The Annals of Mathematical Statistics

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

C. G. Khatri

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Abstract

Let $X:p \times n$ 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 $Y' (XLX')^{-1}Y$ where $Y:p \times m$ 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 $_pF_q(S)$ in matrix arguments. These functions are defined by \begin{equation*}\tag{1}_pF_q(a_1, \cdots, a_p; b_1, \cdots, b_q; Z) \end{equation*} $= \sum^\infty_{k = 0} \sum_\kappa \lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(Z)/k!\rbrack$ where $C_\kappa(Z)$ 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), $\kappa = (k_1, \cdots, k_p), k_1 \geqq k_2 \geqq \cdots \geqq k_p \geqq 0, k_1 + k_2 + \cdots + k_p = k; a_1, \cdots, a_p, b_1, \cdots, b_q$ are real or complex constants, none of the $b_j$ is an integer or half integer $\leqq \frac{1}{2}(m - 1)$ (otherwise some of the denominators in (1) will vanish), \begin{equation*}\tag{2}(a)_\kappa = \prod^m_{j = 1} (a - \frac{1}{2}(j - 1))_{kj} = \Gamma_m(a,\kappa)/\Gamma_m(a), \end{equation*} (x)_n = x(x + 1) \cdots (x + n - 1), (x)_0 = 1$ and \begin{equation*}\tag{3}\Gamma_m(a) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a - \frac{1}{2}(j - 1)) \end{equation*} and $\Gamma_m(a, \kappa) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a + k_j - \frac{1}{2}(j - 1)).$$ In (1), $Z$ is a complex symmetric $m \times m$ matrix, and it is assumed that $p \leqq q + 1$, 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 $p \leqq q$, the series converge for all $Z$. Similarly we define \begin{equation*}\tag{2b}_pF^{(m)}_q (a_1, a_2, \cdots, a_p; b_1, \cdots, b_q; S, R)\end{equation*} $ = \sum^\infty_{k = 0} \sum_\kappa\lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(S)C_\kappa(R)/C_\kappa(I_m)k!\rbrack.$ 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.

Article information

Source
Ann. Math. Statist., Volume 37, Number 2 (1966), 468-479.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699530

Digital Object Identifier
doi:10.1214/aoms/1177699530

Mathematical Reviews number (MathSciNet)
MR190965

Zentralblatt MATH identifier
0138.14502

JSTOR
links.jstor.org

Citation

Khatri, C. G. On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors. Ann. Math. Statist. 37 (1966), no. 2, 468--479. doi:10.1214/aoms/1177699530. https://projecteuclid.org/euclid.aoms/1177699530


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