# Multivariate Likelihood Ratio Order for Skew-Symmetric Distributions with a Common Kernel

Authors: Werner Hürlimann
Journal: ISRN Probability and Statistics
 (2013)

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## Abstract

The multivariate likelihood ratio order comparison of skew-symmetric distributions with a common kernel is considered. Two multivariate likelihood ratio perturbation invariance properties are derived.


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## Body

## 1. Introduction

According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernelf0(x), absolutely continuous univariate skewing distributionG0(x)with an even densityg0(x)=G0′(x), and multivariate odd skewing weightw(x), is defined byf(x)=2f0(x)G0{w(x)}. The SSD depends on the skewing distribution and the skewing weight only through the perturbation functionG(x)=G0{w(x)}such thatG(x)≥0and the reflective propertyG(x)+G(-x)=1holds. Conversely, any functionG(x)that satisfies these conditions ensures thatf(x)=2f0(x)G(x)is a density, which represents the SSD formulation adopted by Wang et al. [2]. In fact, any probability density function admits a uniquely defined SSD representation, as shown first by Wang et al. [2], Proposition3. Azzalini and Regoli [3] refine this result to the representation of a density with arbitrary support in Proposition2.The present note considers the multivariate likelihood ratio order for multivariate skew symmetric distributions with a common kernel. We obtain two general sufficient conditions in terms of a reverse hazard rate order (Theorem4) and a weak reverse hazard rate order (Theorem 7) between perturbation functions. The second sufficient condition is related to Theorem 6.B.8 in Shaked and Shanthikumar [4], which establishes a sufficient condition for the stochastic order. It is simpler and implies even the likelihood ratio order.


## 2. Multivariate Likelihood Ratio Order

Unless otherwise stated,X=(X1,…,Xn)andY=(Y1,…,Yn) denote throughout real random vectors on some probability space with supportsS(X)andS(Y). We assume thatXandYhave skew-symmetric distributions (SSD) in the sense of Azzalini and Capitanio [1] and Wang et al. [2] as unified in Azzalini and Regoli [3]. Our analysis is restricted to SSDs with a common kernel. This means that there exists a continuous centrally symmetric (about 0) density functionf0(x), called kernel, and (reflective) perturbation functionsGX(x) andGY(x)satisfying the conditions
(1)GX(x)≥0,GX(-x)=G-X(x),GY(x)≥0,GY(-x)=G-Y(x),
where the notationG-(x)=1-G(x)is used throughout, such that the probability density functions (pdf) ofXandYare given by
(2)fX(x)=2f0(x)GX(x),x∈S(X),fY(y)=2f0(y)GY(y),y∈S(Y).Equivalently to (2), there exists a continuous skewing distributionG0(x)with an even densityg0(x)=G0′(x)and odd skewing weightswX(x),wY(x)such that
(3)fX(x)=2f0(x)G0{wX(x)},x∈S(X),fY(y)=2f0(y)G0{wY(y)},y∈S(Y).Following Azzalini and Regoli [3, equation (9)], this equivalence is underpinned by the standard choice (made throughout) of a uniformU(-1/2,1/2)random variable with distribution
(4)G0(t)=(t+12)·1(-1,1)(2t)+1[1,∞)(2t),t∈R,w(x)=G(x)-12,
where1A(·)denotes the indicator function of the setA. With this, properties of perturbation functions directly translate into properties of skewing weights and vice versa.It is instructive to begin with the univariate case.Proposition 1 (univariate likelihood ratio order characterization).
Suppose thatXandYare SSD random variables with a common kernel, absolutely continuous perturbation functionsGXandGYwith derivativesgX(x)=GX′(x),gY(y)=GY′(y). The following conditions are equivalent:(LR1)
X≤lrY(XprecedesYin likelihood ratio order).(LR2)
The ratio of the perturbation functionsGY(t)/GX(t)is monotone increasing intover the union of the supportsS(X)∪S(Y).(LR3)
The reverse hazard rates of the perturbation functions satisfy the inequalities(5)rX(t)=gX(t)GX(t)≤rY(t)=gY(t)GY(t),t∈S(X)∪S(Y).Proof.
By definitionX≤lrYif, and only if (e.g. Shaked and Shanthikumar [4, equation (1.C.2)]), one has
(6)fX(x)fY(y)≥fX(y)fY(x),∀x≤y,
or equivalently, for a SSD with a common kernel,
(7)GY(x)GX(x)≤GY(y)GX(y),
that is, (LR2). On the other hand, in caseGY(t)/GX(t)is monotone increasing, so is its logarithmic. The equivalence of (LR2) and (LR3) follows from the relation
(8)ddtln{GY(t)GX(t)}=gY(t)GY(t)-gX(t)GX(t)=rY(t)-rX(t).Though perturbation functions are more general than distribution functions, it is convenient to order them in a similar fashion. With this convention in mind, the univariate likelihood ratio orderX≤lrYbetween two SSD’s with a common kernel is equivalent to the reverse hazard rate orderGX≤rhGYbetween its perturbation functions. This follows from Proposition 1 by noting that (LR2) and (LR3) identify with the conditions (1.B.40) and (1.B.42) in Shaked and Shanthikumar [4] for the reverse hazard rate order.In the multivariate case, the stated unaltered characterization of the likelihood ratio order does not hold. However, when the kernel densityf0is multivariate totally positive of order 2 (MTP2 property) andGX≤rhGYin the multivariate reverse hazard rate order, then the likelihood ratio order is fulfilled, as shown in Theorem 4.Definition 2.
The random vectorXis said to be smaller thanYin the likelihood ratio order or TP2order, written asX≤lrYorX≤TP2Y, if
(9)fX(x)fY(y)≤fX(x∧y)fY(x∨y),∀x,y∈Rn,
where one setsx∧y=(x1∧y1,…,xn∧yn),x∨y=(x1∨y1,…,xn∨yn), andxi∧yi=min(xi,yi),xi∨yi=max(xi,yi).
It is important to remark that the multivariate order≤lr(as well as the reverse hazard rate order introduced later) is not an order in the usual sense because it does not satisfy the reflexive property (e.g. Shaked and Shanthikumar [4], p.291 and 298). In fact,X≤lrXmeans that the densityfX(x)is multivariate totally positive of order 2 (MTP2) such that
(10)fX(x)fX(y)≤fX(x∧y)fX(x∨y),∀x,y∈Rn,
a property discussed in Karlin and Rinott [5] and Whitt [6].Applied to SSD random vectorsXandYwith a common kernelf0and perturbation functionsGXandGY, condition (9) means that
(11)f0(x)f0(y)GX(x)GY(y)≤f0(x∧y)f0(x∨y)GX(x∧y)GY(x∨y),0000000000000000000000000000000000∀x,y∈Rn.Now, if the random vectorX0associated with the kernel densityf0satisfies the reflexive propertyX0≤lrX0, or equivalentlyf0is MTP2, and the following inequalities
(12)GX(x)GY(y)≤GX(x∧y)GY(x∨y),∀x,y∈Rn,
hold, then clearly (11) is satisfied. But (12) can be used to define a multivariate reverse hazard rate order (for perturbation functions) by paraphrasing the definition of the hazard rate order in Shaked and Shanthikumar [4], Section 6.D, as done for the corresponding univariate orders (Sections 1.B.1 and 1.B.6).Definition 3.
The perturbation functionGXis said to be smaller thanGYin the reverse hazard rate order, written asGX≤rhGY, if the property (12) is satisfied.The above discussion leads to the following sufficient condition.Theorem 4 (first sufficient condition for likelihood ratio order).
LetX and Y be SSD random vectors with a common kernel f0 and perturbation functions GX and GY. If f0 is MTP2 and GX≤rhGY, then X≤lrY.A strong incentive for the use of the likelihood ratio order, which in many situations is easy to verify, is its automatic implication of the multivariate stochastic order (Shaked and Shanthikumar [4], Theorem6.E.8). Another possible generalization of the univariate likelihood ratio order is to require instead of (9) the condition
(13)fX(y)fY(x)≤fX(x)fY(y),∀x≤y.
Here x≤y denotes the usual componentwise partial order between vectors, which is defined as follows. If x=(x1,…,xn) and y=(y1,…,yn) are two n-dimensional vectors, then x≤y if xi≤yi for i=1,…,n. It is well known that the simpler condition (13) does not necessarily imply the multivariate stochastic order X≤stY (for counterexamples consult Lehmann [7] and Whitt [6]). However, if besides monotonicity of the ratio fY(x)/fX(x) one requires that the random vector X is associated, then X≤stY (Shaked and Shanthikumar [4], Theorem6.B.8, p.270). Recall that a random vector X is positively associated if Cov[f(X),g(X)]≥0 for all increasing functions f, g for which the covariance exists. Similarly, the random vector is negatively associated if Cov[f(Xi,i∈A),g(Xj,j∈B)]≤0 for every pair of disjoint subsets A, B of {1,…,n}, and for all coordinatewise increasing (or decreasing) functions f, g for which the covariance exists. Positive association has been studied by Esary et al. [8] and negative association by Alam and Saxena [9] and Joag-Dev and Proschan [10] (see also Gerasimov et al. [11] for a recent contribution). Now, for SSD random vectors X and Y with a common kernel, it is possible to establish X≤stY from (13) under an alternative simpler assumption to the association of X, which even implies likelihood ratio order (Theorem 7).Condition (13) for SSDs with a common kernel means that
(14)GX(y)GY(x)≤GX(x)GY(y),∀x≤y.
Similarly to Shaked and Shanthikumar [4], Section6.D, and the discussion preceding Definition 3, condition (14) motivates the following ordering between perturbation functions.Definition 5.
The perturbation functionGXis said to be smaller thanGYin the weak reverse hazard rate order, written asGX≤wrhGYif the property (14) is satisfied.
From (12) and (14), it follows immediately that
(15)GX≤rhGY⟹GX≤wrhGY.
To establish a converse to (15), a result similar to Theorem6.D.1 in Shaked and Shanthikumar [4] is required.Lemma 6.
Assume that the perturbation functionsGXandGYhave a common supportS, which is a lattice (that is, ifx,y∈Sthenx∧y,x∨y∈S). IfGXand/orGYis MTP2, then
(16)GX≤wrhGY⟹GX≤rhGY.Proof.
The orderingGX≤wrhGYimplies by (14) that
(17)GX(x∨y)GY(y)≤GX(y)GY(x∨y),∀x,y∈Rn.
IfGXis MTP2, then by the defining property (10) one has
(18)GX(x)GX(y)≤GX(x∧y)GX(x∨y),∀x,y∈Rn.
Multiplication of these two inequalities yields
(19)GX(x∨y)GY(y)GX(x)GX(y)≤GX(y)GY(x∨y)GX(x∧y)GX(x∨y),000000000000000000000000000000∀x,y∈Rn.
Now, sinceSis a lattice, it follows that ifGX(x)GX(y)>0, thenGX(y)andGX(x∨y)are positive. Cancelling these quantities in the preceding inequality implies that (12) holds in this case. IfGX(x)GX(y)=0then (12) is trivially fulfilled. Together, this shows thatGX≤rhGY. IfGYis MTP2, one shows similarly thatGX≤rhGY. The proof is complete.Theorem 7 (second sufficient condition for likelihood ratio order).
LetXandYbe SSD random vectors with a common kernelf0and perturbation functionsGXandGY, which have a common lattice support. Iff0is MTP2, at least one ofGXandGYis MTP2, andGX≤wrhGY; thenX≤lrY.Proof.
The result follows from Lemma6 and Theorem 4.As a consequence, we derive twoperturbation invariant stochastic order results for the SSD class. Recall that under perturbation invariance one understands general statements about the random vectorsX0andXassociated with the kernel densitiesf0(x)and the SSD densitiesfX(x)=2f0(x)GX(x)that remain valid over a large class of perturbation functions. For example, the well-known even transformation invariance states that
(20)T(X)=dT(X0)(equalityindistribution),
for all even real functionsT(x), whatever the perturbation functionGX(x)is (e.g., Azzalini and Regoli [3], Proposition1).Corollary 8 (likelihood ratio order invariance).
LetfX(x)=2f0(x)GX(x)be the density function of the SSD random vectorX, and letX0be the random vector associated with the kernel densityf0. Assume thatf0is MTP2 and the support ofS(X)is a lattice. Then, the following likelihood ratio order invariant properties hold:(LRI1)
ifGXis an increasing perturbation function thenX0≤lrX;(LRI2)
ifGXis decreasing onS(X)∩R+n, thenX≤lrX0+=X0·1R+n(X0).Proof.
To show (LRI1) it suffices to observe that the perturbation functionGX0(x)≡1/2is MTP2 and generates the kernel densityf0(x)ofX0. The statement follows from Theorem 7. Similarly, the choiceGX0+(x)=1R+n(x)generates the random variableX0+and (LRI2) follows also from Theorem 7.


## 3. Discussion and Conclusions

It appears useful to discuss what has been obtained. So far, in the already large literature on skew-symmetric distributions, only a limited number of results establish in advance formal properties of the SSD by given qualitative properties of the kernel, skewing distribution and skewing weight, or equivalently by given kernel and perturbation function. The even transformation invariance property (20) is the most prominent result of this kind. Azzalini and Regoli [3], Proposition3, derive the following remarkable new characterization result. If even transformation invariance holds between random vectorsXandYfor all evenT(x)that is,T(X)=dT(Y), then the corresponding densities admit necessarily a representation with a common kernel. In the present note, we have studied a bit further the latter class of SSDs with a common kernel. We have derived two general sufficient conditions for the multivariate likelihood ratio order in Theorems 4 and 7. Since the validity of such an ordering relationship automatically implies the multivariate stochastic order, it has potential for applications in statistics. Also, a novel likelihood ratio order perturbation invariance property has been displayed in Corollary 8. Finally, we remark that the sufficient conditionsGX≤wrhGYin Theorem 7 andGX≤rhGYin Theorem 4 can be viewed as multivariate extensions of the criteria (LR2) and (LR3) in Proposition 1. It can be asked whether (in generalization to the univariate case) converses to these results also hold.



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