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

According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernelf 0 (x), absolutely continuous univariate skewing distribution G 0 (x) with an even density g 0 (x) = G 0 (x), and multivariate odd skewing weight w(x), is defined by f(x) = 2f 0 (x)G 0 {w(x)}. The SSD depends on the skewing distribution and the skewing weight only through the perturbation functionG(x) = G 0 {w(x)} such that G(x) ≥ 0 and the reflective property G(x) + G(−x) = 1 holds. Conversely, any function G(x) that satisfies these conditions ensures thatf(x) = 2f 0 (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], Proposition 3. Azzalini and Regoli [3] refine this result to the representation of a density with arbitrary support in Proposition 2. 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 (Theorem 4) 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

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 (Theorem 4) 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.

Multivariate Likelihood Ratio Order
Unless otherwise stated,  = ( 1 , . . .,   ) and  = ( 1 , . . .,   ) denote throughout real random vectors on some probability space with supports () and ().We assume that  and  have 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 function  0 (), called kernel, and (reflective) perturbation functions   () and   () satisfying the conditions where the notation () = 1 − () is used throughout, such that the probability density functions (pdf) of  and  are given by Equivalently to (2), there exists a continuous skewing distribution  0 () with an even density  0 () =   0 () and odd skewing weights   (),   () such that Following Azzalini and Regoli [3, equation (9)], this equivalence is underpinned by the standard choice (made throughout) of a uniform (−1/2, 1/2) random variable with distribution

(LR2)
The ratio of the perturbation functions   ()/  () is monotone increasing in  over the union of the supports () ∪ ().
(LR3) The reverse hazard rates of the perturbation functions satisfy the inequalities or equivalently, for a SSD with a common kernel, that is, (LR2).On the other hand, in case   ()/  () is monotone increasing, so is its logarithmic.The equivalence of (LR2) and (LR3) follows from the relation 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 order  ≤ lr  between two SSD's with a common kernel is equivalent to the reverse hazard rate order   ≤ rh   between 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 density  0 is multivariate totally positive of order 2 (MTP 2 property) and   ≤ rh   in the multivariate reverse hazard rate order, then the likelihood ratio order is fulfilled, as shown in Theorem 4.
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,  ≤ lr  means that the density   () is multivariate totally positive of order 2 (MTP 2 ) such that a property discussed in Karlin and Rinott [5] and Whitt [6].
Applied to SSD random vectors  and  with a common kernel  0 and perturbation functions   and   , condition (9)  Now, if the random vector  0 associated with the kernel density  0 satisfies the reflexive property  0 ≤ lr  0 , or equivalently  0 is MTP 2 , and the following inequalities The above discussion leads to the following sufficient condition.
Theorem 4 (first sufficient condition for likelihood ratio order).Let  and  be SSD random vectors with a common kernel  0 and perturbation functions   and   .If  0 is MTP 2 and   ≤ rh   , then  ≤ lr .
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], Theorem 6.E.8).Another possible generalization of the univariate likelihood ratio order is to require instead of (9) the condition   ()   () ≤   ()   () , ∀ ≤ . (13) Here  ≤  denotes the usual componentwise partial order between vectors, which is defined as follows.If  = ( 1 , . . .,   ) and  = (  [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  and  with a common kernel, it is possible to establish  ≤ st  from (13) under an alternative simpler assumption to the association of , which even implies likelihood ratio order (Theorem 7).Condition (13) for SSDs with a common kernel means that   ()   () ≤   ()   () , ∀ ≤ . ( Similarly to Shaked and Shanthikumar [4], Section 6.D, and the discussion preceding Definition 3, condition (14) motivates the following ordering between perturbation functions.
Definition 5.The perturbation function   is said to be smaller than   in the weak reverse hazard rate order, written as   ≤ wrh   if the property ( 14) is satisfied.From ( 12) and ( 14), it follows immediately that To establish a converse to (15), a result similar to Theorem 6.D.1 in Shaked and Shanthikumar [4]  Proof.The result follows from Lemma 6 and Theorem 4.
As a consequence, we derive two perturbation invariant stochastic order results for the SSD class.Recall that under perturbation invariance one understands general statements about the random vectors  0 and  associated with the kernel densities  0 () and the SSD densities   () = 2 0 ()  () that remain valid over a large class of perturbation functions.For example, the well-known even transformation invariance states that for all even real functions (), whatever the perturbation function   () is (e.g., Azzalini and Regoli [3], Proposition 1).
Corollary 8 (likelihood ratio order invariance).Let   () = 2 0 ()  () be the density function of the SSD random vector , and let  0 be the random vector associated with the kernel density  0 .Assume that  0 is MTP 2 and the support of () is a lattice.Then, the following likelihood ratio order invariant properties hold: Proof.To show (LRI1) it suffices to observe that the perturbation function   0 () ≡ 1/2 is MTP 2 and generates the kernel density  0 () of  0 .The statement follows from Theorem 7.

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], Proposition 3, derive the following remarkable new characterization result.If even transformation invariance holds between random vectors  and  for all even () that is, ()=  (), 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 conditions   ≤ wrh   in Theorem 7 and   ≤ rh   in 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 these results also hold.
[4]), 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).The perturbation function   is said to be smaller than   in the reverse hazard rate order, written as   ≤ rh   , if the property (12) is satisfied.