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

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