Multivariate Likelihood Ratio Order for Skew-Symmetric Distributions with a Common Kernel
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.
According to Azzalini and Capitanio , the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernel , absolutely continuous univariate skewing distribution with an even density, and multivariate odd skewing weight , is defined by. The SSD depends on the skewing distribution and the skewing weight only through the perturbation function such thatand the reflective property holds. Conversely, any functionthat satisfies these conditions ensures thatis a density, which represents the SSD formulation adopted by Wang et al. . In fact, any probability density function admits a uniquely defined SSD representation, as shown first by Wang et al. , Proposition. Azzalini and Regoli  refine this result to the representation of a density with arbitrary support in Proposition.
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 , 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,and denote throughout real random vectors on some probability space with supportsand. We assume thatandhave skew-symmetric distributions (SSD) in the sense of Azzalini and Capitanio  and Wang et al.  as unified in Azzalini and Regoli . Our analysis is restricted to SSDs with a common kernel. This means that there exists a continuous centrally symmetric (about 0) density function, called kernel, and (reflective) perturbation functions andsatisfying the conditions where the notationis used throughout, such that the probability density functions (pdf) ofandare given by
Equivalently to (2), there exists a continuous skewing distributionwith an even densityand odd skewing weights,such that
Following Azzalini and Regoli [3, equation (9)], this equivalence is underpinned by the standard choice (made throughout) of a uniformrandom variable with distribution wheredenotes the indicator function of the set. 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 thatandare SSD random variables with a common kernel, absolutely continuous perturbation functionsandwith derivatives,. The following conditions are equivalent:(LR1) ( precedes in likelihood ratio order).(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
Proof. By definitionif, and only if (e.g. Shaked and Shanthikumar [4, equation (1.C.2)]), one has
or equivalently, for a SSD with a common kernel,
that is, (LR2). On the other hand, in caseis 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 orderbetween two SSD’s with a common kernel is equivalent to the reverse hazard rate orderbetween 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  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 densityis multivariate totally positive of order 2 (MTP2 property) andin the multivariate reverse hazard rate order, then the likelihood ratio order is fulfilled, as shown in Theorem 4.
Definition 2. The random vectoris said to be smaller thanin the likelihood ratio order or TP2 order, written asor, if
where one sets,, and,.
It is important to remark that the multivariate order(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 , p.291 and 298). In fact,means that the densityis multivariate totally positive of order 2 (MTP2) such that a property discussed in Karlin and Rinott  and Whitt .
Applied to SSD random vectorsandwith a common kerneland perturbation functionsand, condition (9) means that
Now, if the random vectorassociated with the kernel densitysatisfies the reflexive property, or equivalentlyis MTP2, and the following inequalities 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 , Section 6.D, as done for the corresponding univariate orders (Sections 1.B.1 and 1.B.6).
Definition 3. The perturbation functionis said to be smaller thanin the reverse hazard rate order, written as, if the property (12) is satisfied.
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 and perturbation functions and . If is MTP2 and , then .
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 , Theorem.E.8). Another possible generalization of the univariate likelihood ratio order is to require instead of (9) the condition Here denotes the usual componentwise partial order between vectors, which is defined as follows. If and are two -dimensional vectors, then if for . It is well known that the simpler condition (13) does not necessarily imply the multivariate stochastic order (for counterexamples consult Lehmann  and Whitt ). However, if besides monotonicity of the ratio one requires that the random vector is associated, then (Shaked and Shanthikumar , Theorem.B.8, p.270). Recall that a random vector is positively associated if for all increasing functions , for which the covariance exists. Similarly, the random vector is negatively associated if for every pair of disjoint subsets , of , and for all coordinatewise increasing (or decreasing) functions , for which the covariance exists. Positive association has been studied by Esary et al.  and negative association by Alam and Saxena  and Joag-Dev and Proschan  (see also Gerasimov et al.  for a recent contribution). Now, for SSD random vectors and with a common kernel, it is possible to establish 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 , Section.D, and the discussion preceding Definition 3, condition (14) motivates the following ordering between perturbation functions.
Definition 5. The perturbation functionis said to be smaller thanin the weak reverse hazard rate order, written asif the property (14) is satisfied.
From (12) and (14), it follows immediately that To establish a converse to (15), a result similar to Theorem.D.1 in Shaked and Shanthikumar  is required.
Lemma 6. Assume that the perturbation functionsandhave a common support, which is a lattice (that is, if,then,). Ifand/oris MTP2, then
Proof. The orderingimplies by (14) that
Ifis MTP2, then by the defining property (10) one has
Multiplication of these two inequalities yields
Now, sinceis a lattice, it follows that if, thenandare positive. Cancelling these quantities in the preceding inequality implies that (12) holds in this case. Ifthen (12) is trivially fulfilled. Together, this shows that. Ifis MTP2, one shows similarly that. The proof is complete.
Theorem 7 (second sufficient condition for likelihood ratio order). Letandbe SSD random vectors with a common kerneland perturbation functionsand, which have a common lattice support. Ifis MTP2, at least one ofandis MTP2, and; then.
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 vectorsandassociated with the kernel densitiesand the SSD densitiesthat 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 functionis (e.g., Azzalini and Regoli , Proposition).
Corollary 8 (likelihood ratio order invariance). Letbe the density function of the SSD random vector, and letbe the random vector associated with the kernel density. Assume thatis MTP2 and the support ofis a lattice. Then, the following likelihood ratio order invariant properties hold:(LRI1)ifis an increasing perturbation function then;(LRI2)ifis decreasing on, then.
Proof. To show (LRI1) it suffices to observe that the perturbation functionis MTP2 and generates the kernel densityof. The statement follows from Theorem 7. Similarly, the choicegenerates the random variableand (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 , Proposition, derive the following remarkable new characterization result. If even transformation invariance holds between random vectorsandfor all eventhat 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 conditionsin Theorem 7 andin 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.