Canonical Equation K ₁₉

Abstract

The main application of the theory of random matrices is the statistical analysis of some functions of observations of random vector, where the dimension of variables m is large and comparable with the sample size n. In this chapter we continue to analyze the normalized spectral functions of empirical covariance matrices in general case when it is difficult to deduce canonical equations. To demonstrate the main ideas and for simplification of calculations we deduce at first equation for the function of empirical expectation. Suppose that f(x) is a Borel function in ${R^{{m_n}}}$ having partial derivatives of the third order. Let ${\vec x_1}, \ldots ,\;{\vec x_n}$ be independent observations of an m _n - dimensional vector $\vec \xi ,\;E\vec \xi = \vec a.$ We need a consistent estimator of the value $f(\vec a).$ Many problems of multivariate statistical analysis can be formulated in these terms. If f is a continuous function we take

$$ \mathop a\limits^{\hat \to } = \;{n^{ - 1}}\sum\limits_{i = 1}^n {{{\vec x}_i}} $$

as the estimator of $\vec a.$ Then, obviously, for fixed m, $p\;{\lim _{n \to \infty }}f(\mathop a\limits^{\hat \to } ) = \;f(\mathop a\limits^{\hat \to } ).$ But the application of this method in solving practical problems is unsatisfactory due to the fact that the number of observations n necessary to solve the problem with a given accuracy increases sharply with m. It is possible to reduce significantly the number of observations n by making use of the fact that under some conditions, including lim _n→∞ mn ⁻¹ = c, 0 < c < ∞, the G-assertion

$$ p\;\mathop {{\rm{lim}}}\limits_{n \to \infty } \;[f(\mathop a\limits^{\hat \to } ) = E\;f(\mathop a\limits^{\hat \to } )] = 0 $$

holds. We call G-assertion and similar identities the basic relations of the G-analysis of large dimensional observations.