Асимптотические свойства робастных оценок масштабного параметра

Изучаются свойства робастных оценок масштабного параметра. Показано, что оценка медианы абсолютных разностей имеет асимтотически нормальное распределение, является B-робастной и имеет ограниченную функцию влияния. Приводятся результаты сравнения оценок масштабного параметра в рамках гауссовской модели с засорением.
This paper deals with asymptotic robust properties of some estimators of scale parameter by the e -contamination of the model distributions: F = Фе x (x) = (1 е)Ф(x) + еФ(x /x), e is a known proportion of contamination (0 1,...,Xn is a random sample with distribution function F(x) and F has a density f (x), x е R1. Let T(F), F еЗ, is a generic scale functional and Tn(X1,...,Xn) = T(Fn) is its sample estimator. We consider the functional T(F) defined by J [ F (x + T (F)) F (x T (F)) F (x)]dF (x) = 0 and the location invariance and scale equivariance sample estimators of the functional T (F), F еЗ. The sample estimator of this functional T (F) is given by Tn (X1,..., Xn) = med {| Xi Xj |, 1 < i, j < n }. This estimator is also named as the median of the absolute differences. The purpose of this article is to study asymptotic robust properties Tn estimators for different models distributions. The formal calculation of the Influence Function IF(x; F, T) is given by IF(x; F, T) = dJ(F; Д F) = 1 + 2F(x T) 2F(x + T), x е R1. 2J [ f (x + T) + f (x T)] dF (x) Note that Influence Function IF(x;F,T) is bounded and looks like as the U-shaped curve. If j[f (x + T) + f (x T)]dF(x) > 0, then the random variable *Jn{T T(F)}/ ct(F, Tn) has asymptotically standard normal distribution, where the asymptotic variance of JnT is given by the following formula: да2 j [1 + 2 F (x T) 2 F (x + T )]2 dF (x) a2(F, T„) = j IF2(x; F, T)dF(x) = 1---. -да 4 (j [ f (x + T) + f (x T)] dF (x)) The paper contains numerical comparisons for some estimators of scale parameters by e contamination of the model distribution for different values of e and x. It is shown that for normal distribution asymptotic relative efficiency Tn estimator with respect to S1 having the classical standard deviation is equal: ARE(I) (Tn : S1) = 0.86 and ARE(I) (Tn : S2) = 0.98, where S2 has the average absolute deviation.