On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors

Let X:p × n be a matrix of random real variates such that the column vectors of X are independently and identically distributed as multivariate normals with zero mean vectors. Then a positive definite quadratic function in normal vectors is defined as $XLX$' where L is a symmetric positive definite (p.d.) matrix with real elements. In the analysis of variance, such functions appear. In the previous study, Khatri [14], [16], has established the necessary and sufficient conditions for the independence and the Wishartness of such functions. In this paper, we study the distribution of a positive definite quadratic function and the distribution of Y' (XLX')⁻¹Y where Y:p × m is independently distributed of X and its columns are independently and identically distributed as multivariate normals with zero mean vectors. Moreover, we study the distribution of the characteristic (ch.) roots of (YY')(XLX')⁻¹ and the similar related problems. When $p = 1$, the distribution of a p.d. quadratic function in normal variates (central or noncentral) has been studied by a number of people (see references). In the study of the above and related topics in multivariate distribution theory, we are using zonal polynomials. A. T. James [10], [11], [12], [13], and Constantine [1], [2], have used them successfully and have given the final results in a very compact form, using hypergeometric functions ₚFq(S) in matrix arguments. These functions are defined by {equation*}{1}_pF_q(a_1, ⋯, a_p; b_1, ⋯, b_q; Z) {equation*} = ∑^∞k = 0 ∑_κ (a₁)_κ ⋯ (aₚ)_κ/(b₁)_κ ⋯ (bq)_κ C_κ(Z)/k! where C_κ(Z) is a symmetric homogeneous polynomial of degree k in the latent roots of Z, called zonal polynomials (for more detail study of zonal polynomials, see the references of A. T. James and Constantine), κ = (k₁, ⋯, kₚ), k₁ k₂ ⋯ kₚ 0, k₁ + k₂ + ⋯ + kₚ = k; a₁, ⋯, aₚ, b₁, ⋯, bq are real or complex constants, none of the bⱼ is an integer or half integer 1/2(m - 1) (otherwise some of the denominators in (1) will vanish), {equation*}{2}(a)_κ = ∏^mj = 1 (a - 1/2(j - 1))ₖⱼ = Γ_m(a,κ)/Γ_m(a), {equation*} (x)_n = x(x + 1) ⋯ (x + n - 1), (x)_0 = 1and equation*3Γₘ(a) = π1/4m(m - 1) ∏ᵐj = 1 Γ(a - 1/2(j - 1)) equation* andΓ_m(a, κ) = π1/4m(m - 1) ∏^mj = 1 Γ(a + k_j - 1/2(j - 1)).$$ In (1), $Z$ is a complex symmetric $m × m$ matrix, and it is assumed that $p q + 1$, otherwise the series may converge for $Z = 0$. For $p = q + 1$, the series converge for $\|Z\| < 1$, where $\|Z\|$ denote the maximum of the absolute value of ch. roots of $Z$. For $p q$, the series converge for all $Z. Similarly we define equation*2bₚF⁽ᵐ⁾q (a₁, a₂, ⋯, aₚ; b₁, ⋯, bq; S, R)equation* = ∑^∞k = 0 ∑_κ (a_1)_κ ⋯ (a_p)_κ/(b_1)_κ ⋯ (b_q)_κ C_κ(S)C_κ(R)/C_κ(I_m)k!.$ The Section 2 gives some results on integration with the help of zonal polynomials, the Section 3 derives the distributions based on p.d. quadratic functions, the Section 4 gives the moments of certain statistics arising in the study of multivariate distributions, and the Section 5 gives the results for complex multivariate Gaussian variates.