A note on inconsistent families of discrete multivariate distributions

Transcription

1 Ghosh et al. Journal of Statistical Distriutions and Applications :7 DOI /s RESEARCH Open Access A note on inconsistent families of discrete multivariate distriutions Sugata Ghosh 1, Suhajit Dutta 1 and Marc G. Genton 2* *Correspondence: marc.genton@kaust.edu.sa 2 CEMSE Division, King Adullah University of Science and Technology, Thuwal , Saudi Araia Full list of author information is availale at the end of the article Astract We construct a d-dimensional discrete multivariate distriution for which any proper suset of its components elongs to a specific family of distriutions. However, the joint d-dimensional distriution fails to elong to that family and in other words, it is inconsistent with the distriution of these susets. We also address preservation of this inconsistency property for the symmetric Binomial distriution, and some discrete distriutions arising from the multivariate discrete normal distriution. Keywords: Binomial distriution, Discrete normal and skew-normal, Multivariate distriutions, Sign function, Symmetric distriutions AMS Suject Classification: 62E10; 62H05 1 Introduction If a multivariate distriution is parametrically specified, often the lower-dimensional marginals follow the same distriution of an appropriate dimension. For example, all the lower-dimensional distriutions of a multivariate Gaussian distriution are Gaussian. The converse is however not necessarily true. For example, Dutta and Genton 2014 gave a construction of a non-gaussian multivariate distriution with all lower-dimensional Gaussians and considered some generalizations of this for a certain class of elliptical and skew-elliptical distriutions. In the discrete case, all the lower-dimensional marginals of a ivariate Binomial distriution Bairamov and Gultekin 2010 follow the Binomial distriution. Conversely, given a set of marginal distriutions, different dependence structures can give rise to different joint distriutions. In a more general setting, Hoeffding 1940 and Fréchet 1951 independently otained a characterization of the class of ivariate distriutions with given univariate marginals. Characterization prolems for discrete distriutions using conditional distriutions and their expectations have een investigated y several authors; see, e.g., Dahiya and Korwar 1977, Ruiz and Navarro 1995, and Nguyen et al The paper y Conway 1979 also discussed some additional properties and derived appropriate relationships for such systems, while a method of constructing multivariate distriutions with specified univariate marginals and a given correlation matrix was studied y Cuadras The main aim of this paper is the converse question for the discrete multivariate setup. More specifically, we construct a discrete multivariate distriution see, e.g., Johnson et al. 1997, all of whose lower-dimensional marginals follow the same symmetric distriution ut the joint distriution does not conform to that pattern and has a different distriution. TheAuthors. 2017Open Access This article is distriuted under the terms of the Creative Commons Attriution 4.0 International License which permits unrestricted use, distriution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 2 of 13 We say that the joint distriution is inconsistent with the distriution of its marginals. The main idea of this construction is ased on a transformation using the sign function applied to a class of discrete symmetric random variales. The sign function plays a key role in yielding a distriution with restricted support. The structure of the paper is as follows. In Section 2, we first motivate our construction for the ivariate set-up with a simple case starting from a symmetric Binomial distriution. Generalization of this result to symmetric discrete distriutions with a specific support in higher dimensions is explored in Section 3 where our main result is stated. We also address preservation of this inconsistency property for the symmetric Binomial distriution in Section 4, a class of symmetric discrete distriutions constructed from symmetric continuous distriutions in Section 5, and the multivariate discrete skew-normal distriution in Section 6. Proofs of all the theorems are provided in the Appendix. 2 A motivating ivariate example Consider a random variale U that follows a Binomial distriution with parameters n = 2andp = 1 2. The support of this distriution is the set N 2 ={0, 1, 2}, and it has the following proaility mass function pmf: q 0 u = 2 u 1 4 if u N 2. Let U 1 and U 2 e two independent copies of U. The support of the joint distriution of U 1 and U 2 is N2 2 ={0, 1, 2} {0, 1, 2}.Thus,thejointpmfofU 1 and U 2 is q 0 u 1, u 2 = 16 if u 1, u 2 N2 2. u 1 u 2 Define X = U 1. The support of the distriution of X is the set Z 1 ={ 1, 0, 1},andit has the pmf: 2 1 qx = x if x Z 1. Clearly, this is a version of the Binomial distriution which is symmetric aout 0. Let X 1 and X 2 e two independent copies of X. The support of the joint distriution X 1, X 2 T is Z1 2 ={ 1, 0, 1} { 1, 0, 1}.Thus,thejointpmfofX 1 and X 2 is 2 qx 1, x 2 = x x Recall the univariate sign function, namely, 1, if x < 0, Sx = 0, if x = 0, 1, if x > if x 1, x 2 Z We use a modified version of this sign function. Consider the following transformation: X 1 = X 1S 2,1 and X 2 = X 2S 1,2, where S 1,2 and S 2,1 are defined as follows:

3 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 3 of 13 1, if X 1 < 0, or X 1 = 0andX 2 = 0, S 1,2 = 0, if X 1 = 0andX 2 = 0, 1, if X 1 > 0, and 1, if X 2 < 0, or X 2 = 0andX 1 = 0, S 2,1 = 0, if X 2 = 0andX 1 = 0, 1, if X 2 > 0. We now study the distriution of X1. and PX1 = 1 = PX 1S 2,1 = 1 = PX 1 = 1, S 2,1 = 1 + PX 1 = 1, S 2,1 = 1 = PX 1 =1, X < 0+ PX 1 = 1, X 2 = 0, X 1 = 0 + PX 1 = 1, X 2 > 0 = PX 1 = 1, X 2 = 1 + PX 1 = 1, X 2 = 0 + PX 1 = 1, X 2 = 1 = = 1 4, PX1 = 0 = PX 1S 2,1 = 0 = PX 1 = 0, S 2,1 = 0 + PX 1 = 0, S 2,1 = 0 + PX 1 = 0, S 2,1 = 0 = { PX 1 = 0, X 2 < 0 + PX 1 = 0, X 2 > 0 } PX 1 = 0, X 2 = 0 = PX 1 = 0, X 2 = 1 + PX 1 = 0, X 2 = 1 + PX 1 = 0, X 2 = 0 = = 1 2. This now implies that PX1 = 1 = 1/4. Hence, the distriution of X 1 is the same as that of the distriution of X. Similarly, one can show that X2 is identically distriuted with X. By our construction, the support of the joint distriution is on a restricted set, i.e., X 1 X 2 = X 1S 2,1 X 2 S 1,2 = X 1 S 1,2 X 2 S 2,1 which is non-negative with proaility one. The joint distriution of X 1, X 2 T is as follows: PX 1 = 1, X 2 = 1=PX 1 = 1, X 2 = 1 + PX 1 = 1, X 2 = 1 = = 1 8, PX 1 = 1, X 2 = 0 = PX 1 = 1, X 2 = 0 = 1 8, PX 1 = 0, X 2 = 1 = PX 1 = 0, X 2 = 1 = 1 8, PX 1 = 0, X 2 = 0 = PX 1 = 0, X 2 = 0 = 1 4, PX 1 = 0, X 2 = 1 = PX 1 = 0, X 2 = 1 = 1 8, PX 1 = 1, X 2 = 0 = PX 1 = 1, X 2 = 0 = 1 8,and PX 1 = 1, X 2 = 1 = PX 1 = 1, X 2 = 1 + PX 1 = 1, X 2 = 1 = = 1 8. Let I A x e the indicator function, which is defined as

4 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 4 of 13 Fig. 1 3D plots of the two proaility mass functions qx 1, x 2 and q x 1, x 2 on Z 2 2 I A x = { 1, if x A, 0, if x / A. It is easy to check that the joint pmf of X 1, X 2 T canere-writteninaconsiseformas follows: q x 1, x 2 = qx 1, x 2 I {0} x 1 x 2 + 2qx 1, x 2 I 0, x 1 x 2 for x 1, x 2 Z 2 1, where the expression of qx 1, x 2 is defined in 1. The joint distriution of X1, X 2 T is asymmetric, and clearly does not elong to the family of distriutions containing the joint distriution of X 1, X 2 T as the support of X1, T 2 X is restricted to two quadrants only, although the univariate distriutions of X1 and X 2 are the same as that of the distriution of X. The plot in Fig. 1 demonstrates the asic idea of shifting of proaility mass to only two quadrants see the right panel of Fig. 1, and clearly shows the difference etween the pmfs qx 1, x 2 and q x 1, x 2. This motivates us to construct a d-dimensional random vector such that the distriution of any proper suset of its components elongs to a certain family of distriutions, ut the joint distriution, with all the d components taken together, fails to conform to that family of distriutions. 3 General symmetric discrete multivariate distriutions We now consider a general discrete distriution symmetric aout the point 0 with support on a finite or countaly infinite set S,givenythepmf qx = δx jm j, 2 j S where 0 m j 1andm j = m j for all j S with j S m j = 1. Let X 1,..., X d e independent copies of X q. Here, the function δ is defined as follows: δx = { 1, if x = 0, 0, if x = 0.

5 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 5 of 13 We have I {0} x = δx. The joint pmf of X 1,..., X d T is given y qx = qx i = δx i jm j, j S where x = x 1,..., x d T S d. Again, consider the transformation X 1,..., X d X 1,..., X d such that X 1 = X 1S 2,1,..., X i 1 = X i 1S i,i 1,..., X d 1 = X d 1S d,d 1, X d = X ds 1,d, 3 where the modified sign function is defined as 1, if X i < 0orX i = 0andX i 1 = 0, S i,i 1 = 0, if X i = 0andX i 1 = 0, 1, if X i > 0, for i = 2,..., d,and 1, if X 1 < 0orX 1 = 0andX d = 0, S 1,d = 0, if X 1 = 0andX d = 0, 1, if X 1 > 0. Now, d X i = X 1 S 2,1 X 2 S 3,2 X d S 1,d = X 1 S 1,d X 2 S 2,1 X d S d,d 1.Fixi {2,..., d}. Again,X i < 0 S i,i 1 = 1, hence X i S i,i 1 > 0, while X i > 0 S i,i 1 = 1, hence X i S i,i 1 > 0. Also, X i = 0 X i S i,i 1 = 0. To summarize, X i S i,i 1 0 for any i = 2,..., d. Similarly, X 1 S 1,d 0. Hence, we otain d X i 0. Theorems 1 and 2 elow state the joint distriution of X 1,..., X d T as well as the distriution of the lower-dimensional vectors. Theorem 1 The joint pmf of X1,..., T d X is q x = qxi {0} x i + 2qxI 0, Theorem 2 Any su-vector T Xk 1,..., Xk d of X 1,..., Xd T with d < d is component- T Xk 1,..., Xk d is the same as that of wise independent, and the joint distriution of T. Xk1,..., X kd x i for x S d. In particular, this result holds true if S = Z ={..., 1, 0, 1,...} or any proper suset of Z which is symmetric aout 0, i.e., x S x S. 4 The symmetric inomial distriution We started our investigation in Section 2 using a discrete distriution symmetric aout 0 withsupportonthesetn 2, and extended it to the case of a general symmetric discrete distriution symmetric aout 0, supported on a finite or countaly infinite suset S of R. However, it is interesting to see whether such inconsistency results continue to hold for other symmetric distriutions for which the point of symmetry is not necessarily 0. Suppose that U follows a symmetric distriution with a finite or countaly infinite support and point of symmetry u 0.DefineX = U u 0.Then,X follows a symmetric distriution aout 0 with a finite or countaly infinite support as mentioned in 2.

6 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 6 of 13 Assume U Binomialn,1/2, and consider u 0 to e n/2. We now have versions of Theorems 1 and 2 for symmetric Binomial distriutions. 5 Discrete symmetric distriutions Given any continuous distriution with distriution function df F on R, wecandefine its discrete analogue see, e.g., Alzaatreh et al with pmf q d as follows: q d x = Fx + 1 Fx, x Z, with q d x 0and x Z q dx = 1. If U F, then this implies that [U] q d.here, [u] denotes the largest integer less than or equal to u. Assume F to e symmetric aout the point 0, i.e., Fx + F x = 1 for any x R. Now, note that q d 0 = q d 1 and the point of symmetry of [ U] is clearly 1 2.Define X =[U] Then,X is a discrete variate with support on a countaly infinite set S which is symmetric aout the point 0, say, X q ds. Inparticular,ifwetakeFx = x, where is the df of the standard normal distriution, then we otain the discrete normal distriution dn Roy The pmf of X say, q dn simplifies to e x + 1/2 x 1/2 with x S. Using this discretization idea, Chakraorty and Chakravarty 2016 have proposed a discrete logistic distriution starting from the continuous twoparameter logistic distriution. Now, we can construct versions of Theorems 1 and 2 for such symmetric discrete proaility distriutions. 6 The multivariate discrete normal and related distriutions In the multivariate setting say, R d, let U d N d 0, I d e the d-dimensional normal distriution with I d as the d d identity matrix. We can define an analogue of the discrete normal dn distriution on S d as follows: X d =[ U d ] d = [ U 1 ] + 1 2,...,[U d] T. 4 ThejointpmfofX d is the product d q dn x i. We now apply the transformation defined in Eq. 6 on the vector X d stated in Eq. 4 to otain X d.define Y d = A1 d + BX d, where A, B S are discrete random variales independent of X d. The support of the random vector Y d is the set Sd,whereS is a countale set which depends on the joint distriution of A and B.DefineY d 1,d ased on Y d y selecting a suset of size d 1from the set {1,..., d}. Theorem 3 The distriutions of all the d 1-dimensional random vectors Y d 1,d elong to the same family of distriutions, ut that of Y d does not. We now state consequences of Theorem 3 in some popular and important classes of symmetric and asymmetric distriutions derived from the multivariate discrete normal distriution:

7 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 7 of 13 Discrete Scale Mixture Consider A = 0 a degenerate random variale and B = W,whereW is a non-negative, discrete random variale independent of X d. Discrete Skew-Normal Consider a discrete normal variate X d+1 q dn independent of X d. Define A = δ X d+1 and B = 1 δ 2 a degenerate random variale with 1 <δ<1 also see pp of Azzalini Using Theorem 3, we now otain this inconsistency result for a class of discrete skew-normal distriutions. One can also extend this results to a class of skew-elliptical distriutions starting from the discrete scale mixture. More general results using the idea of modulation of symmetric discrete distriutions proposed recently y Azzalini and Regoli 2014 still remain open. Appendix: Proofs and Mathematical Details Proof 1 Proof of Theorem 1 We reak the proof into two parts, namely, Case I and Case II. Case I: d Xi = 0. This fact now implies that at least one of the Xi s is zero. Without loss of generality, let X1 = 0 and consider the event X1 = 0, X 2 = x 2,..., Xd = x d. Under the assumption that X1 = 0, we now estalish the following facts: F1 Xi = 0 X i = 0 for any i = 1,..., d. To show this, we note that if X1 = 0, then X 1 S 2,1 = 0. Now, either X 1 = 0orS 2,1 = 0. If X 1 = 0 then we are done. If S 2,1 = 0, then X 2 = X 1 = 0, in particular X 1 = 0. More generally, Xi = 0 X i = 0 for any i = 1,..., d. F2 If X1 = 0, then X d = x d. To show this, suppose that Xd = 0, then we otain X d S 1,d = 0 and in particular X d = 0. We have assumed that X1 = 0, now using F1 we know that X 1 = 0. Comining the facts that X 1 = 0andX d = 0, we have y definition S 1,d = 1. Again, Xd = x d X d S 1,d = x d and thus we otain X d = x d.onthe other hand, suppose that Xd = 0. In this case x d = 0. By F1, Xd = 0 X d = 0. Thus we trivially have X d = x d oth sides eing equal to 0. Hence, in all the cases, we have X d = x d. F3 We have X i = x i S i+1,i for i = 2,..., d 1. To show this, we start with any i {2,..., d 1}. NowXi = x i, which implies X i S i+1,i = x i.firsttakex i = 0. Then X i = 0 and S i+1,i = 0. Since X i = 0, S i+1,i simplifies to the following function: { 1, if Xi+1 0, S i+1,i = 1, if X i+1 > 0. Thus, S i+1,i only takes the values 1 or 1, which implies that Si+1,i 2 = 1. We can multiply S i+1,i on oth sides of the equation X i S i+1,i = x i to otain X i = x i S i+1,i. On the other hand, suppose x i = 0. Using F1, we otain X i = 0 which trivially satisfies X i = x i S i+1,i as oth sides equal zero. This is true for all i = 2,..., d 1. F4 We have PX i = x i S i+1,i = PX i = x i. To show this, we first consider the case when x i = 0. Then S i+1,i = ±1. Hence PX i = x i S i+1,i = PX i = ±x i, which is PX i = x i y virtue of symmetry. The claim follows trivially when x i = 0. Now, we consider the proaility

8 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 8 of 13 PX1 = 0, X 2 = x 2,..., Xd = x d = PX 1 = 0, X 2 = x 2 S 3,2,..., X d 1 = x d 1 S d,d 1, X d = x d = PX 1 =0PX 2 =x 2 S 3,2 PX d 1 =x d 1 S d,d 1 PX d = x d [ Using independence ] = PX 1 = 0PX 2 = x 2 PX d 1 = x d 1 PX d = x d [ Using F4 ] = PX 1 = 0, X 2 = x 2,..., X d 1 = x d 1, X d = x d In this case, the joint pmf of X 1,..., X d T denoted y q x is given as q x = qx. This completes the first part. Case II: Assume d Xi > 0. This implies that none of the Xi s are zero. Since X i = 0 implies that X i = 0fori = 1,..., d, we can ignore the value of the sign function S at 0 and it simplifies to the following function: { 1, if x < 0, Sx = 1, if x > 0. Thus, in this case we always have Sx =±1. Also, note that here the modified sign function S i,i 1 simplifies to SX i, and hence Xi = X i SX i+1, i = 1,..., d 1andXd = X d SX 1. We now consider enumerating the joint proaility P X1 = x 1,..., Xd = x d with the restriction that d x i > 0. Lemma 1 The event X1 = x 1,..., Xd = x d is equivalent to the occurrence of either of the following two events: dj=2 i X 1 = x 1, X 2 = x 2 Sx 2, X 3 = x 3 Sx 2 x 3,..., X d = x d S,or dj=2 ii X 1 = x 1, X 2 = x 2 Sx 2, X 3 = x 3 Sx 2 x 3,..., X d = x d S. Proof 2 Proof of Lemma 1 Note that X k = Xk for all k = 1,..., d.also,inthiscase, the value of SX 2 can either e 1, or 1. We now consider two separate cases. Case i: Assume SX 2 = 1, and we get X 1 = x 1.RecallthatX2 = X 2SX 3 = X 2 SX 2 SX 3,hencewegetX2 = X 2 SX 3 and X 2 = X2 /SX 3. Now,thefollowing holds: X2 = x 2 X 2 SX 3 = x 2 x 2 SX 3 = x 2 SX 3 = x 2 x 2 = Sx 2. Then X2 = X 2SX 3 X 2 = X 2 SX 3 = X 2 SX 3 = x 2 Sx 2. Here, we use the fact that Su = 1/Su. Again, X3 = X 3SX 4 = X 3 SX 3 SX 4. Using the expression of SX 3 derived aove, we otain X 3 = x 3 X 3 SX 3 SX 4 = x 3 x 3 Sx 2 SX 4 = x 3. Therefore, x 3 SX 4 = x 3 Sx 2 = Sx 3 Sx 2 = Sx 3Sx 2 = Sx 2 x 3. Note that Sxy = SxSy.Then X 3 = X 3SX 4 X 3 = X 3 SX 4 = X 3 SX 4 = x 3 Sx 2 x 3.

9 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 9 of 13 Further, X 4 = X 4SX 5 = X 4 SX 4 SX 5 and using the expression of SX 4 otained aove we have, X 4 = x 4 X 4 SX 4 SX 5 = x 4 x 4 Sx 2 x 3 SX 5.Thisimpliesthat SX 5 = x 4 x 4 Sx 2 x 3 = Sx 4 Sx 2 x 3 = Sx 2x 3 x 4. Therefore, X4 = X 4SX 5 X 4 = X 4 SX 5 = x 4 Sx 2 x 3 x 4 = x 4Sx 2 x 3 x 4. Proceeding in a similar fashion, we otain X d = x d Sx 2 x d. Case ii: The proof follows y taking SX 2 = 1, and repeating the line of arguments stated aove for Case i. Now, we can enumerate the quantity P X1 = x 1, X2 = x 2,..., Xd = x d as follows P X1 = x 1, X2 = x 2,..., Xd = x d = PX 1 SX 2 = x 1, X 2 SX 3 = x 2,..., X d SX 1 = x d i = P X 1 = x 1, X 2 = x 2 Sx 2,..., X i = x i S,..., X d = x d S i + P X 1 = x 1, X 2 = x 2 Sx 2,..., X i = x i S i = PX 1 = x 1 PX 2 = x 2 Sx 2 P X i = x i S,..., X d = x d S i + PX 1 = x 1 PX 2 = x 2 Sx 2 P X i = x i S [ Using independence ] P X d = x d S P X d = x d S = PX 1 = x 1 PX i = x i PX d = x d + PX 1 = x 1 PX i = x i PX d = x d [ Using symmetry aout 0, and the fact that S =±1 ] = 2P X 1 = x 1,..., X d = x d. Hence in this case, the joint pmf of X 1,..., X d T is given y q x = 2qx. Thiscompletestheproofofthesecondpart. Comining these two cases, we may write the joint pmf of X1,..., T d X as follows: q x = qxi {0} x i + 2qxI 0, x i. 5 This completes the proof. Proof 3 Proof of Theorem 2 First we consider the univariate distriutions, namely, when d = 1 and compute the proaility PXt = x t, denoted y q x t for a fixed t {1,..., d}. Now,supposethatx t = 0. Note that Xt = X t S t+1,t = 0 X t = 0since S t+1,t = 0 also requires X t to e zero. We trivially have the reverse, i.e., X t = 0 Xt = 0. So, we have Xt = 0 X t = 0 and hence P Xt = 0 = P X t = 0. Thus,weotain q x t = qx t for any t = 1,..., d. On the other hand, if x t = 0, then S t+1,t simplifies to the following function

10 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 10 of 13 S t+1,t = and we get { 1, if X t+1 0, 1, if X t+1 > 0, PXt = x t = PX t S t+1,t = x t = PX t = x t, S t+1,t = 1 + PX t = x t, S t+1,t = 1 = PX t = x t, X t+1 > 0 + PX t = x t, X t+1 0 = PX t = x t PX t+1 > 0 + PX t = x t PX t+1 0 = PX t = x t {PX t+1 > 0 + PX t+1 0} = PX t = x t. Thus, in all these cases, we otain q x t = qx t for t = 1,..., d. 6 Define the following random vectors: X 1 =X 2,..., X d T, X t =X 1,..., X t 1, X t+1,..., X d T, X d =X 1,..., X d 1 T and X 1 = X 2,..., X d T, X t = X 1,..., X t 1, X t+1,..., X d T, X d = X 1,..., X d 1 T, and the sets I 1 ={2,..., d}, I t ={1,..., t 1, t + 1,..., d}, I d ={1,..., d 1}. Let x t =x 1,..., x t 1, x t+1,..., x d T,andqx t =P X t = x t for t =1,..., d. We want to compute the proaility P X t = x t, and denote it y q x t.further, we denote the joint proaility P X t = x t, X t = x t y qx t, x t, which is nothing ut P X 1 = x 1,..., X d = x d, and the joint proaility P X t = x t, Xt = x t y q x t, x t, which is nothing ut P X 1 = x 1,..., Xd = x d. We now consider three separate cases. Case I: i I t x i = 0 Note that i I t x i = 0 d x i = 0. Hence, we get q x = qx = qx t qx t. This now gives the following: q x t = q x=q x t qx t =q x t = qx i = q x i [Using 6]. x t S x t S i I t i I t Case II: i I t x i > 0 Consider three separate su-cases i If x t = 0, then d x i = 0, and q x = qx = q x t q0. So,q x t,0 = q x t q0. ii If x t > 0, then d x i > 0. Thus, q x = 2qx = 2q x t qxt.hence, q x t, x t = 2q x t t:x t >0 qx t. iii If x t < 0, then d x i < 0, which cannot happen y the construction of X i s. Hence, q x t, x t = 0. Considering these three su-cases, we have

11 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 11 of 13 q x t = t:x t <0 q x t, x t + q x t,0 + t:x t >0 q x t, x t = 0 + q x t q0 + 2q x t qx t t:x t >0 = q x t qx t [Using symmetry aout 0] x t S = q x t = qx i = q x i [Using 6]. i I t i I t Case III: i I t x i < 0 Again, consider three separate su-cases i If x t = 0, then d x i = 0. Hence, q x = qx = q x t q0, i.e., q x t,0 = q x t q0. ii If x t < 0, then d x i > 0. Hence, q x = 2qx = 2q x t qxt, i.e., q x t, x t = 2q x t t:x t <0 qx t. iii If x t > 0, then d x i < 0. Hence, q x t, x t = 0. Comining these three su-cases, we otain q x t = q x t, x t + q x t,0 + q x t, x t x t <0 x t >0 = 2q x t t:x t <0 qx t + q x t q0 + 0 = q x t qx t [ Using symmetry aout 0 ] x t S = q x t = qx i = q [ ] x i Using 6. i I t i I t T In all these cases, the components of a su-vector Xk 1,..., Xk of X d 1 1,..., Xd T are independent of each other. Again, if we take a su-vector with size less than d 1, then the components will also e independent as all the su-vectors of size d 1are T component-wise independent. Thus, we can say that any su-vector Xk 1,..., Xk d of X 1,..., Xd T with d < d is component-wise independent. Now, P Xk 1 = x 1, Xk 2 = x 2,..., Xk = x d d [Using = P X = x k1 1 P X = x k2 2 P Xk ] = x d d independence = P X k1 = x 1 P Xk2 = x 2 P Xkd = x d [ Since the marginal distriutions of X i and X i are identical for i = k 1,..., k d ] = P [ ] X k1 = x 1, X k2 = x 2,..., X kd = x d Again using independence. T Hence, the joint distriution of Xk 1,..., Xk T. d is same as that of Xk1,..., X kd Proof 4 Proof of Theorem 3 Let us denote the joint distriution of A, B y the pmf Ga, with a, S. Recall that we have assumed A, B to e independent of X d.so, the conditional distriution of Y d given A = a, B = is same as its unconditional distriution. We will use this fact throughout the proof of this theorem. The joint distriution of Y d is given y

12 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 12 of 13 r y d = a, S = a, S a, S P Y d = y d a, Ga, P X d = y d a1 d Ga, q yd a1 d Ga, = a, S = q yd a1 d I y i a = 0 yd a1 d + 2q I y i a > 0 Ga,. Recall the expression of q x from Eq. 5. Define Y 1 = Y2, Y 3,..., Y d T with I 1 ={2, 3,..., d}; Y t = Y1,..., Y t 1, Yt+1,..., Y d T with I t = {1,..., t 1, t + 1,..., d}, t = 2,...,d 1; andy d = T Y1, Y 2 d 1,..., Y with I d = {1, 2,..., d}. Also, define y t = y 1,..., y t 1, y t+1,..., y d.consider r y t = k S a, S P Y 1 = y 1,..., Y t Now, for any fixed a, S,wehave P Y 1 =y 1,..., Y t =k,..., Y d =y d Then, r y t = P a, S k S = a, S = a, S = a, S P X 1 = y 1 a X 1 = y 1 a q y t a1 d 1 y t a1 d 1 q =P X 1 = y 1 a = k,..., Y d = y d a, Ga,.,..., Xt = k a,..., X d = y d a.,..., Xt = k a,..., X d = y d a Ga,,..., Xt 1 = y t 1 a, Xt+1 = y t+1 a,..., Xd = y d a Ga, [ Recall the expression of q x t ] Ga, = a, S i I t q yi a Ga,. Ga, Thus, the random vectors Y t for t = 2,..., d 1 possess the same joint distriution. Similarly, one may argue that the random vectors Y 1 and Y d also follow the same T joint distriution as well. Further, we can argue that any su-vector Yk 1, Yk 2,..., Yk d of Y 1, Y 2,..., Y d T,whered < d, has the same joint distriution. Acknowledgements We thank the Reviewers and Editors for comments that improved the paper. Funding This research was supported y the King Adullah University of Science and Technology KAUST. Availaility of data and materials Not applicale. Authors contriutions SG, SD, MG contriuted equally to the research. All authors read and approved the final manuscript. Competing interests No competing interests.

13 Ghosh et al. Journal of Statistical Distriutions and Applications :7 Page 13 of 13 Consent for pulication Not applicale. Ethics approval and consent to participate Not applicale. Pulisher s Note Springer Nature remains neutral with regard to jurisdictional claims in pulished maps and institutional affiliations. Author details 1 Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur , India. 2 CEMSE Division, King Adullah University of Science and Technology, Thuwal , Saudi Araia. Received: 13 Feruary 2017 Accepted: 31 May 2017 References Alzaatreh, A, Lee, C, Famoye, F: On the discrete analogues of continuous distriutions. Stat. Methodol. 9, Azzalini, A: With the collaoration of A. Capitanio. The Skew-Normal and Related Families. Camridge University Press, UK 2014 Azzalini, A, Regoli, G: Modulation of symmetry for discrete variales and some extensions. Stat 3, Bairamov, I, Gultekin, OE: Discrete distriutions connected with the ivariate Binomial. Hacettepe J. Math. Stat. 39, Chakraorty, S, Chakravarty, D: A new discrete proaility distriution with integer support on,. Commun. Stat. Theory Methods. 45, Conway, DA: Multivariate distriutions with specified marginals Availale at default/files/olk%20nsf%20145.pdf. Accessed 10 June 2017 Cuadras, CM: Proaility distriutions with given multivariate marginals and given dependence structure. J. Multivar. Anal. 42, Dahiya, R, Korwar, R: On characterizing some ivariate discrete distriutions y linear regression. Sankhyā: Indian J. Stat. Series A. 39, Dutta, S, Genton, MG: A non-gaussian multivariate distriution with all lower-dimensional Gaussians and related families. J. Multivar. Anal. 132, Fréchet, M: Sur les taleaux de corrélation dont les marges sont données. Ann. Univ. de Lyon Sect. A, Series 3. 14, Hoeffding, W: Massatainvariate korrelations-theorie. Scriften Math. Inst. Univ. Berlin. 5, Johnson, NL, Kotz, S, Balakrishnan, N: Discrete Multivariate Distriutions. Wiley, New York 1997 Nguyen, TT, Gupta, AK, Wang, Y: A characterization of certain discrete exponential families. Ann. Inst. Stat. Math. 48, Roy, D: The discrete normal distriution. Commun. Stat. Theory Methods. 32, Ruiz, JM, Navarro, J: Characterization of discrete distriutions using expected values. Stat Papers. 36,