradiction.

Stochastic orders of univariate distribution functions
Now, we consider the set F 2 := F × F equipped with the topology induced by the metric d L × d L . For a given stochastic order ≤ * on F we denote by the set of all pairs of distribution functions that are comparable according to the given order. Clearly, the complement set of F 2 * in F 2 is given by Loosely speaking, we are going to show that, in a "typical" pair in F ×F , the two distribution functions are not comparable with respect to any of the stochastic orders introduced in Sect. 2. Now, we consider some preliminary results.
We claim that both (F 2 * ) 1 and (F 2 * ) 2 are closed, hence F 2 * is closed. We only prove that Assume that ≤ * =≤ icx . In view of the characterization of ≤ icx provided in Müller and Stoyan [7, Theorem 1.5.7], it holds that, for every fixed a ∈ I and for every n ∈ N, Thus, by the Dominated Convergence Theorem,

Remark 1
Notice that, in the space of all distribution functions on R, the stochastic order ≤ icx is not closed with respect to weak convergence (see, for instance, Müller and Stoyan [7, Example 1.5.8]).

Lemma 3 The set F 2 icx is nowhere dense.
Proof We need to prove that F 2 icx cannot contain any interior points. So let us assume, by way of contradiction, that Without loss of generality, we can assume that F ≤ icx G. Now, for every, δ, h ∈ I, one can define two functions F δ : R → I and G h : R → I in the following way: and Notice that both F δ and G h belong to F for all δ, h ∈ I.
We claim that it is possible to choose δ and h such that and for a suitable choice of δ and h. Indeed, fix any α ∈ (0, ε). Note that the left-hand inequality in (7) is true, since F δ (x) ≤ F(x) for every x ∈ I. To prove the right-hand inequality, it is enough to choose δ such that 1 − δ + α ≥ 1, i.e., δ ≤ α, that is the only restriction one has on the choice of δ. As far as h, for every x, the right-hand inequality in (8) is always true; the left-hand inequality holds if and only if, for every Moreover, notice that, for every possible choice of δ and for every a ∈ I, Now, without loss of generality, we can assume F = G, so that there exists some t ∈ (0, 1) such that Furthermore, we can assume that since we have showed that F and G can be approximated arbitrarily well by pairs of functions that satisfy all of these properties and that belong to that is, F icx G h . In order to prove that G h icx F, we set (9)), and without loss of generality we assume h > max(t, 1 − ε 0 ) (otherwise, just increase h). Hence we have:

A bivariate extension for copulas
Now, we consider an extension to the bivariate case by considering the Fréchet class F (F 1 , F 2 ) of continuous two-dimensional distribution functions with fixed univariate marginals F 1 , F 2 . As known (see, e.g., Rüschendorf [10]), each element F of such a class can be uniquely represented as F = C (F 1 , F 2 ), where C is a copula, i.e. a two-distribution function with standard uniform marginals (see, e.g., Durante and Sempi [4]). Moreover, it holds that: • for every F, G ∈ F (F 1 , F 2 ), F ≤ lo G if, and only if, C F (x, y) ≥ C G (x, y) for every (x, y) ∈ I 2 , where C F and C G are the copulas associated with F and G, respectively; • weak convergence in F (F 1 , F 2 ) is equivalent to uniform convergence of the corresponding copulas (see, e.g., Sempi [13]).
In view of the one-to-one correspondance between elements of the Fréchet class and copulas, we shall only consider the space of all bivariate copulas, which as usual we denote by C 2 , equipped with the metric d, defined for every A, B ∈ C 2 as We recall that the metric space (C 2 , d) is complete (and compact); see, e.g., Durante and Sempi [4]. Given A, B ∈ C 2 , A is less than or equal to B in the pointwise order (A ≤ B, in symbols), whenever A(x, y) ≤ B(x, y) for every (x, y) ∈ I 2 ; hence A ≥ B if and only if A ≤ lo B. In particular, for every C ∈ C 2 , it holds for every (x, y) ∈ I 2 . Notice that, the upper orthant order could be considered as well.
For every C ∈ C 2 , we shall denote by δ C the diagonal section of C, that is, the function δ C : I → I defined by δ C (t) := C(t, t), for every t ∈ I. Note that, for every t ∈ I and every copula C ∈ C 2 , one has δ C (t) = C(t, t) ≤ C(t, 1) = t. We can now state the following result.

Theorem 5 Let C be a fixed copula in C 2 . The following statements hold:
(a) If there exists some ε ∈ (0, 1) such that either δ C (t) < t holds for all t ∈ (1 − ε, 1) or for all t ∈ (0, ε), then the set then the set A C is of second category in (C 2 , d).
Proof First of all, it is straightforward to verify that A C is closed with respect to the metric d. Thus, in (a), the goal is to show that A C does not contain any interior points, whereas in the proof of (b) we need to prove that A C contains at least an interior point (with respect to the metric d). In fact, in this latter case, the assertion will follow from the fact that the metric space (C 2 , d) is a Baire space. As for (a), let us first assume that δ C (t) < t holds for all t ∈ (1 − ε, 1) for some ε ∈ (0, 1) and that, by way of contradiction, there exist G ∈ A C and δ > 0 such that B d (G, δ) ⊂ A C . Without loss of generality, we can assume δ < ε/2. Note that every copula D ∈ A C fulfills δ D (t) ≤ δ C (t) for every t ∈ I, and δ D (t) < t for all t ∈ (1 − ε, 1). Having set γ := δ/2, we define D as the ordinal of G and M with respect to the intervals [0, 1 − γ ] and [1 − γ, 1] (for an overview of ordinal sums, see, e.g., Durante et al. [5]), i.e., D is the copula given, for every (x, y) ∈ I 2 , by min(x, y), elsewhere.
Since γ < ε, we have 1 − γ > 1 − ε, hence which implies that D / ∈ A C . We shall prove that D ∈ B d (G, δ) ⊂ A C and that will be the contradiction we need. Let (x, y) ∈ I 2 . We have to distinguish several cases.
We shall prove that D ∈ B d (G, δ) ⊂ A C and that will be the contradiction we need. Let (x, y) ∈ I 2 . Again, we have to distinguish several cases.
W (x, y) = 0. In this subcase, (12) yields G(x, y) < ε/2. Moreover: W (x, y) > 0. In this subcase, (12) yields Moreover, because of the Lipschitz property of C, we have • If min(x, y) > ε and max(x, y) ≥ 1 − ε, then, as in the first case, we shall prove that C(x, y) = M(x, y). Indeed, having set z := max(x, y), because of the Lipschitz property of C one has Since C(x, y) = M(x, y) ≥ G(x, y), the proof of (b) is completed.
The following result focuses on C 2 × C 2 and basically states that, in a "typical" pair of copulas (C 1 , C 2 ), C 1 and C 2 are not comparable.

Theorem 6 The set
Proof First of all, it is easy to prove that P is closed with respect to the metric d × d: indeed, note that both P 1 and P 2 being closed with respect to the product distance d × d. The assertion will follow from proving that P does not have any interior points, i.e., every product of open balls of the kind B d (C 1 , γ ) × B d (C 2 , γ ) contains at least one pair (S 1 , S 2 ) of copulas that are not comparable (that is, S 1 lo S 2 and S 1 lo S 2 ), for every possible choice of (C 1 , C 2 ) ∈ P and γ ∈ (0, 1]. Fix any pair (C 1 , C 2 ) ∈ P and choose any ρ ∈ (0, γ /3). We define S 1 as the ordinal sum of W , C 1 , and M with respect to the intervals [0, ρ], [ρ, 1 − ρ] and [1 − ρ, 1], i.e., for every (x, y) ∈ I 2 , min(x, y), elsewhere, and S 2 as the ordinal sum of M, C 2 and W with respect to the same intervals, , i.e., for every (x, y) ∈ I 2 , min(x, y), elsewhere.
Note that hence S 1 S 2 , and hence S 1 S 2 . Thus, the copulas S 1 and S 2 are not comparable. We need to prove that For both the proofs of S 1 ∈ B d (C 1 , γ ) and S 2 ∈ B d (C 2 , γ ), we have to consider several cases, depending on the "position" of an arbitrarily chosen pair (x, y) ∈ I 2 . STEP 1: • If (x, y) ∈ (0, ρ) 2 , then , y), max(x, y)) < ρ < γ.
Assume (x, y) ∈ (ρ, 1/2] 2 . Since in this subcase we have: Assume y ∈ (ρ, 1/2] whereas x > 1/2. Since in this subcase we have, because of the Lipschitz property of C 1 : Assume x ∈ (ρ, 1/2] whereas y > 1/2. This subcase is analogous to the previous one, so the reader can easily verify that, again, it holds Assume (x, y) ∈ (1/2, 1 − ρ) 2 . Since in this subcase we have, because of the Lipschitz property of C 1 : • Assume (x, y) ∈ (ρ, 1 − ρ) 2 and S 1 (x, y) < C 1 (x, y). In this case: and again we need to distinguish four subcases, in which we apply the monotonicity, with respect to both arguments, as well as the Lipschitz property, of C 1 .
min(x, y) ≤ ρ. In this subcase: min(x, y) > ρ. Since in this subcase we necessarily have max(x, y) ≥ 1 − ρ, and because of the Lipschitz property of C 1 , it holds Since we covered all the possible cases and subcases, we have finally proved that d(S 1 , C 1 ) < γ , so that S 1 ∈ B d (C 1 , γ ). STEP 2: S 2 ∈ B d (C 2 , γ ). Let (x, y) ∈ I 2 .
min(x, y) > ρ. Since in this subcase we necessarily have max(x, y) ≥ 1 − ρ, and because of the Lipschitz property of C 2 , it holds Since we covered all the possible cases and subcases, we have finally proved that d(S 2 , C 2 ) < γ , so that S 2 ∈ B d (C 2 , γ ). Hence which means that every possible product of open balls includes pairs of copulas that are not comparable. This proves the assertion, that is, P is nowhere dense in C 2 × C 2 with respect to the metric d × d.
Thus the following result holds.
Corollary 7 (P) c is co-meager. Thus, two elements of the same Fréchet class F (F 1 , F 2 ) of continuous distribution functions are typically not comparable in the ≤ lo sense.