\(G_ 2\)-estimators of the spectral functions of covariance matrices. (English. Russian original) Zbl 0634.62052

Girko, V. L.

Theory Probab. Math. Stat. 35, 27-30 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 28-31 (1986).

The following G-estimator G is introduced for the resolvents of covariance matrices \(m_ n^{-1}Tr(I+tR_{m_ n})^{-1}\), \(t\geq 0:\) \((t,\tilde R_{m_ n})={\tilde \phi}({\tilde \theta}_ n(t))\), where \[ {\tilde\phi}(t)=m_ n^{-1}Tr(I+t\tilde R_{m_ n})^{-1}, \]
\[ \tilde R_{m_ n}=(n-1)^{-1}\sum^{n}_{k-1}(x_ k-\tilde x)(x_ k-\tilde x)',\quad \tilde x=n^{-1}\sum^{n}_{k-1}x_ k, \] the \(x_ k\) are observations from an \(m_ n\)-dimensional random vector \(\xi\) distributed according to a normal distribution \(N(a,R_{m_ n})\), \({\tilde \theta}_ n\) is the solution of the equation \[ \theta (1-m_ n(n-1)^{-1}+m_ n(n-1)^{-1}{\tilde \phi}(\theta))=t, \]
\[ \overline{\lim}_{n\to \infty}m_ n^{-1}n^{-1}<1,\quad 0<c_ 1\leq \lambda_ i\leq c_ 2<\infty, \] and the \(\lambda_ i\) are the eigenvalues of the matrix \(R_{m_ n}\). Under these conditions it is shown that the distributions of the random variables \[ [G(t,\tilde R_{m_ n})-\phi (t,R_{m_ n})]\sqrt{(n-1)m_ n}an(t)+c_ n(t) \] are asymptotically normal \([a_ n(t)\) and \(c_ n(t)\) are bounded].

MSC:

62H12	Estimation in multivariate analysis
62E20	Asymptotic distribution theory in statistics
62F12	Asymptotic properties of parametric estimators

Keywords:

asymptotic normality; G-estimator; resolvents of covariance matrices