Output distributional influence function

When selecting a filter for an application, it is often essential to know the behaviour of the filter in presence of contamination. This robustness of a filter is traditionally explored by means of influence function (IF) and change-of-variance function (CVF). However, as these are asymptotic measures there is uncertainty of the applicability of the obtained results to the finite length filters used in the real world applications. This paper disperses this uncertainty by presenting a new method, called output distributional influence function (ODIF), for examining the robustness of the finite length filters. The method gives extensive information about the robustness of any filter with known output distribution function. As examples the ODIFs for distribution function, density function, expectation, and variance are given for the mean and the median filters and interpreted in detail. 1. INFLUENCE FUNCTION Influence function (IF) is a useful heuristic tool of robust statistics introduced by Hampel [2, 3] under the name influence curve (IC) for studying the performance of filters under noisy conditions. Definition 1 . The IF of estimator T at underlying probability distribution F is given by IF(x;T; F ) = lim t!0+ T ((1 t)F + t x) T (F ) t for those x where this limit exists. In this definition x is the probability measure which puts mass 1 at the point x. The IF gives the effect that an infinitesimal contamination at point x has on the estimator T when divided by the mass of the contamination. So the IF gives asymptotic bias caused by the contamination and thus characterizes properties of the estimator as the number of observations approaches infinity. We denote by and the distribution and the density functions of the standard normal distribution. The influence functions for the mean and the median are shown in Figure 1 where the underlying distributionF = . For the mean the gross error sensitivity, i.e., the worst influence which a small amount of contamination of fixed size can have on the value of the estimator, equals infinity and for the median it is finite and equals p 2 1:253. So for the mean single outlier can carry the estimate over all bounds but for the median an outlier has a fixed influence. -4 -2 2 4 -2 -1.5 -1 -0.5 0.5 1 1.5 2 Figure 1: The IF of the mean (–) and the median (-) at F = . 2. CHANGE-OF-VARIANCE FUNCTION The IF gives only one aspect of robustness of an estimator, namely local robustness of the asymptotic value of the estimator. Another important aspect is the local robustness of asymptotic variance. The asymptotic variance of estimator T at F denoted by V (T; F ) is defined to be the variance of p N [T (FN ) T (F )] as N !1, where FN is the empirical distribution of sample (X1;X2; : : : ; XN ). Local robustness of the asymptotic variance can be characterized by the change-of-variance function (CVF) defined as follows, [4]. Definition 2. The CVF of estimator T at F is defined as CVF(x;T; F ) = lim t!0+ V (T; (1 t)F + t x) V (T;F ) t for those x where this limit exists. If F = , the CVF of the mean is x 1 which is displayed in Figure 2. In the same figure is also shown the CVF of the median at F = . It has a constant value p 2 1:253 elsewhere but at zero, where the graph has a negative delta function. If the CVF is negative, the asymptotic variance of the estimator has decreased, and if positive, the asymptotic variance has increased. So for the mean the asymptotic variance decreases if the contamination is in the interval ( 1; 1). The further the contamination is from this interval the more the variance is increased and a single outlier can carry the asymptotic variance over all bounds. For the median contamination at the origin reduces the asymptotic variance significantly and the contamination anywhere else causes only a constant increase to the variance of the median. So the median is robust also in this sense.