- Email: [email protected]
Elementary multivariate rearrangements and stochastic dominance on a Fréchet class
Available online at www.sciencedirect.com
Journal of Economic Theory 147 (2012) 1450–1459 www.elsevier.com/locate/jet
Elementary multivariate rearrangements and stochastic dominance on a Fréchet class Koen Decancq a,b,c,∗,1 a Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven, Belgium b CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium c Herman Deleeck Centre for Social Policy, University of Antwerp, B-2000 Antwerp, Belgium
Received 31 March 2010; final version received 21 February 2011; accepted 21 June 2011 Available online 4 November 2011
Abstract A Fréchet class collects all multivariate joint distribution functions that have the same marginals. Members of a Fréchet class only differ with respect to the interdependence between their marginals. In this paper, I study orders of interdependence on a Fréchet class using two multivariate generalizations of the bivariate rearrangement proposed by Epstein and Tanny (1980) [4] and Tchen (1980) [16]. I show how these multivariate rearrangements are underlying multivariate first order stochastic dominance in terms of the joint distribution function and the survival function. A combination of both rearrangements is shown to be equivalent to the concordance order proposed by Joe (1990) [9]. © 2011 Elsevier Inc. All rights reserved. JEL classification: C14 Keywords: Concordance order; Fréchet class; Multivariate rearrangements; Multivariate stochastic dominance; Orthant dependence order; Supermodular order
* Correspondence to: Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven,
Belgium. E-mail address: [email protected]. 1 This is a revised version of a chapter of my PhD dissertation. I gratefully thank André Decoster, James Foster, Thibault Gajdos, Luc Lauwers, Margaret Meyer, Alfred Müller, Erwin Ooghe, Marco Scarsini, Erik Schokkaert, Dirk Van de gaer, John Weymark, Claudio Zoli, three anonymous referees and participants at seminars in Leuven, Montreal, Oxford, Louvain-la-Neuve, Verona, Paris and Moscow for very helpful comments and suggestions to this or an earlier version of the paper. Remaining errors are all mine. 0022-0531/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2011.11.001
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1451
1. Introduction A Fréchet class consists of multivariate joint distribution functions with the same marginals. A joint distribution function is said to be more interdependent when its marginals tend to be large or small simultaneously. The difference between members of the same Fréchet class is the interdependence between their marginals. Hence, Fréchet classes are a natural environment to study multivariate interdependence. In a wide range of economic problems one is interested in comparing interdependence of joint distribution functions. In financial and actuarial analysis, comparing portfolios with the same individual asset returns or risks with respect to their interdependence provides a measure of the degree of its diversification and systemic risk (Denuit et al. [3]). In welfare economics, the notion of interdependence between income distributions at different points in time is inversely related to (exchange) income mobility (D’Agostino and Dardanoni [2]). Atkinson and Bourguignon [1], Tsui [17] and Gravel and Moyes [5] have argued that an order of multidimensional inequality should be sensitive to the interdependence between the dimensions.2 A central role in the (economic) literature on bivariate interdependence has been played by the intuitive bivariate elementary rearrangement suggested by Hamada [7]. The bivariate rearrangement shifts probability mass on the vertices of a rectangle, such that the probability mass on the vertices with both high or both low values is increased and the probability mass on the vertices with one high and one low value is decreased. It is intuitive to say that a bivariate distribution function after such a rearrangement shows more interdependence between its marginals. Furthermore, Epstein and Tanny [4] and Tchen [16] have shown that in the bivariate case, a joint distribution function obtained by a bivariate rearrangement is stochastically dominated by the initial one to the first order and vice versa. This important result combines two equivalent perspectives: one in terms of elementary rearrangements that are interpretable in terms of interdependence and another which is empirically implementable. Unfortunately, this equivalence no longer holds in a multivariate setting once there are three or more dimensions. This paper proposes a generalization of the bivariate elementary rearrangement which has some theoretical and intuitive appeal and which allows one to recover the equivalence between both perspectives in a multivariate setting. The paper contains three related results. First, I characterize multivariate first order stochastic dominance on a Fréchet class in terms of elementary rearrangements which shift probability mass on the vertices of a hyperbox (rather than a rectangle). Second, I show that an alternative geometric generalization of the bivariate elementary rearrangement which is a priori equally justifiable characterizes multivariate first order stochastic dominance in terms of the survival function rather than the joint distribution function. In fact, both generalizations are underlying the lower and upper orthant orders as studied in the literature on stochastic orders.3 Third, combining both types of rearrangements characterizes the concordance order proposed by Joe [9]. I argue that the almost exclusive focus of the economic literature on multivariate stochastic dominance in terms of the joint distribution function seems unwarranted based on an analysis of the underlying rearrangements. The paper is organized as follows. Section 2 introduces notation. Section 3 surveys the literature on bivariate elementary rearrangements and stochastic dominance on a bivariate Fréchet 2 Hennessy and Lapan [8] and Meyer and Strulovici [11] provide an overview of the notion of interdependence in various economic problems. 3 See Müller and Stoyan [13] for an overview.
1452
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 1 Rearranged probability mass for a positive 2-rearrangement with δ > 0. −δ
+δ
(x 1 , x 2 ) (x 1 , x 2 )
(x 1 , x 2 ) (x 1 , x 2 )
class. In Section 4, the bivariate rearrangements are generalized into the multivariate setting and I investigate how they are underlying multivariate stochastic dominance. Section 5 concludes. 2. Notation Let m be a natural number larger than or equal to 2. Let F and G be two (discrete) mdimensional joint distribution functions with S being the smallest product set containing all points in the joint support. Their corresponding density functions are f and g and the corresponding survival functions are denoted by F and G.4 We say that G is stochastically dominated by F according to the first order if and only if G(x) F (x) for all x in S. Assume furthermore that F and G have m non-degenerate marginals F1 , . . . , Fm . Denote by F (F1 , . . . , Fm ) the set of all joint distribution functions with given marginals F1 , . . . , Fm . This is called the Fréchet class of F1 , . . . , Fm . Vector inequalities are denoted by , <, . We write x
s x}, and F (x) =
f (s), where the sum is taken over all elements of {s ∈ S | s > x}.
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1453
When δ > 0, we call the rearrangement a positive 2-rearrangement, when δ < 0, we call the rearrangement a negative 2-rearrangement. The positive 2-rearrangement goes back to the early work of Hamada [7] who argues that it is intuitive to say that the marginals of a joint distribution function are more interdependent after such a rearrangement. Epstein and Tanny [4] and Tchen [16] have shown that on a bivariate Fréchet class, a finite sequence of positive 2rearrangements leads to stochastic dominance of the first order expressed in terms of the joint distribution functions as well as stochastic dominance in terms of the survival functions. Moreover, they also show that the converse is true. The result, proven independently by these authors, can be stated as follows. Proposition 1. Suppose that m = 2. For joint distribution functions F and G in F(F1 , F2 ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of positive 2-rearrangements, 2. G(x1 , x2 ) F (x1 , x2 ) for all (x1 , x2 ) in S, 3. G(x1 , x2 ) F (x1 , x2 ) for all (x1 , x2 ) in S. Proof. See Theorem 1 of Epstein and Tanny [4] or Tchen [16] on the equivalence between statements 1 and 2. The implication that 1 implies 2 is straightforward. The converse implication that 2 implies 1 is proven in a constructive way by providing an algorithm to identify an appropriate sequence of positive 2-rearrangements. The equivalence between 2 and 3 is well known in the bivariate case (see, for instance, Theorem 3 of Epstein and Tanny [4]). 2 This result combines two perspectives. The first statement takes the perspective of an elementary rearrangement which can be interpreted as leading to more interdependence between both marginals of a bivariate joint distribution function. The other two statements involve a finite number of tests which are empirically implementable. The second statement is a test of bivariate first order stochastic dominance as studied by Hadar and Russell [6]. The authors show a further equivalence with unanimous agreement amongst all expected utility maximizers with a supermodular (or superadditive) utility function that G is preferred to F .5 The equivalence between these perspectives is a powerful result which gives a clear interpretation to bivariate stochastic dominance on a Fréchet class as a test of “more interdependence”. (See, for instance, Atkinson and Bourguignon [1].) 4. The multivariate case Let us relax the assumption on m and move to the multivariate case. Consider the following rearrangement (Table 2) which involves three dimensions. Let δ be negative so that positive probability mass −δ is added to the vectors in the left column of the table and probability mass −δ is taken away from the vectors in the right column of the table. It can be checked that after this rearrangement is carried out, positive probability mass is added to the vertex which is simultaneously low in all dimensions and that the new joint distribution function is first order stochastically dominated by the initial one. However, from the table it is clear that this rearrangement cannot be obtained by a finite sequence of positive 2-rearrangements as it consists of 5 For any sufficiently differentiable supermodular function U , it holds that ∂ 2 U (x)/∂x ∂x 0 for all x in S. j1 j2
1454
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 2 Rearranged probability mass for an alternating 3-rearrangement with δ < 0. −δ
+δ
(x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 )
(x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 )
a positive 2-rearrangement (in the plane where xj1 = x j1 , the first two rows of Table 2) and a negative one (in the plane where xj1 = x j1 , the last two rows of Table 2). Hence, by relaxing the assumption on m, statement 2 of Proposition 1 no longer implies statement 1. Furthermore, it is well established that the equivalence between statement 2 and statement 3 no longer holds once there are three or more dimensions. (See, for instance, Müller and Stoyan [13].) Unfortunately, the equivalence between the statements in Proposition 1 breaks down in the multivariate case. Each of the three statements of Proposition 1 inspired a separate multivariate stochastic order in the statistical literature. The first statement and its equivalence with a unanimous agreement amongst all expected utility maximizers with a supermodular utility function has lead to the supermodular order. (See Müller and Scarsini [12] and Meyer and Strulovici [11].) The second statement is the definition of the lower orthant order and the last statement defines the upper orthant order.6 The supermodular order has the advantage of being built upon the bivariate elementary rearrangement with its clear interpretation, but is hard to test empirically. The orthant orders can be easily tested, but cannot be interpreted in terms of 2-rearrangements. The aim of this paper is to recover at least some aspects of the bivariate equivalence by first generalizing the notion of a positive 2-rearrangement in the multivariate setting and then by showing how this generalization is underlying the orthant orders. To this end, some additional notation is necessary. Recall that x = (x 1 , . . . , x m ) and x = (x 1 , . . . , x m ) are two different vectors in S with x
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1455
It can easily be checked that a k-rearrangement indeed generalizes a 2-rearrangement. Note that a k-rearrangement can be applied to any multivariate function f with S as support, irrespective of whether f is a joint density function or not. Two specific k-rearrangements are considered in this paper. When δ > 0, we call the rearrangement a positive k-rearrangement and when (−1)k δ > 0, we call it an alternating k-rearrangement. For any even k, an alternating k-rearrangement is also a positive k-rearrangement and vice versa. In general, there is a top-tobottom symmetry between an alternating k-rearrangement and a positive k-rearrangement. Let us return to the example given by Table 2. Positive probability mass is added to the vertices with either 1 or 3 values of xj equal to x j . Hence, this rearrangement is an alternating 3-rearrangement. The inductive observation of stochastic dominance (in terms of the joint distribution function, not in terms of the survival function) can be generalized further, as will be shown in Proposition 2. Let ≺ be the lexicographic ordering on S, so that y ≺ x if y1 < x1 or y1 = x1 and y2 < x2 or y1 = x1 and y2 = x2 and y3 < x3 and so on. The following lemma will be helpful in proving the proposition. Lemma 1. Suppose that m 2 and l 0. Further suppose that F l and G have the same marginals F1 , . . . , Fm with G(x) F l (x) for all x in S. Let t = (t1 , . . . , tm ) be the minimal element of S according to ≺ for which g(t) > f l (t). Then there exists an alternating k-rearrangement R k with k in {2, . . . , m} such that f l+1 = R k (f l ) and: 1. 2. 3. 4.
f l+1 (t) = g(t), F l+1 has the same marginals as F l and G, G(x) F l+1 (x) F l (x) for all x in S, F l+1 (x) = F l (x) for all x ≺ t .
Proof. I will select a particular alternating k-rearrangement that meets the conditions stated in the lemma. Define therefore for all j in {1, . . . , m}: min{Zj } if Zj = ∅, yj = if Zj = ∅, tj where the sets Zj ⎧ ⎪ ⎨ Z1 = z1 ⎪ ⎩ ⎧ ⎪ ⎨ Z2 = z2 ⎪ ⎩ .. .
⎧ ⎪ ⎨ Zj = zj ⎪ ⎩ .. .
are recursively defined as follows: ⎫ ∃z with z1 = z1 such that ⎪ ⎬ t1 < z1 , t2 z2 , . . . , tm zm and , ⎪ ) > g(z , . . . , z ) ⎭ f l (z1 , . . . , zm m 1 ∃z with z2 = z2 such that
and t1 z1 y1 , t2 < z2 , t3 z3 , . . . , tm zm
) > g(z , . . . , z ) f l (z1 , . . . , zm m 1
⎫ ⎪ ⎬ ⎪ ⎭
,
⎫ ∃z with zj = zj such that ⎪ ⎬ and , t1 z1 y1 , . . . , tj < zj , tj +1 zj +1 , . . . , tm zm ⎪ ⎭ l f (z1 , . . . , zm ) > g(z1 , . . . , zm )
1456
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
⎫ = z such that ∃z with zm ⎪ m ⎬ and ym−1 , tm < zm . Zm = zm t1 z1 y1 , . . . , tm−1 zm−1 ⎪ ⎪ ⎭ ⎩ l f (z1 , . . . , zm ) > g(z1 , . . . , zm ) ⎧ ⎪ ⎨
Consider the following alternating k-rearrangement:
R k = R k (t1 , . . . , tj , . . . , tm ), (y1 , . . . , yj , . . . , ym ), (−1)k δ .
(1)
Let δ = g(t) − f l (t) so that f l+1 (t) = f l (t) + δ = g(t) and that indeed condition 1 is fulfilled. Note that this rearrangement shifts probability mass towards t. The number of dimensions for which yj = tj (that is the number of the non-empty sets Zj ) equals k, the dimensionality of the rearrangement. To show that k is in {2, . . . , m} and that F l+1 has the same marginals as F l and G, I consider the following three cases. Assume first that k = 0. Observe that for all the vectors of S which are preceding t according to ≺ it holds that g(t) = f l (t). If k = 0, it holds that g(z) f l (z) for all z > t by the definition of the sets Zj . Together with the observation that g(t) > f l (t), this contradicts that F l and G have the same marginals F1 , . . . , Fm . In particular, there exists at least one j in {1, . . . , m}: Gj (xj ) > Fjl (xj ) for xj tj . Second, assume that k = 1. Without loss of generality, assume that the single nonempty set is Zj , so that the obtained alternating k-rearrangement is R k ((t1 , . . . , tj , . . . , tm ), (t1 , . . . , yj , . . . , tm ), (−1)k δ). This leads to Gj (xj ) > Fjl (xj ) for all xj ∈ [tj , yj ], which contradicts the equal marginals between G and F l . Third, assume that k is in {2, . . . , m}. By the definition of an alternating k-rearrangement, all marginals of the obtained F l+1 contain an equal number of vertices which belong to Bek as to Bok so that F l+1 has the same marginals as F l (and G). To establish the third and fourth conditions, we define V = {v ∈ S | t v < y}, the hyperbox B k (t, y) excluding its upper boundaries. From the definition of an alternating k-rearrangement and an appropriate choice for δ, we have that F l+1 (x) = F l (x) for all x in S/V. Hence, condition 4 holds. Furthermore, it follows that G(x) = F l+1 (x) for all x in V. Given that G(x) F l (x) for all x in S, we obtain that G(x) F l+1 (x) for all x in S. By the definition of an alternating k-rearrangement, we obtain additionally that F l+1 (x) F l (x) for all x in S so that also condition 3 holds. 2 Proposition 2. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) for all x in S. Proof. It follows immediately from the definition of an alternating k-rearrangement that if g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}, then G(x) F (x) for all x in S. To show the converse, an explicit algorithm is provided by repeatedly applying the lemma; first with the joint distribution function F 0 = F and the minimal element of S according to ≺ for which g > f l to obtain an alternating k-rearrangement with k in {2, . . . , m} and F 1 , then with F 1 and the new minimal element of S according to ≺ for which g > f l to obtain another alternating k-rearrangement with k in {2, . . . , m} and F 2 and so on. After every step, the obtained
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1457
Table 3 A sequence of alternating rearrangements for the example of Müller and Scarsini [12]. l
t
Rearrangement
Probability mass ε added to
Probability mass ε removed from
1 2 3 4 5 6 7 8 9
(0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (0, 2, 0) (0, 2, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0)
R 3 ((0, 0, 0), (1, 1, 1), ε) R 3 ((0, 0, 1), (1, 1, 2), ε) R 3 ((0, 1, 0), (1, 2, 1), ε) R 3 ((0, 1, 1), (1, 2, 2), ε) R 2 ((0, 2, 0), (2, 2, 1), ε) R 2 ((0, 2, 1), (2, 2, 2), ε) R 3 ((1, 0, 0), (2, 1, 1), ε) R 3 ((1, 0, 1), (2, 1, 2), ε) R 3 ((1, 1, 0), (2, 2, 1), ε)
(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) (0, 0, 1), (0, 1, 2), (1, 0, 2), (1, 1, 1) (0, 1, 0), (0, 2, 1), (1, 1, 1), (1, 2, 0) (0, 1, 1), (0, 2, 2), (1, 1, 2), (1, 2, 1) (0, 2, 0), (2, 2, 1) (0, 2, 1), (2, 2, 2) (1, 0, 0), (1, 1, 1), (2, 0, 1), (2, 1, 0) (1, 0, 1), (1, 1, 2), (2, 0, 2), (2, 1, 1) (1, 1, 0), (1, 2, 1), (2, 1, 1), (2, 2, 0)
(1, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1) (1, 1, 2), (1, 0, 1), (0, 1, 1), (0, 0, 2) (1, 2, 1), (1, 1, 0), (0, 2, 0), (0, 1, 1) (1, 2, 2), (1, 1, 1), (0, 2, 1), (0, 1, 2) (0, 2, 1), (2, 2, 0) (0, 2, 2), (2, 2, 1) (2, 1, 1), (2, 0, 0), (1, 1, 0), (1, 0, 1) (2, 1, 2), (2, 0, 1), (1, 1, 1), (1, 0, 2) (2, 2, 1), (2, 1, 0), (1, 2, 0), (1, 1, 1)
function F l shares the same univariate marginals with F and G. Moreover, compared to F l , F l+1 coincides with G for an additional element of S. After a finite number L steps, G equals F L over the complete support S and its corresponding density function g is obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}.8 2 To illustrate the algorithm, consider the following example, which is inspired by Müller and Scarsini [12]. Let g and f be two three-dimensional density functions such that f has equal probability mass ε = 1/6 on the six points (2, 2, 1);
(2, 1, 2);
(1, 2, 2);
(1, 1, 1);
(0, 0, 2)
and
(2, 0, 0)
and
(0, 0, 0).
and g has equal probability mass ε = 1/6 on the six points (2, 2, 2);
(2, 1, 1);
(1, 2, 1);
(1, 1, 2);
(2, 0, 2)
It is easy to see that F and G have indeed the same marginals. Müller and Scarsini [12] show that G(x) F (x) for all x in S. By applying the algorithm of the proof of Proposition 2, I provide a sequence of alternating k-rearrangement with k in {2, . . . , m} which lead from f to g. These rearrangements are summarized in Table 3. Proposition 2 recovers the two perspectives of the bivariate result. It connects a test of multivariate stochastic dominance with a sequence of elementary multivariate rearrangements. Furthermore, Hadar and Russel [6] and Atkinson and Bourguignon [1] have shown the equivalence between multivariate first order stochastic dominance and unanimity within the class of utility functions with alternating signs of their cross derivatives. Scarsini [15] calls such a utility function an m-variate risk averse utility function. Can we interpret this test of multivariate stochastic dominance as a test of “more interdependence” as we did in the bivariate case? Yes and no. When k is even, the k-rearrangement rearranges probability on 2k vertices and shifts thereby probability mass towards the vertices which are simultaneously high or low in all dimensions. Although these rearrangements might be less appealing to one’s intuitions given the large number of involved vertices, it seems reasonable to extend the intuition of Hamada [7] and say they indeed increase interdependence. 8 Although F and G are joint distribution functions, the intermediate function F l for l = 1, . . . , L − 1 obtained after each of these alternating k-rearrangements is not necessarily a “proper” joint distribution function. It is not excluded, for instance, that more probability mass is removed from a vertex than there was initially present. Gravel and Moyes [5] introduce in these cases a so-called “phantom”.
1458
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 4 A sequence of positive rearrangements for the example of Müller and Scarsini [12]. l
t
Rearrangement
Probability mass ε added to
Probability mass ε removed from
2 1
(2, 2, 2) (2, 0, 2)
R 3 ((1, 1, 1), (2, 2, 2), ε) R 2 ((0, 0, 0), (2, 0, 2), ε)
(2, 2, 2), (2, 1, 1), (1, 2, 1), (1, 1, 2) (0, 0, 0), (2, 0, 2)
(2, 2, 1), (2, 1, 2), (1, 2, 2), (1, 1, 1) (2, 0, 0), (0, 0, 2)
However, when k is odd things become more intricate (as in the example in Table 2). An alternating k-rearrangement shifts probability mass to the vertex with all low values, but probability mass is shifted away from the vertex with all high values. One could even argue that such a rearrangement does not increase interdependence, but rather decreases it. So that in case k is odd, a positive k-rearrangement increases dependence. This leads to the following result. Proposition 3. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of positive k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) for all x in S. Proof. The proof has the same structure as the one of Proposition 2 and is omitted. In particular, the algorithm needs to be adjusted in a top-to-bottom symmetrical way. 2 Returning to the example of Müller and Scarsini [12], the authors show that G(x) F (x) also holds for all x in S. Here, a sequence of positive k-rearrangement with k in {2, . . . , m} can be provided to obtain g from f . In particular, the algorithm leads to the sequence of positive rearrangements given in Table 4. Observe that the last statement of Proposition 3 involves a test of first order stochastic dominance in terms of the survival function. A priori there seems no reason to prioritize alternating k-rearrangements to positive k-rearrangements, yet most results in the economic literature on multivariate stochastic dominance focus exclusively on the results obtained by imposing alternating k-rearrangements. This seems unwarranted. Given the hard choice between both types of k-rearrangements, a natural approach is to require both initial statements of Propositions 2 and 3 to be fulfilled. Only when a sequence of alternating k-rearrangements can be found that leads from f to g as well as a sequence of positive k-rearrangements, we say that G shows “more interdependence” than F (as in the example by Müller and Scarsini [12]). Proposition 4. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m} and g can be obtained from f by a finite sequence of positive k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) and G(x) F (x) for all x in S. Proof. The proof concatenates the findings of Proposition 1 and Proposition 2.
2
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1459
The last statement of Proposition 4 defines the so-called concordance order as proposed in the statistical literature by Joe [9]. The above result provides additional insights in the elementary rearrangements underlying this concordance order. 5. Conclusion In this paper, the results on bivariate stochastic dominance and elementary rearrangements are extended to the multivariate setting. I have proposed two geometric multivariate generalizations of the elementary bivariate rearrangement and I have shown how they are underlying multivariate tests of stochastic dominance in terms of the joint distribution function and in terms of the survival function. Based on these underlying rearrangements, I argue that an exclusive focus on multivariate stochastic dominance in terms of the joint distribution function does not suffice to define an order of “more interdependence”. Stochastic dominance in terms of the survival function provides an equally justifiable test. Preferably both tests are combined as in the definition of the concordance order. The results of this paper can be applied to any pair of discrete joint distribution functions with given marginal distribution functions. In particular, it can be applied to discrete joint distribution functions whose distribution functions are standard uniform, the so-called discrete copula functions (Joe [10] or Nelsen [14]). The copula function has recently become increasingly popular as a practical and useful tool to decompose the information in the joint distribution function in information about the distributional profile within each dimension and the dependence structure between the dimensions. Comparing the underlying copula functions, rather than joint distribution functions, allows the extension of the present results to joint distribution functions which do not belong to the same Fréchet class. References [1] A.B. Atkinson, F. Bourguignon, The comparison of multi-dimensioned distributions of economic status, Rev. Econ. Stud. 49 (1982) 183–201. [2] M. D’Agostino, V. Dardanoni, The measurement of rank mobility, J. Econ. Theory 144 (2009) 1783–1803. [3] M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas, Actuarial Theory for Dependent Risks, Wiley, Chichester, 2005. [4] L.G. Epstein, S.M. Tanny, Increasing generalized correlation: a definition and some economic consequences, Canad. J. Econ. (Revue canadienne d’Economique) 13 (1980) 16–34. [5] N. Gravel, P. Moyes, Ethically robust comparisons of distributions of two individual attributes, J. Econ. Theory 147 (4) (2012) 1384–1426, in this issue. [6] J. Hadar, W. Russell, Stochastic dominance in choice under uncertainty, in: M. Balch, D. McFadden, S. Wu (Eds.), Essays on Economic Behavior under Uncertainty, North-Holland, Amsterdam, 1974, pp. 133–149. [7] K. Hamada, Comment on Hadar and Russel (1974), in: M. Balch, D. McFadden, S. Wu (Eds.), Essays on Economic Behavior under Uncertainty, North-Holland, Amsterdam, 1974, pp. 150–153. [8] D.A. Hennessy, H.E. Lapan, A definition of ‘more systematic risk’ with some welfare implications, Economica 70 (2003) 493–507. [9] H. Joe, Multivariate concordance, J. Multivariate Anal. 35 (1990) 12–30. [10] H. Joe, Multivariate Models and Dependence Concepts, Chapman and Hall, London, 1997. [11] M. Meyer, B. Strulovici, Increasing interdependence of multivariate distributions, J. Econ. Theory 147 (4) (2012) 1460–1489, in this issue. [12] A. Müller, M. Scarsini, Some remarks on the supermodular order, J. Multivariate Anal. 73 (2000) 107–119. [13] A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, Chichester, 2002. [14] R.B. Nelsen, Introduction to Copulas, Springer-Verlag, New York, 2006. [15] M. Scarsini, Dominance conditions for multivariate utility functions, Manag. Sci. 34 (1988) 454–460. [16] A. Tchen, Inequalities for distributions with given marginals, Ann. Probab. 8 (1980) 814–827. [17] K.Y. Tsui, Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation, Soc. Choice Welfare 16 (1999) 145–157.
Journal of Economic Theory 147 (2012) 1450–1459 www.elsevier.com/locate/jet
Elementary multivariate rearrangements and stochastic dominance on a Fréchet class Koen Decancq a,b,c,∗,1 a Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven, Belgium b CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium c Herman Deleeck Centre for Social Policy, University of Antwerp, B-2000 Antwerp, Belgium
Received 31 March 2010; final version received 21 February 2011; accepted 21 June 2011 Available online 4 November 2011
Abstract A Fréchet class collects all multivariate joint distribution functions that have the same marginals. Members of a Fréchet class only differ with respect to the interdependence between their marginals. In this paper, I study orders of interdependence on a Fréchet class using two multivariate generalizations of the bivariate rearrangement proposed by Epstein and Tanny (1980) [4] and Tchen (1980) [16]. I show how these multivariate rearrangements are underlying multivariate first order stochastic dominance in terms of the joint distribution function and the survival function. A combination of both rearrangements is shown to be equivalent to the concordance order proposed by Joe (1990) [9]. © 2011 Elsevier Inc. All rights reserved. JEL classification: C14 Keywords: Concordance order; Fréchet class; Multivariate rearrangements; Multivariate stochastic dominance; Orthant dependence order; Supermodular order
* Correspondence to: Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven,
Belgium. E-mail address: [email protected]. 1 This is a revised version of a chapter of my PhD dissertation. I gratefully thank André Decoster, James Foster, Thibault Gajdos, Luc Lauwers, Margaret Meyer, Alfred Müller, Erwin Ooghe, Marco Scarsini, Erik Schokkaert, Dirk Van de gaer, John Weymark, Claudio Zoli, three anonymous referees and participants at seminars in Leuven, Montreal, Oxford, Louvain-la-Neuve, Verona, Paris and Moscow for very helpful comments and suggestions to this or an earlier version of the paper. Remaining errors are all mine. 0022-0531/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2011.11.001
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1451
1. Introduction A Fréchet class consists of multivariate joint distribution functions with the same marginals. A joint distribution function is said to be more interdependent when its marginals tend to be large or small simultaneously. The difference between members of the same Fréchet class is the interdependence between their marginals. Hence, Fréchet classes are a natural environment to study multivariate interdependence. In a wide range of economic problems one is interested in comparing interdependence of joint distribution functions. In financial and actuarial analysis, comparing portfolios with the same individual asset returns or risks with respect to their interdependence provides a measure of the degree of its diversification and systemic risk (Denuit et al. [3]). In welfare economics, the notion of interdependence between income distributions at different points in time is inversely related to (exchange) income mobility (D’Agostino and Dardanoni [2]). Atkinson and Bourguignon [1], Tsui [17] and Gravel and Moyes [5] have argued that an order of multidimensional inequality should be sensitive to the interdependence between the dimensions.2 A central role in the (economic) literature on bivariate interdependence has been played by the intuitive bivariate elementary rearrangement suggested by Hamada [7]. The bivariate rearrangement shifts probability mass on the vertices of a rectangle, such that the probability mass on the vertices with both high or both low values is increased and the probability mass on the vertices with one high and one low value is decreased. It is intuitive to say that a bivariate distribution function after such a rearrangement shows more interdependence between its marginals. Furthermore, Epstein and Tanny [4] and Tchen [16] have shown that in the bivariate case, a joint distribution function obtained by a bivariate rearrangement is stochastically dominated by the initial one to the first order and vice versa. This important result combines two equivalent perspectives: one in terms of elementary rearrangements that are interpretable in terms of interdependence and another which is empirically implementable. Unfortunately, this equivalence no longer holds in a multivariate setting once there are three or more dimensions. This paper proposes a generalization of the bivariate elementary rearrangement which has some theoretical and intuitive appeal and which allows one to recover the equivalence between both perspectives in a multivariate setting. The paper contains three related results. First, I characterize multivariate first order stochastic dominance on a Fréchet class in terms of elementary rearrangements which shift probability mass on the vertices of a hyperbox (rather than a rectangle). Second, I show that an alternative geometric generalization of the bivariate elementary rearrangement which is a priori equally justifiable characterizes multivariate first order stochastic dominance in terms of the survival function rather than the joint distribution function. In fact, both generalizations are underlying the lower and upper orthant orders as studied in the literature on stochastic orders.3 Third, combining both types of rearrangements characterizes the concordance order proposed by Joe [9]. I argue that the almost exclusive focus of the economic literature on multivariate stochastic dominance in terms of the joint distribution function seems unwarranted based on an analysis of the underlying rearrangements. The paper is organized as follows. Section 2 introduces notation. Section 3 surveys the literature on bivariate elementary rearrangements and stochastic dominance on a bivariate Fréchet 2 Hennessy and Lapan [8] and Meyer and Strulovici [11] provide an overview of the notion of interdependence in various economic problems. 3 See Müller and Stoyan [13] for an overview.
1452
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 1 Rearranged probability mass for a positive 2-rearrangement with δ > 0. −δ
+δ
(x 1 , x 2 ) (x 1 , x 2 )
(x 1 , x 2 ) (x 1 , x 2 )
class. In Section 4, the bivariate rearrangements are generalized into the multivariate setting and I investigate how they are underlying multivariate stochastic dominance. Section 5 concludes. 2. Notation Let m be a natural number larger than or equal to 2. Let F and G be two (discrete) mdimensional joint distribution functions with S being the smallest product set containing all points in the joint support. Their corresponding density functions are f and g and the corresponding survival functions are denoted by F and G.4 We say that G is stochastically dominated by F according to the first order if and only if G(x) F (x) for all x in S. Assume furthermore that F and G have m non-degenerate marginals F1 , . . . , Fm . Denote by F (F1 , . . . , Fm ) the set of all joint distribution functions with given marginals F1 , . . . , Fm . This is called the Fréchet class of F1 , . . . , Fm . Vector inequalities are denoted by , <, . We write x
s x}, and F (x) =
f (s), where the sum is taken over all elements of {s ∈ S | s > x}.
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1453
When δ > 0, we call the rearrangement a positive 2-rearrangement, when δ < 0, we call the rearrangement a negative 2-rearrangement. The positive 2-rearrangement goes back to the early work of Hamada [7] who argues that it is intuitive to say that the marginals of a joint distribution function are more interdependent after such a rearrangement. Epstein and Tanny [4] and Tchen [16] have shown that on a bivariate Fréchet class, a finite sequence of positive 2rearrangements leads to stochastic dominance of the first order expressed in terms of the joint distribution functions as well as stochastic dominance in terms of the survival functions. Moreover, they also show that the converse is true. The result, proven independently by these authors, can be stated as follows. Proposition 1. Suppose that m = 2. For joint distribution functions F and G in F(F1 , F2 ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of positive 2-rearrangements, 2. G(x1 , x2 ) F (x1 , x2 ) for all (x1 , x2 ) in S, 3. G(x1 , x2 ) F (x1 , x2 ) for all (x1 , x2 ) in S. Proof. See Theorem 1 of Epstein and Tanny [4] or Tchen [16] on the equivalence between statements 1 and 2. The implication that 1 implies 2 is straightforward. The converse implication that 2 implies 1 is proven in a constructive way by providing an algorithm to identify an appropriate sequence of positive 2-rearrangements. The equivalence between 2 and 3 is well known in the bivariate case (see, for instance, Theorem 3 of Epstein and Tanny [4]). 2 This result combines two perspectives. The first statement takes the perspective of an elementary rearrangement which can be interpreted as leading to more interdependence between both marginals of a bivariate joint distribution function. The other two statements involve a finite number of tests which are empirically implementable. The second statement is a test of bivariate first order stochastic dominance as studied by Hadar and Russell [6]. The authors show a further equivalence with unanimous agreement amongst all expected utility maximizers with a supermodular (or superadditive) utility function that G is preferred to F .5 The equivalence between these perspectives is a powerful result which gives a clear interpretation to bivariate stochastic dominance on a Fréchet class as a test of “more interdependence”. (See, for instance, Atkinson and Bourguignon [1].) 4. The multivariate case Let us relax the assumption on m and move to the multivariate case. Consider the following rearrangement (Table 2) which involves three dimensions. Let δ be negative so that positive probability mass −δ is added to the vectors in the left column of the table and probability mass −δ is taken away from the vectors in the right column of the table. It can be checked that after this rearrangement is carried out, positive probability mass is added to the vertex which is simultaneously low in all dimensions and that the new joint distribution function is first order stochastically dominated by the initial one. However, from the table it is clear that this rearrangement cannot be obtained by a finite sequence of positive 2-rearrangements as it consists of 5 For any sufficiently differentiable supermodular function U , it holds that ∂ 2 U (x)/∂x ∂x 0 for all x in S. j1 j2
1454
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 2 Rearranged probability mass for an alternating 3-rearrangement with δ < 0. −δ
+δ
(x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 )
(x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 ) (x j1 , x j2 , x j3 )
a positive 2-rearrangement (in the plane where xj1 = x j1 , the first two rows of Table 2) and a negative one (in the plane where xj1 = x j1 , the last two rows of Table 2). Hence, by relaxing the assumption on m, statement 2 of Proposition 1 no longer implies statement 1. Furthermore, it is well established that the equivalence between statement 2 and statement 3 no longer holds once there are three or more dimensions. (See, for instance, Müller and Stoyan [13].) Unfortunately, the equivalence between the statements in Proposition 1 breaks down in the multivariate case. Each of the three statements of Proposition 1 inspired a separate multivariate stochastic order in the statistical literature. The first statement and its equivalence with a unanimous agreement amongst all expected utility maximizers with a supermodular utility function has lead to the supermodular order. (See Müller and Scarsini [12] and Meyer and Strulovici [11].) The second statement is the definition of the lower orthant order and the last statement defines the upper orthant order.6 The supermodular order has the advantage of being built upon the bivariate elementary rearrangement with its clear interpretation, but is hard to test empirically. The orthant orders can be easily tested, but cannot be interpreted in terms of 2-rearrangements. The aim of this paper is to recover at least some aspects of the bivariate equivalence by first generalizing the notion of a positive 2-rearrangement in the multivariate setting and then by showing how this generalization is underlying the orthant orders. To this end, some additional notation is necessary. Recall that x = (x 1 , . . . , x m ) and x = (x 1 , . . . , x m ) are two different vectors in S with x
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1455
It can easily be checked that a k-rearrangement indeed generalizes a 2-rearrangement. Note that a k-rearrangement can be applied to any multivariate function f with S as support, irrespective of whether f is a joint density function or not. Two specific k-rearrangements are considered in this paper. When δ > 0, we call the rearrangement a positive k-rearrangement and when (−1)k δ > 0, we call it an alternating k-rearrangement. For any even k, an alternating k-rearrangement is also a positive k-rearrangement and vice versa. In general, there is a top-tobottom symmetry between an alternating k-rearrangement and a positive k-rearrangement. Let us return to the example given by Table 2. Positive probability mass is added to the vertices with either 1 or 3 values of xj equal to x j . Hence, this rearrangement is an alternating 3-rearrangement. The inductive observation of stochastic dominance (in terms of the joint distribution function, not in terms of the survival function) can be generalized further, as will be shown in Proposition 2. Let ≺ be the lexicographic ordering on S, so that y ≺ x if y1 < x1 or y1 = x1 and y2 < x2 or y1 = x1 and y2 = x2 and y3 < x3 and so on. The following lemma will be helpful in proving the proposition. Lemma 1. Suppose that m 2 and l 0. Further suppose that F l and G have the same marginals F1 , . . . , Fm with G(x) F l (x) for all x in S. Let t = (t1 , . . . , tm ) be the minimal element of S according to ≺ for which g(t) > f l (t). Then there exists an alternating k-rearrangement R k with k in {2, . . . , m} such that f l+1 = R k (f l ) and: 1. 2. 3. 4.
f l+1 (t) = g(t), F l+1 has the same marginals as F l and G, G(x) F l+1 (x) F l (x) for all x in S, F l+1 (x) = F l (x) for all x ≺ t .
Proof. I will select a particular alternating k-rearrangement that meets the conditions stated in the lemma. Define therefore for all j in {1, . . . , m}: min{Zj } if Zj = ∅, yj = if Zj = ∅, tj where the sets Zj ⎧ ⎪ ⎨ Z1 = z1 ⎪ ⎩ ⎧ ⎪ ⎨ Z2 = z2 ⎪ ⎩ .. .
⎧ ⎪ ⎨ Zj = zj ⎪ ⎩ .. .
are recursively defined as follows: ⎫ ∃z with z1 = z1 such that ⎪ ⎬ t1 < z1 , t2 z2 , . . . , tm zm and , ⎪ ) > g(z , . . . , z ) ⎭ f l (z1 , . . . , zm m 1 ∃z with z2 = z2 such that
and t1 z1 y1 , t2 < z2 , t3 z3 , . . . , tm zm
) > g(z , . . . , z ) f l (z1 , . . . , zm m 1
⎫ ⎪ ⎬ ⎪ ⎭
,
⎫ ∃z with zj = zj such that ⎪ ⎬ and , t1 z1 y1 , . . . , tj < zj , tj +1 zj +1 , . . . , tm zm ⎪ ⎭ l f (z1 , . . . , zm ) > g(z1 , . . . , zm )
1456
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
⎫ = z such that ∃z with zm ⎪ m ⎬ and ym−1 , tm < zm . Zm = zm t1 z1 y1 , . . . , tm−1 zm−1 ⎪ ⎪ ⎭ ⎩ l f (z1 , . . . , zm ) > g(z1 , . . . , zm ) ⎧ ⎪ ⎨
Consider the following alternating k-rearrangement:
R k = R k (t1 , . . . , tj , . . . , tm ), (y1 , . . . , yj , . . . , ym ), (−1)k δ .
(1)
Let δ = g(t) − f l (t) so that f l+1 (t) = f l (t) + δ = g(t) and that indeed condition 1 is fulfilled. Note that this rearrangement shifts probability mass towards t. The number of dimensions for which yj = tj (that is the number of the non-empty sets Zj ) equals k, the dimensionality of the rearrangement. To show that k is in {2, . . . , m} and that F l+1 has the same marginals as F l and G, I consider the following three cases. Assume first that k = 0. Observe that for all the vectors of S which are preceding t according to ≺ it holds that g(t) = f l (t). If k = 0, it holds that g(z) f l (z) for all z > t by the definition of the sets Zj . Together with the observation that g(t) > f l (t), this contradicts that F l and G have the same marginals F1 , . . . , Fm . In particular, there exists at least one j in {1, . . . , m}: Gj (xj ) > Fjl (xj ) for xj tj . Second, assume that k = 1. Without loss of generality, assume that the single nonempty set is Zj , so that the obtained alternating k-rearrangement is R k ((t1 , . . . , tj , . . . , tm ), (t1 , . . . , yj , . . . , tm ), (−1)k δ). This leads to Gj (xj ) > Fjl (xj ) for all xj ∈ [tj , yj ], which contradicts the equal marginals between G and F l . Third, assume that k is in {2, . . . , m}. By the definition of an alternating k-rearrangement, all marginals of the obtained F l+1 contain an equal number of vertices which belong to Bek as to Bok so that F l+1 has the same marginals as F l (and G). To establish the third and fourth conditions, we define V = {v ∈ S | t v < y}, the hyperbox B k (t, y) excluding its upper boundaries. From the definition of an alternating k-rearrangement and an appropriate choice for δ, we have that F l+1 (x) = F l (x) for all x in S/V. Hence, condition 4 holds. Furthermore, it follows that G(x) = F l+1 (x) for all x in V. Given that G(x) F l (x) for all x in S, we obtain that G(x) F l+1 (x) for all x in S. By the definition of an alternating k-rearrangement, we obtain additionally that F l+1 (x) F l (x) for all x in S so that also condition 3 holds. 2 Proposition 2. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) for all x in S. Proof. It follows immediately from the definition of an alternating k-rearrangement that if g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}, then G(x) F (x) for all x in S. To show the converse, an explicit algorithm is provided by repeatedly applying the lemma; first with the joint distribution function F 0 = F and the minimal element of S according to ≺ for which g > f l to obtain an alternating k-rearrangement with k in {2, . . . , m} and F 1 , then with F 1 and the new minimal element of S according to ≺ for which g > f l to obtain another alternating k-rearrangement with k in {2, . . . , m} and F 2 and so on. After every step, the obtained
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1457
Table 3 A sequence of alternating rearrangements for the example of Müller and Scarsini [12]. l
t
Rearrangement
Probability mass ε added to
Probability mass ε removed from
1 2 3 4 5 6 7 8 9
(0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (0, 2, 0) (0, 2, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0)
R 3 ((0, 0, 0), (1, 1, 1), ε) R 3 ((0, 0, 1), (1, 1, 2), ε) R 3 ((0, 1, 0), (1, 2, 1), ε) R 3 ((0, 1, 1), (1, 2, 2), ε) R 2 ((0, 2, 0), (2, 2, 1), ε) R 2 ((0, 2, 1), (2, 2, 2), ε) R 3 ((1, 0, 0), (2, 1, 1), ε) R 3 ((1, 0, 1), (2, 1, 2), ε) R 3 ((1, 1, 0), (2, 2, 1), ε)
(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) (0, 0, 1), (0, 1, 2), (1, 0, 2), (1, 1, 1) (0, 1, 0), (0, 2, 1), (1, 1, 1), (1, 2, 0) (0, 1, 1), (0, 2, 2), (1, 1, 2), (1, 2, 1) (0, 2, 0), (2, 2, 1) (0, 2, 1), (2, 2, 2) (1, 0, 0), (1, 1, 1), (2, 0, 1), (2, 1, 0) (1, 0, 1), (1, 1, 2), (2, 0, 2), (2, 1, 1) (1, 1, 0), (1, 2, 1), (2, 1, 1), (2, 2, 0)
(1, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1) (1, 1, 2), (1, 0, 1), (0, 1, 1), (0, 0, 2) (1, 2, 1), (1, 1, 0), (0, 2, 0), (0, 1, 1) (1, 2, 2), (1, 1, 1), (0, 2, 1), (0, 1, 2) (0, 2, 1), (2, 2, 0) (0, 2, 2), (2, 2, 1) (2, 1, 1), (2, 0, 0), (1, 1, 0), (1, 0, 1) (2, 1, 2), (2, 0, 1), (1, 1, 1), (1, 0, 2) (2, 2, 1), (2, 1, 0), (1, 2, 0), (1, 1, 1)
function F l shares the same univariate marginals with F and G. Moreover, compared to F l , F l+1 coincides with G for an additional element of S. After a finite number L steps, G equals F L over the complete support S and its corresponding density function g is obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m}.8 2 To illustrate the algorithm, consider the following example, which is inspired by Müller and Scarsini [12]. Let g and f be two three-dimensional density functions such that f has equal probability mass ε = 1/6 on the six points (2, 2, 1);
(2, 1, 2);
(1, 2, 2);
(1, 1, 1);
(0, 0, 2)
and
(2, 0, 0)
and
(0, 0, 0).
and g has equal probability mass ε = 1/6 on the six points (2, 2, 2);
(2, 1, 1);
(1, 2, 1);
(1, 1, 2);
(2, 0, 2)
It is easy to see that F and G have indeed the same marginals. Müller and Scarsini [12] show that G(x) F (x) for all x in S. By applying the algorithm of the proof of Proposition 2, I provide a sequence of alternating k-rearrangement with k in {2, . . . , m} which lead from f to g. These rearrangements are summarized in Table 3. Proposition 2 recovers the two perspectives of the bivariate result. It connects a test of multivariate stochastic dominance with a sequence of elementary multivariate rearrangements. Furthermore, Hadar and Russel [6] and Atkinson and Bourguignon [1] have shown the equivalence between multivariate first order stochastic dominance and unanimity within the class of utility functions with alternating signs of their cross derivatives. Scarsini [15] calls such a utility function an m-variate risk averse utility function. Can we interpret this test of multivariate stochastic dominance as a test of “more interdependence” as we did in the bivariate case? Yes and no. When k is even, the k-rearrangement rearranges probability on 2k vertices and shifts thereby probability mass towards the vertices which are simultaneously high or low in all dimensions. Although these rearrangements might be less appealing to one’s intuitions given the large number of involved vertices, it seems reasonable to extend the intuition of Hamada [7] and say they indeed increase interdependence. 8 Although F and G are joint distribution functions, the intermediate function F l for l = 1, . . . , L − 1 obtained after each of these alternating k-rearrangements is not necessarily a “proper” joint distribution function. It is not excluded, for instance, that more probability mass is removed from a vertex than there was initially present. Gravel and Moyes [5] introduce in these cases a so-called “phantom”.
1458
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
Table 4 A sequence of positive rearrangements for the example of Müller and Scarsini [12]. l
t
Rearrangement
Probability mass ε added to
Probability mass ε removed from
2 1
(2, 2, 2) (2, 0, 2)
R 3 ((1, 1, 1), (2, 2, 2), ε) R 2 ((0, 0, 0), (2, 0, 2), ε)
(2, 2, 2), (2, 1, 1), (1, 2, 1), (1, 1, 2) (0, 0, 0), (2, 0, 2)
(2, 2, 1), (2, 1, 2), (1, 2, 2), (1, 1, 1) (2, 0, 0), (0, 0, 2)
However, when k is odd things become more intricate (as in the example in Table 2). An alternating k-rearrangement shifts probability mass to the vertex with all low values, but probability mass is shifted away from the vertex with all high values. One could even argue that such a rearrangement does not increase interdependence, but rather decreases it. So that in case k is odd, a positive k-rearrangement increases dependence. This leads to the following result. Proposition 3. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of positive k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) for all x in S. Proof. The proof has the same structure as the one of Proposition 2 and is omitted. In particular, the algorithm needs to be adjusted in a top-to-bottom symmetrical way. 2 Returning to the example of Müller and Scarsini [12], the authors show that G(x) F (x) also holds for all x in S. Here, a sequence of positive k-rearrangement with k in {2, . . . , m} can be provided to obtain g from f . In particular, the algorithm leads to the sequence of positive rearrangements given in Table 4. Observe that the last statement of Proposition 3 involves a test of first order stochastic dominance in terms of the survival function. A priori there seems no reason to prioritize alternating k-rearrangements to positive k-rearrangements, yet most results in the economic literature on multivariate stochastic dominance focus exclusively on the results obtained by imposing alternating k-rearrangements. This seems unwarranted. Given the hard choice between both types of k-rearrangements, a natural approach is to require both initial statements of Propositions 2 and 3 to be fulfilled. Only when a sequence of alternating k-rearrangements can be found that leads from f to g as well as a sequence of positive k-rearrangements, we say that G shows “more interdependence” than F (as in the example by Müller and Scarsini [12]). Proposition 4. Suppose that m 2. For joint distribution functions F and G in F (F1 , . . . , Fm ) with corresponding joint density functions f and g and survival functions F and G, the following statements are equivalent: 1. g can be obtained from f by a finite sequence of alternating k-rearrangements with all k in {2, . . . , m} and g can be obtained from f by a finite sequence of positive k-rearrangements with all k in {2, . . . , m}, 2. G(x) F (x) and G(x) F (x) for all x in S. Proof. The proof concatenates the findings of Proposition 1 and Proposition 2.
2
K. Decancq / Journal of Economic Theory 147 (2012) 1450–1459
1459
The last statement of Proposition 4 defines the so-called concordance order as proposed in the statistical literature by Joe [9]. The above result provides additional insights in the elementary rearrangements underlying this concordance order. 5. Conclusion In this paper, the results on bivariate stochastic dominance and elementary rearrangements are extended to the multivariate setting. I have proposed two geometric multivariate generalizations of the elementary bivariate rearrangement and I have shown how they are underlying multivariate tests of stochastic dominance in terms of the joint distribution function and in terms of the survival function. Based on these underlying rearrangements, I argue that an exclusive focus on multivariate stochastic dominance in terms of the joint distribution function does not suffice to define an order of “more interdependence”. Stochastic dominance in terms of the survival function provides an equally justifiable test. Preferably both tests are combined as in the definition of the concordance order. The results of this paper can be applied to any pair of discrete joint distribution functions with given marginal distribution functions. In particular, it can be applied to discrete joint distribution functions whose distribution functions are standard uniform, the so-called discrete copula functions (Joe [10] or Nelsen [14]). The copula function has recently become increasingly popular as a practical and useful tool to decompose the information in the joint distribution function in information about the distributional profile within each dimension and the dependence structure between the dimensions. Comparing the underlying copula functions, rather than joint distribution functions, allows the extension of the present results to joint distribution functions which do not belong to the same Fréchet class. References [1] A.B. Atkinson, F. Bourguignon, The comparison of multi-dimensioned distributions of economic status, Rev. Econ. Stud. 49 (1982) 183–201. [2] M. D’Agostino, V. Dardanoni, The measurement of rank mobility, J. Econ. Theory 144 (2009) 1783–1803. [3] M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas, Actuarial Theory for Dependent Risks, Wiley, Chichester, 2005. [4] L.G. Epstein, S.M. Tanny, Increasing generalized correlation: a definition and some economic consequences, Canad. J. Econ. (Revue canadienne d’Economique) 13 (1980) 16–34. [5] N. Gravel, P. Moyes, Ethically robust comparisons of distributions of two individual attributes, J. Econ. Theory 147 (4) (2012) 1384–1426, in this issue. [6] J. Hadar, W. Russell, Stochastic dominance in choice under uncertainty, in: M. Balch, D. McFadden, S. Wu (Eds.), Essays on Economic Behavior under Uncertainty, North-Holland, Amsterdam, 1974, pp. 133–149. [7] K. Hamada, Comment on Hadar and Russel (1974), in: M. Balch, D. McFadden, S. Wu (Eds.), Essays on Economic Behavior under Uncertainty, North-Holland, Amsterdam, 1974, pp. 150–153. [8] D.A. Hennessy, H.E. Lapan, A definition of ‘more systematic risk’ with some welfare implications, Economica 70 (2003) 493–507. [9] H. Joe, Multivariate concordance, J. Multivariate Anal. 35 (1990) 12–30. [10] H. Joe, Multivariate Models and Dependence Concepts, Chapman and Hall, London, 1997. [11] M. Meyer, B. Strulovici, Increasing interdependence of multivariate distributions, J. Econ. Theory 147 (4) (2012) 1460–1489, in this issue. [12] A. Müller, M. Scarsini, Some remarks on the supermodular order, J. Multivariate Anal. 73 (2000) 107–119. [13] A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, Chichester, 2002. [14] R.B. Nelsen, Introduction to Copulas, Springer-Verlag, New York, 2006. [15] M. Scarsini, Dominance conditions for multivariate utility functions, Manag. Sci. 34 (1988) 454–460. [16] A. Tchen, Inequalities for distributions with given marginals, Ann. Probab. 8 (1980) 814–827. [17] K.Y. Tsui, Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation, Soc. Choice Welfare 16 (1999) 145–157.