Matrix Variate Pareto Distribution of the Second Kind

We generalize the univariate Pareto distribution of the second kind to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of the Pareto matrix. Some of these results are expressed in terms of special functions of matrix arguments, zonal, and invariant polynomials.


Introduction
The Lomax distribution, also called the Pareto distribution of the second kind is given by the p.d.f.
where shape parameter β > 0 and location parameter λ > 0. The Lomax distribution, named after Lomax, is a heavy-tail probability distribution often used in business, economics, and actuarial modeling. The standard Pareto Distribution of the second kind has λ 1 with the p.d.f.

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Although a wealth of results on Pareto distribution is available in the literature see Johnson et al. 1 nothing appears to have been done to define and study matrix variate Pareto distribution. Therefore, in this paper, we define matrix variate Pareto distribution and study several of its properties.
We will use the following standard notations cf. Gupta and Nagar 2 . Let A a ij be an m × m matrix. Then, A T denotes the transpose of A; tr A a 11 · · · a mm ; etr A exp tr A ; det A determinant of A; A norm of A; A > 0 means that A is symmetric positive definite and A 1/2 denotes the unique symmetric positive definite square root of A > 0. The submatrices A α and A α , 1 ≤ α ≤ m, of the matrix A are defined as A α a ij , 1 ≤ i, j ≤ α, and A α a ij , α ≤ i, j ≤ m, respectively. The multivariate gamma function which is frequently used in multivariate statistical analysis is defined by , Re a > m − 1 2 .

1.3
The multivariate generalization of the beta function is given by The beta type 1 and beta type 2 families of distributions are defined by the density functions Johnson et al. 1 : respectively, where α > 0, β > 0, and Recently, Cardeño et al. 3 have defined and studied the family of beta type 3 distributions. A random variable w is said to follow a beta type 3 distribution if its density function is given by If a random variable u has the p.d.f. 1.6 , then we will write u ∼ B1 α, β , and if the p.d.f. of the random variable v is given by 1.7 , then v ∼ B2 α, β . The distribution given by the density 1.9 will be designated by w ∼ B3 α, β . The matrix variate generalizations of 1.6 , 1.7 , and 1.9 are defined as follows Gupta and Nagar 2, 4, 5 .

The Density Function
First we define the matrix variate Pareto distribution of the second kind.

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where Λ is an m × m symmetric positive definite matrix and β > m − 1 /2.
For m 1, the matrix variate Pareto distribution and matrix variate Lomax distribution reduce to their respective univariate forms.
The matrix variate Pareto distribution can be derived by using independent gamma matrices. A random matrix Y is said to have a matrix variate gamma distribution with parameters Ψ > 0 and κ > m − 1 /2 , denoted by Y ∼ Ga m, κ, Ψ , if its p.d.f. is given by Theorem 2.3. Let Y 1 and Y 2 be independent, Y 1 ∼ Ga m, m 1 /2, I m and Y 2 ∼ Ga m, β, I m .
Proof. The joint density function of Y 1 and Y 2 is given by density of Y 1 and Y 2 , we obtain the joint density of W and Y 2 as Now, the desired result is obtained by integrating Y 2 using 1.3 .

5
The cumulative distribution function of V is obtained as where the last line has been obtained by substituting W the above expression is rewritten as

2.8
Finally, using the integral representation of the Gauss hypergeometric function Herz 6 , Constantine 7 , James 8 , and Gupta and Nagar 2 , namely, where Re a > m − 1 /2, Re c − a > m − 1 /2, and X < I m , we obtain

2.10
The moment generating function of V is derived as

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where Z m × m 1 δ ij z ij /2 . Now, evaluating the above integral, we obtain where the confluent hypergeometric function Ψ, with m×m symmetric matrix X as argument, is defined by the integral valid for Re X > 0 and Re a > m − 1 /2.

Properties
In this section, we give several properties of the matrix variate Pareto distribution of the second kind defined in the previous section.

Theorem 3.1. Let V ∼ P m β and let A be an m × m constant nonsingular matrix. Then, the density of X AV A T is
Proof. From the partition of V , we have we get the joint density of V 11 , V 22·1 , and X as From the above factorization, it is clear that V 11 and V 22·1 are independent, V 11 ∼ B2 q, m 1 /2, β − m − q /2 and V 22·1 ∼ P m−q β . The second part is similar.

3.5
Proof. Write A M I q , 0 G, where M is a q × q nonsingular matrix and G is an m × m orthogonal matrix. Now,

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where Y Now, noting that MM T AA T and making the transformation S AA T 1/2 Z AA T 1/2 with the Jacobian J Z → S det AA T − m 1 /2 in the above density, we get the desired result. The proof of the second part is similar.
From Theorem 3.5, it is clear that if V ∼ P m β and a ∈ R m , a / 0, then a T a a T V −1 a −1 ∼ P β . Further, if y m × 1 is a random vector, independent of W, and P y / 0 1, then it follows that y T y y T V −1 y −1 ∼ P β and y T y −1 y T V y ∼ B2 m 1 /2, β − m − 1 /2 .
From the above results, it is straightforward to show that if c m × 1 is a nonzero constant vector or a random vector independent of V with P c / 0 1, then

3.8
The expectation of V , E V , can easily be obtained from the above result. For any fixed c ∈ R m , c / 0, where which has the same form as the density 3.12 with m replaced by m − 1. Repeating the argument given above on the density function of W 22 , we observe that w 2 22 ∼ B2 m/2, β − m − 2 /2 and is independent of w 33 , . . . , w mm . Continuing further with the same argument, we get the desired result. Corollary 3.7. If V ∼ P m β , then the distribution of det V is the same as the distribution of the product of m independent beta type 2 variables, that is,

3.23
From the above factorization, it is clear that W 11 , w mm , and y are all independent, w 2 mm ∼ B2 m 1 /2, β − m − 1 /2 and the density of W 11 is proportional to which has the same form as the density 3.19 with m replaced by m − 1. Repeating the argument given above on the density function of W 11 , we observe that w 2 m−1,m−1 ∼ B2 m/2, β − m − 2 /2 and is independent of w m−2,m−2 , . . . , w 11 . Continuing further with the same argument, we get the desired result.
We conclude this section by deriving moments of det V and det I m V −1 .

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Proof. By definition Now, evaluating the above integral using 1.5 , we get the result.

3.28
By writing multivariate gamma functions in terms of ordinary gamma functions, expressions E det V r and E det I m V −s can be simplified as Substituting r, s 1, 2, the first and second order moments of det V and det I m V −1 are calculated as 3.31 where the Pochhammer notation a k is defined by a k a a 1 · · · a k − 1 , k 1, 2, . . . with a 0 1.

Results Involving Zonal and Invariant Polynomials
Let C κ X be the zonal polynomial of an m × m symmetric matrix X corresponding to the partition κ k 1 , . . . , k m , k 1 · · · k m k, k 1 ≥ · · · ≥ k m ≥ 0. Then, for small values of k, explicit formulas for C κ X are available as James 8 tr X 3 6 tr X tr X 2 8 tr X 3 ,

4.1
From the above results, it is straightforward to show that

4.4
Further, det I m − X −a , in terms of zonal polynomials, can be expanded as where κ denotes summation over all ordered partitions κ of k.

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For properties and further results, the reader is referred to Constantine 7 and Gupta and Nagar 2 .

4.6
Davis 9, 10 introduced a class of polynomials C κ,λ φ X, Y of m × m symmetric matrix arguments X and Y , which are invariants under the transformation X, Y → HXH T , HY H T , H ∈ O m . For properties and applications of invariant polynomials, we refer to Davis 9, 10 , Chikuse 11 , and Nagar and Gupta 12 . Let κ, λ, φ, and ρ be ordered partitions of the nonnegative integers k, , f k , and r, respectively. Then where φ ∈ κ · λ denotes that irreducible representation of Gl m, R , the group of m × m real invertible matrices, indexed by 2φ, appears in the decomposition of the tensor product 2κ⊗2λ of the irreducible representation indexed by 2κ and 2λ. Further,

4.10
From the density of V , we have where the last line has been obtained by using 4.6 .
Using results 4.1 -4.2 on zonal polynomials, it is easy to see that

4.13
Further, using the invariance of the distribution of V and the above results, one obtains 4.14