Two estimates of general statistical analysis. (Russian) Zbl 0645.62047

Girko, V. L.

Teor. Veroyatn. Mat. Stat., Kiev 37, 26-36 (1987).

Let \(X_ 1,...,X_ n\) be independent and identically distributed random vectors of dimension \(m_ n\). It is supposed that the vector components are independent and \(m_ n\) tends to infinity at some rate as n does. The paper is concerned with an asymptotic study of so-called G-estimators of two quantities which are functions of \(R_{m_ n}\), the covariance matrix of \(X_ i\). In the first case the quantity estimated is \(c_ n^{-1}\ell n(\det R_{m_ n})\), while in the second one it is the Stieltjes transform \[ \phi (z,R_{m_ n})=\int^{\infty}_{0}(z- x)^{-1}d\mu_{m_ n} \] of a normalized spectral function \(\mu_{m_ n}=m_ n^{-1}\sum^{m_ n}_{k=1}\chi (\lambda_ k<\infty)\) where \(\lambda_ k\) are eigenvalues of \(R_{m_ n}\). In the both cases the author shows the asymptotic normality of the estimators.

Reviewer: T.Bednarski

MSC:

62G05	Nonparametric estimation
62E20	Asymptotic distribution theory in statistics

Keywords:

generalized dispersion; G-estimators; covariance matrix; Stieltjes transform; spectral function; asymptotic normality