Gaussian representation of a class of Riesz probability distributions

Transcription

1 arxiv:763v [mathpr] 8 Dec 7 Gaussian representation of a class of Riesz probability distributions A Hassairi Sfax University Tunisia Running title: Gaussian representation of a Riesz distribution Abstract : The Wishart probability distribution on symmetric matrices has been initially defined by mean of the multivariate Gaussian distribution as an of the chi-square distribution A more general definition is given using results for harmonic analysis Recently a probability distribution on symmetric matrices called the Riesz distribution has been defined by its Laplace transform as a generalization of the Wishart distribution The aim of the present paper is to show that some Riesz probability distributions which are not necessarily Wishart may also be presented by mean of the Gaussian distribution using Gaussian samples with missing data Keywords: Chi-square distribution, Gaussian distribution, Wishart probability distribution, Riesz probability distribution, Laplace transform AMS Classification : 6B, 6B5, 6B Introduction Let V be the space of real symmetric r r matrices, Ω be the cone of positive definite and Ω the cone of positive semi-definite elements of V We denote the identity matrix by e, the trace of an element x of V by trx and its determinant by x We equip V by the scalar product θ,x = trθx Denoting L µ θ = V e θ,x µdx the Laplace transform of a measure µ, it is well known see Faraut and Korányi 994 that there exists a positive measure µ p on Ω such that the Laplace transform of µ p exists for θ Ω and is equal to exptrθxµ p dx = p θ, Ω Corresponding author address: abdelhamidhassairi@fssrnutn

2 if and only if p is in the so called Gindikin set Λ = { } r r,,,,+ When p > r, the measure µ p is absolutely continuous with respect to the Lebesgue measure and is given by where µ p dx = Γ Ω p p r+ x Ω xdx, Γ Ω p = π n r r k= Γp k, where n is the dimension of V And when p = j with j integer, j r, the measure µ p is singular concentrated on the set of elements of rank j of Ω For p Λ and σ Ω, implies that the measure W p,σ on Ω defined by W p,σ dx = p σexp trxσ µ p dx 3 is a probability distribution It is called the Wishart distribution with shape parameter p and scale parameter σ Its Laplace transform exists for θ σ Ω and is equal to Ω exptrθxw p,σ dx = p e σθ 4 When p > r, the Wishart distribution W p,σ is given by W p,σ dx = p σ Γ Ω p exp trxσ p r+ xω xdx 5 Further details on the Wishart distribution on Ω may be obtained from Casalis and Letac 996 or from Letac and Massam 998 It is however important to mention that the Wishart probability distribution has initially been defined as a multivariate extension of the chi-square distribution In fact, taking U,,U p random vectors in IR r with distribution N,σ, the distribution of the matrix X = p i= Ut i U i is a Wishart matrix This kind of Wishart probability distributions may be defined using the Gaussian random matrices Let u = t u,,u r be a random vector with distribution N r,σ Suppose that we have a random sample of u ie, a sequence of independent and identically distributed random variables with distribution equal to the distribution of u, and consider the r s Gaussian matrix U = u, u,s u r, u r,s

3 The distribution of U is N r,s,σ equal to where u,σ u = trσ u t u We have π rs σ s e u,σ u, Proposition If U N r,s,σ, then the matrix X = U t U has the Wishart probability distribution W r s, σ Proof The Laplace transform of X estimated in θ σ Ω is given by Ee θ,x u = π rs σ s IR rs e θ,ut u,σ u du = π rs σ s e u,σ θu du rs IR σ θ = s s σ = s σ θ s σ It is then the Laplace transform of a W r s, σ distribution This distribution is absolutely continuous with respect to the Lebesgue measure if and only if s > r, that is if and only if the size s of the sample is greater or equal to the dimension r of the space of observations If we consider p independent random matrices U,,U p with distribution N r,s,σ, then the distribution of the matrix X = p i= U i t U i W r ps, σ Of course the class of Wishart distributions defined by mean of the Gaussian vectors or the Gaussian matrices doesn t cover all the Wishart distributions, however the fact that this random matrix is expressed in terms of Gaussian random vectors allows to establish many important properties of this subclass of the class of Wishart probability distribution Using a theorem due to Gindikin which relayes on the notion of generalized power, Hassairi and Lajmi [4] have introduced an important generalization of the Wishart distribution that they have called Riesz distribution This distribution in its general form is defined by its Laplace transform We will show that some among the Riesz distributions may be represented by the Gaussian distribution using samples of Gaussian vectors with missing data Riesz distributions For i,j r, we define the matrix µ ij = γ kl k,l r such that γ ij = and the other entries are equal to We also set c i = µ ii for i r, and e ij = µ ij +µ ji, for i < j r 3

4 With these notations, an element x of V may be written r x = i c i i=x + x ij e ij i<j In particular, for k r, we set e k = c + +c k Now consider the map P k from V into V defined by r x = i c i i=x + x ij e ij P k x = i<j k x i c i + x ij e ij i= i<j k Then the determinant k P k x of the k k bloc P k x is denoted by k x, it is the principal minor of x of order k The generalized power of an element x of Ω is then defined for s = s,,s r IR r, by s x = x s s x s s3 r x sr Note that s x = p x if s = p,,p with p IR It is also easy to see that s+s x = s x s x In particular, if m IR and s+m = s +m,,s r +m, we have s+m x = s x m x Now, we introduce the set Ξ of elements s of IR r defined as follows: For a real u, we put εu = if u = and εu = if u > For u,u,,u r, we define s = u and s k = u k + εu + + εu k, for k r We have the following result due to Gindikin [3], it is also proved in the monograph by Faraut and Korányi [] Theorem There exists a positive measure R s on V such that the Laplace transform L Rs is defined on Ω and is equal to s θ if and only if s is in Ξ The measures R s, defined in the previous theorem in terms of their Laplace transforms are divided into two classes according to the position of s in Ξ In the first class, the measures are absolutely continuous with respect to the Lebesgue measure on Ω More precisely, we have see [4] that when s = s,,s r in Ξ is such that for all i, s i > i, where Γ Ω s = π n r r R s dx = Γ Ω s s r+ x Ω xdx Γs j j j= The second class corresponds to s in Ξ \ r i= ] i,+ [ In this case the Riesz measure R s is concentrated on the boundary Ω of Ω, it has a complicated form which is explicitly described in [5] For s = s,,s r in Ξ and σ in Ω, we define the Riesz distribution R r s,σ by R r s,σdx = e σ,x s σ R sdx 4

5 The Laplace transform of the Riesz distribution is defined for θ in σ Ω by L Rrs,σθ = sσ θ s σ 6 If s = p,,p such that p Λ = {, r,, } ]r,+ [, then R rs,σ is nothing but the Wishart distribution with shape parameter p and scale parameter σ The definition of the absolutely continuous Riesz distribution on the cone of positive definite symmetric matrices Ω relies on the notion of generalized power It is defined for σ in Ω and s = s,,s r IR r such that s i > i by 3 Main result Rs, σdx = Γ Ω s s σ e σ,x s r+x Ω xdx We now show that some of the Riesz distributions have a representation in terms of the Gaussian distribution generalizing what accours in the case of the Wishart distribution For simplicity, we take σ equal to the identity matrix Suppose that we have s r independent observations of the vector u,,u r with Gaussian distribution N r,i r, and that some data in the observations are missing The sample is then partitioned according to the number of available components in the observation More precisely, we suppose that there exist integers < s s s r such that we have s observationsonu,,u r whereallthecomponentsoftheobservationareavailable, s s observations on u,,u r, the first component of the observation is missing, s i+ s i observations on u i+,,u r, the i first components of the observation are missing, i r The number s i is in particular equal to the number of times the component u i appears in the data Define p = sup{i ;s i = s }, p = sup{i > p ;s i = s p + }, p l+ = sup{i > p l ;s i = s pl +}, p k = inf{i;s i = s r }, so that we have s = = s p, s p s p +, s p + = = s p, s p s p + s pl + = = s pl+, s pk s r 5

6 s pk + = = s r We have that p k+ = r Replacing the missing components by the theoretical mean that is by zero, we obtain the matrix U = u, u,sp u p, u p,s p u p +, u p +,s p u p +,s p u p, u p,s p u p,s p u pk +, u pk +,s r u r, u r,sr This matrix U consists of blocks defined recursively in the following way : 37 with U l+ = U l U l+ = u i,j pl + i p l+, j s pl U l+ = u i,j pl + i p l+, s pl + j s pl+ We have that U k+ = U U = u i,j i p j s p l+ U l+ U = U l+ U l+, l k 38 Theorem 3 The distribution of X = U t U is Rs, Ir with s = s,, sr Proof We use the following Peirse decomposition of an r r matrix ω For l k, we set ω l = ω i,j i pl j p l, and we write ω l+ as ω l+ = ω l t ω l+ ω l+ ω l+ From Proposition, we have that U t U has the W p sp other hand, since s = = s p, we have that sp W p, I p s = R p,, s p 6 39, Ip, I p distribution On the

7 It suffises to show that if U l t U l is R s pl,, sp l, Ip l, then U l+ t U l+ is s R pl+,, sp l+, Ip l+ Again we use the Laplace transform For θ Ir Ω and with the decomposition 39, we have L U l+ t U l+θl+ = E exp θ l+,u l+ t U l+ = E exp tr θ l+ U l+ t U l+ E exp tr θ l U l t U l + t θ l+ U l+ t U l + U l t U l+ θ l+ +θ l+ U l+ t U l+ We have that U l+ is a p l+ p l s pl+ s pl Gaussian matrix with distribution N,I pl+ p l According to Proposition, the matrix U l+ W r sp l+ sp l, I p l+ p l Hence E exp tr θ l+ U l+ sp l+ sp l t U l+ = Concerning the second expectation, we have that tr t U l+ has the Wishart distribution sp l+ sp l I p l+ p l t θ l+ U l+ t U l +U l t U l+ θ l+ +θ l+ U l+ = U l+, θ l+ U l + U l+, θ l+ U l+ I pl+ p l t U l+ As U l+ is a p l+ p l s pl matrix with distribution N,I pl+ p l, we obtain that E exp sp l = exp U l+, θ l+ U l + U l+, θ l+ I θl+ pl+ pl θ l+ U l, I pl+ p l U l+ U l θ l+ U l Now multiplying this by exp tr θ l U l t U l, weneedthentocalculate theexpectation E exp θ l + t θ l+ Ipl+ p l θ l+,u l t U l As from the hypothesis, U l t U l is R s pl,, sp l, Ip l, using 6 this is equal to s,, sp l I pl θ l + t θ l+ Ipl+ p l s,, sp l I pl 7 θ l+

8 Finally, we obtain that is L U l+ t U l+θl+ = Ip sp l l+ p l sp l s,, sp l I pl+ p l L U l+ t U l+θl+ = s,, sp l But s pl + = = s pl+, then sp l+ sp l+ sp l Ip sp l+ sp l l+ p l I pl+ p l I pl θl t θ l+ Ipl+ p l s,, sp l I pl Ip sp l+ l+ p l sp l+ I pl+ p l I pl θl t θ l+ Ipl+ p l s,, sp l I pl θ l+ θ l+ I pl+ p l = s pl +,, sp I l+ pl+ p l,, and sp l+ Ipl+ p l = s pl +,, sp l+ Ipl+ p l As we have s,, sp l I pl s pl +,, sp I l+ pl+ p l = s,, sp I l+ pl+, it follows that L U l+ t U l+θl+ = Ipl+ s,, sp l+ s,, sp I l+ pl+ Therefore U l+ t U l+ is a Riesz random matrix with distribution s R pl+,, s p l+, I p l+ 8

9 Given that X = U k+ t U k+, the result follows References [] Casalis, M and Letac, G, The Lukacs-Olkin-Rubin characterization of the Whishart distributions on the symmetric cones Ann Statist ,996 [] Faraut, J; Korányi, A, Analysis on Symmetric Cones Oxford Univ Press 994 [3] Gindikin, SG, Analysis on homogenous domains Russian Math, Surveys 9, -89, 964 [4] Hassairi, A, Lajmi, S, Riesz exponential families on symmetric cones J Theoret Probab 4, [5] Hassairi, A, Lajmi, S, Singular Riesz measures on symmetric cones Adv Oper Theory 3, no, -5, 8 [6] Letac, G and Massam, H, Quadratic and inverse regressions for Wishart distributions Ann Statist 6, , 998 9