Asymptotic properties of robust estimators of scale parameters

This paper deals with asymptotic robust properties of some estimators of scale parameter by the e -contamination of the model distributions: F = Ф_{е x} (x) = (1 - е)Ф(x) + еФ(x /x) , e is a known proportion of contamination (0 1,...,X_n is a random sample with distribution function F(x) and F has a density f (x) , x е R. Let T(F) , F еЗ, is a generic scale functional and T_n(X₁,...,X_n) = T(F_n) 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 T_n (X₁,..., X_n) = med {| X_i - X_j |, 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 T_n - 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) = + F(x - T) - 2F(x + ) , 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, T_n) has asymptotically standard normal distribution, where the asymptotic variance of JnT is given by the following formula: ₂ j [1 + 2 F (x - T) - 2 F (x + T )] dF (x) a(F, T„) = j IF(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 T_n - estimator with respect to S₁ having the classical standard deviation is equal: ARE_(I) (T_n : S₁) = 0.86 and ARE_(I) (T_n : S₂) = 0.98 , where S₂ has the average absolute deviation.

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