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

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.

1. Introduction

According to Azzalini and Capitanio [1], the density function of the multivariate skew-symmetric distribution (SSD) with centrally symmetric (about 0) density kernel 𝑓₀(𝑥), absolutely continuous univariate skewing distribution 𝐺₀(𝑥)with an even density𝑔₀(𝑥)=𝐺^′₀(𝑥), and multivariate odd skewing weight 𝑤(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 function 𝐺(𝑥)=𝐺₀{𝑤(𝑥)}such that𝐺(𝑥)≥0and the reflective property 𝐺(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 (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

Unless otherwise stated,𝑋=(𝑋₁,…,𝑋_𝑛)and𝑌=(𝑌₁,…,𝑌_𝑛) 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 functions 𝐺_𝑋(𝑥) and𝐺_𝑌(𝑥)satisfying the conditions 𝐺_𝑋(𝑥)≥0,𝐺_𝑋(−𝑥)=𝐺_𝑋(𝑥),𝐺_𝑌(𝑥)≥0,𝐺_𝑌(−𝑥)=𝐺_𝑌(𝑥), (1) where the notation𝐺(𝑥)=1−𝐺(𝑥)is used throughout, such that the probability density functions (pdf) of𝑋and𝑌are given by 𝑓_𝑋(𝑥)=2𝑓₀(𝑥)𝐺_𝑋(𝑥)(𝑋),𝑓,𝑥∈𝑆_𝑌𝑦=2𝑓₀𝑦𝐺_𝑌𝑦,𝑦∈𝑆(𝑌). (2)

Equivalently to (2), there exists a continuous skewing distribution𝐺₀(𝑥)with an even density𝑔₀(𝑥)=𝐺^′₀(𝑥)and odd skewing weights𝑤_𝑋(𝑥),𝑤_𝑌(𝑥)such that 𝑓_𝑋(𝑥)=2𝑓₀(𝑥)𝐺₀𝑤_𝑋(𝑥(𝑋),𝑓),𝑥∈𝑆_𝑌𝑦=2𝑓₀𝑦𝐺₀𝑤_𝑌𝑦,𝑦∈𝑆(𝑌). (3)

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 𝐺₀(𝑡)=1𝑡+2⋅1_(−1,1)(2𝑡)+1_[1,∞)()𝑤(𝑥)(𝑥)−12𝑡,𝑡∈𝑅,=𝐺2, (4) where1_𝐴(⋅)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 functions𝐺_𝑋and𝐺_𝑌with derivatives𝑔_𝑋(𝑥)=𝐺^′_𝑋(𝑥),𝑔_𝑌(𝑦)=𝐺^′_𝑌(𝑦). The following conditions are equivalent:(LR1)𝑋≤_𝑙𝑟𝑌 (𝑋 precedes 𝑌 in likelihood ratio order).(LR2)The ratio of the perturbation functions 𝐺_𝑌(𝑡)/𝐺_𝑋(𝑡) is monotone increasing in 𝑡 over the union of the supports 𝑆(𝑋)∪𝑆(𝑌).(LR3)The reverse hazard rates of the perturbation functions satisfy the inequalities𝑟_𝑋(𝑡)=𝑔_𝑋(𝑡)𝐺_𝑋(𝑡)≤𝑟_𝑌(𝑡)=𝑔_𝑌(𝑡)𝐺_𝑌(𝑡),𝑡∈𝑆(𝑋)∪𝑆(𝑌). (5)

Proof. By definition𝑋≤_lr𝑌if, and only if (e.g. Shaked and Shanthikumar [4, equation (1.C.2)]), one has 𝑓_𝑋(𝑥)𝑓_𝑌𝑦≥𝑓_𝑋𝑦𝑓_𝑌(𝑥),∀𝑥≤𝑦, (6) or equivalently, for a SSD with a common kernel, 𝐺_𝑌(𝑥)𝐺_𝑋(𝑥)≤𝐺_𝑌𝑦𝐺_𝑋𝑦, (7)
that is, (LR2). On the other hand, in case𝐺_𝑌(𝑡)/𝐺_𝑋(𝑡)is monotone increasing, so is its logarithmic. The equivalence of (LR2) and (LR3) follows from the relation 𝑑𝐺𝑑𝑡ln^𝑌(𝑡)𝐺_𝑋(𝑡)=𝑔_𝑌(𝑡)𝐺_𝑌(𝑡)−𝑔_𝑋(𝑡)𝐺_𝑋(𝑡)=𝑟_𝑌(𝑡)−𝑟_𝑋(𝑡). (8)

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 order𝐺_𝑋≤_rh𝐺_𝑌between 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) and𝐺_𝑋≤_rh𝐺_𝑌in 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 TP₂ order, written as𝑋≤_lr𝑌or𝑋≤_TP2𝑌, if 𝑓_𝑋(𝑥)𝑓_𝑌𝑦≤𝑓_𝑋𝑥∧𝑦𝑓_𝑌𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛, (9) where one sets𝑥∧𝑦=(𝑥₁∧𝑦₁,…,𝑥_𝑛∧𝑦_𝑛),𝑥∨𝑦=(𝑥₁∨𝑦₁,…,𝑥_𝑛∨𝑦_𝑛), and𝑥_𝑖∧𝑦_𝑖=min(𝑥_𝑖,𝑦_𝑖),𝑥_𝑖∨𝑦_𝑖=max(𝑥_𝑖,𝑦_𝑖).
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 density𝑓_𝑋(𝑥)is multivariate totally positive of order 2 (MTP₂) such that 𝑓_𝑋(𝑥)𝑓_𝑋𝑦≤𝑓_𝑋𝑥∧𝑦𝑓_𝑋𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛, (10) a property discussed in Karlin and Rinott [5] and Whitt [6].

Applied to SSD random vectors𝑋and𝑌with a common kernel𝑓₀and perturbation functions𝐺_𝑋and𝐺_𝑌, condition (9) means that 𝑓₀(𝑥)𝑓₀𝑦𝐺_𝑋(𝑥)𝐺_𝑌𝑦≤𝑓₀𝑥∧𝑦𝑓₀𝑥∨𝑦𝐺_𝑋𝑥∧𝑦𝐺_𝑌𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛. (11)

Now, if the random vector𝑋₀associated with the kernel density𝑓₀satisfies the reflexive property𝑋₀≤_lr𝑋₀, or equivalently𝑓₀is MTP₂, and the following inequalities 𝐺_𝑋(𝑥)𝐺_𝑌𝑦≤𝐺_𝑋𝑥∧𝑦𝐺_𝑌𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛, (12) 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 function𝐺_𝑋is said to be smaller than𝐺_𝑌in the reverse hazard rate order, written as𝐺_𝑋≤_rh𝐺_𝑌, 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 𝑋 and 𝑌 be SSD random vectors with a common kernel 𝑓₀ and perturbation functions 𝐺_𝑋 and 𝐺_𝑌. If 𝑓₀ is MTP₂ and 𝐺_𝑋≤_𝑟ℎ𝐺_𝑌, 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 𝑓_𝑋𝑦𝑓_𝑌(𝑥)≤𝑓_𝑋(𝑥)𝑓_𝑌𝑦,∀𝑥≤𝑦. (13) Here 𝑥≤𝑦 denotes the usual componentwise partial order between vectors, which is defined as follows. If 𝑥=(𝑥₁,…,𝑥_𝑛) and 𝑦=(𝑦₁,…,𝑦_𝑛) are two n-dimensional vectors, then 𝑥≤𝑦 if 𝑥_𝑖≤𝑦_𝑖 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 𝑓_𝑌(𝑥)/𝑓_𝑋(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(𝑋_𝑖,𝑖∈𝐴),𝑔(𝑋_𝑗,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 𝐺_𝑋𝑦𝐺_𝑌(𝑥)≤𝐺_𝑋(𝑥)𝐺_𝑌𝑦,∀𝑥≤𝑦. (14) 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 function𝐺_𝑋is said to be smaller than𝐺_𝑌in the weak reverse hazard rate order, written as𝐺_𝑋≤_wrh𝐺_𝑌if the property (14) is satisfied.
From (12) and (14), it follows immediately that 𝐺_𝑋≤_rh𝐺_𝑌⟹𝐺_𝑋≤_wrh𝐺_𝑌. (15) 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 functions𝐺_𝑋and𝐺_𝑌have a common support𝑆, which is a lattice (that is, if𝑥,𝑦∈𝑆then𝑥∧𝑦,𝑥∨𝑦∈𝑆). If𝐺_𝑋and/or𝐺_𝑌is MTP₂, then 𝐺_𝑋≤_wrh𝐺_𝑌⟹𝐺_𝑋≤_rh𝐺_𝑌. (16)

Proof. The ordering𝐺_𝑋≤_wrh𝐺_𝑌implies by (14) that 𝐺_𝑋𝑥∨𝑦𝐺_𝑌𝑦≤𝐺_𝑋𝑦𝐺_𝑌𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛. (17) If𝐺_𝑋is MTP₂, then by the defining property (10) one has 𝐺_𝑋(𝑥)𝐺_𝑋𝑦≤𝐺_𝑋𝑥∧𝑦𝐺_𝑋𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛. (18) Multiplication of these two inequalities yields 𝐺_𝑋𝑥∨𝑦𝐺_𝑌𝑦𝐺_𝑋(𝑥)𝐺_𝑋𝑦≤𝐺_𝑋𝑦𝐺_𝑌𝑥∨𝑦𝐺_𝑋𝑥∧𝑦𝐺_𝑋𝑥∨𝑦,∀𝑥,𝑦∈𝑅^𝑛. (19)
Now, since𝑆is a lattice, it follows that if𝐺_𝑋(𝑥)𝐺_𝑋(𝑦)>0, then𝐺_𝑋(𝑦)and𝐺_𝑋(𝑥∨𝑦)are positive. Cancelling these quantities in the preceding inequality implies that (12) holds in this case. If𝐺_𝑋(𝑥)𝐺_𝑋(𝑦)=0then (12) is trivially fulfilled. Together, this shows that𝐺_𝑋≤_rh𝐺_𝑌. If𝐺_𝑌is MTP₂, one shows similarly that𝐺_𝑋≤_rh𝐺_𝑌. The proof is complete.

Theorem 7 (second sufficient condition for likelihood ratio order). Let𝑋and𝑌be SSD random vectors with a common kernel𝑓₀and perturbation functions𝐺_𝑋and𝐺_𝑌, which have a common lattice support. If𝑓₀is MTP₂, at least one of𝐺_𝑋and𝐺_𝑌is MTP₂, and𝐺_𝑋≤_𝑤𝑟ℎ𝐺_𝑌; then𝑋≤_𝑙𝑟𝑌.

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 vectors𝑋₀and𝑋associated with the kernel densities𝑓₀(𝑥)and the SSD densities𝑓_𝑋(𝑥)=2𝑓₀(𝑥)𝐺_𝑋(𝑥)that remain valid over a large class of perturbation functions. For example, the well-known even transformation invariance states that 𝑇(𝑋)=_𝑑𝑇𝑋₀equalityindistribution, (20) for all even real functions𝑇(𝑥), whatever the perturbation function𝐺_𝑋(𝑥)is (e.g., Azzalini and Regoli [3], Proposition1).

Corollary 8 (likelihood ratio order invariance). Let𝑓_𝑋(𝑥)=2𝑓₀(𝑥)𝐺_𝑋(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)if𝐺_𝑋is an increasing perturbation function then𝑋₀≤_𝑙𝑟𝑋;(LRI2)if𝐺_𝑋is decreasing on𝑆(𝑋)∩𝑅^𝑛₊, then𝑋≤_𝑙𝑟𝑋⁺₀=𝑋₀⋅1_𝑅^𝑛₊(𝑋₀).

Proof. To show (LRI1) it suffices to observe that the perturbation function𝐺_𝑋0(𝑥)≡1/2is MTP₂ and generates the kernel density𝑓₀(𝑥)of𝑋₀. The statement follows from Theorem 7. Similarly, the choice𝐺_𝑋⁺₀(𝑥)=1_𝑅^𝑛₊(𝑥)generates the random variable𝑋⁺₀and (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 conditions𝐺_𝑋≤_wrh𝐺_𝑌in Theorem 7 and𝐺_𝑋≤_rh𝐺_𝑌in 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.