A. A. ZINGER

ON THE DISTRIBUTIONS OF POLYNOMIAL STATISTICS IN SAMPLES FROM A NORMAL POPULATION

(Presented by Academician V. I. Smirnov on 10 X 1962)

Let \(x_1, x_2, \ldots, x_k\) be a repeated homogeneous sample of size \(k \geqslant 1\) from a normal population with mean \(0\) and variance \(1\). A known property (see (1)) of the distribution of a quadratic form in the variables \(x_1, x_2, \ldots, x_k\) is that the characteristic function \(\varphi(t)\) of this distribution has the form

\[ \varphi(t)=\prod_{\nu=1}^{k}(1-2i\lambda_\nu t)^{-1/2}, \tag{1} \]

where \(\lambda_1,\lambda_2,\ldots,\lambda_k\) are the characteristic roots of the quadratic form. From (1) it follows directly that \(\varphi(t)\) satisfies a linear homogeneous differential equation with polynomial coefficients of the form

\[ P(t)\varphi' + {}^{1}/_{2} P'(t)\varphi = 0, \tag{2} \]

where \(P(t)=\prod_{\nu=1}^{k}(1-2i\lambda_\nu t)\). This property of the distributions of quadratic forms in normal variables can be generalized to the case of polynomial statistics with real coefficients of higher degrees.

We shall call a form in the variables \(x_1, x_2, \ldots, x_k\) suitable if its partial derivatives with respect to \(x_1, x_2, \ldots, x_k\) have no common root, except perhaps the trivial one \((0,0,\ldots,0)\). Examples of suitable forms are any definite quadratic form, \(x_1^n+x_2^n+\cdots+x_k^n\), and the like. We shall call a polynomial statistic suitable if the form of highest degree contained in it is suitable.

Let \(X=(X_1,X_2,\ldots,X_k)\) be a random vector having a nondegenerate normal distribution, and let \(P(X)=P(X_1,X_2,\ldots,X_k)\) be a polynomial statistic with real coefficients. For an arbitrary polynomial statistic \(F(X)\), define the function of \(\tau\)

\[ \varphi(\tau,F)=E\,[F(X)\exp(\tau P(X))], \qquad \tau=it\;(-\infty<t<+\infty). \tag{3} \]

For arbitrary positive integers \(N<M\), we shall consider the set \(\Phi=\{\varphi_{j_1,j_2,\ldots,j_k}(\tau)\}\), where

\[ \varphi_{j_1,j_2,\ldots,j_k}(\tau) = E\,[X_1^{j_1}X_2^{j_2}\cdots X_k^{j_k}\exp(\tau P(X))], \qquad \tau=it\;(-\infty<t<+\infty), \tag{4} \]

and \((j_1,j_2,\ldots,j_k)\) runs through all possible sets of nonnegative integers satisfying the relation

\[ N \leqslant j_1+j_2+\cdots+j_k \leqslant M. \]

Denote by \(\mathscr{L}_{NM}(\tau)\) the vector whose components are the elements of the set \(\Phi\), arranged in some fixed order.

Theorem 1. If \(P(X)\) is an appropriate polynomial statistic, then for any polynomial statistic \(F(X)\) the function \(\varphi(\tau,F)\) is a solution of some linear homogeneous differential equation with polynomial coefficients.

Theorem 2. If \(P(X)\) is an appropriate form of degree \(n+1>3\), \(EX=0\), then one can specify such a positive integer \(N_0\) that the vector \(\mathcal{L}_{N_0,N_0+n}(\tau)\) satisfies the differential equation

\[ \frac{d\mathcal{L}}{d\tau}=\Omega(\tau)\mathcal{L}, \tag{5} \]

\[ \tau=it\;(-\infty<t<+\infty), \]

where \(\Omega=A/\tau+B/\tau^2\). Here \(A,B\) are constant square matrices of the corresponding order, and the elements of the matrix \(A\) depend only on the degree of the form \(P\), but do not depend on its coefficients.

Remark. For \(n+1=3\) the theorem remains valid, only in this case \(\Omega\) has the form

\[ \Omega=\frac{A}{\tau}+\frac{B}{\tau^2}+\frac{C}{\tau^3}. \]

From Theorem 1 there follows directly the generalization mentioned at the beginning. Namely:

Theorem 3. The characteristic function of an appropriate polynomial statistic satisfies a linear homogeneous differential equation with polynomial coefficients.

Application of the Fourier transform makes it possible to obtain a similar equation also for the distribution function of an appropriate polynomial statistic. This equation can be used to study certain probabilistic properties of these statistics.

Received
29 IX 1962

CITED LITERATURE

¹ H. Cramér, Mathematical Methods of Statistics, IL, 1948.