High‐breakdown estimation of multivariate mean and covariance with missing observations

Abstract

We consider the problem of outliers in incomplete multivariate data when the aim is to estimate a measure of mean and covariance, as is the case, for example, in factor analysis. The ER algorithm of Little and Smith which combines the EM algorithm for missing data and a robust estimation step based on an M-estimator could be used in such a situation. However, the ER algorithm as originally proposed can fail to be robust in some cases, especially in high dimensions. We propose here two alternatives to avoid the problem. One is to combine a small modification of the ER algorithm with a so-called high-breakdown estimator as the starting point for the iterative procedure, and the other is to base the estimation step of the ER algorithm on a high-breakdown estimator. Among the high-breakdown estimators which are actually built to keep their robustness properties even if the number of variables is relatively large, we consider here the minimum covariance determinant estimator and the t-biweight S-estimator. Simulated and real data are used to compare and illustrate the different procedures.

Aykırı değer varlığında sağlam konum ve ölçek kestiricilerinin faktör analizinde kullanımı

“…Ayrıca aykırı değerlerden dolayı normallik varsayımının sağlanmaması nedeniyle faktör modelinde sağlam kovaryans matrisinin kullanılması gerekliliği önerilmiştir [12]. Çok değişkenli sürekli değişkenler için sağlam konum ve yayılım (dispersion) ölçülerini elde etmek için alternatif yolları inceleyen pek çok çalışma söz konusudur [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Ayrıca sağlam faktör analizi üzerine de çalışmalar mevcuttur [4], [7], [11] [27][28][29][30].…”

Fast Algorithms for Computing High Breakdown Covariance Matrices with Missing Data

“…The iteration step for the covariance matrix (2.6) does not exactly correspond to the same step in the ER algorithm in that the weights w η i are not applied to the correction matrix C i . We will however, in what follows consider this slight modification of the ER algorithm (Cheng and Victoria-Feser, 2002). If the case i is uncontaminated, the data are normal and missing values are missing at random, then (2.8) is asymptotically χ 2 pi where p i = dim y [oi] .…”

“…The choices for ε * and α are to be made by the analyst. The former is the suspected maximal amount of contaminated data and for the latter Cheng and Victoria-Feser (2002) propose choices between 0.1% and 1%.…”

“…However, little has been done to consider also the case of missing data which is in practice very common. Only Little and Smith (1987) and Cheng and Victoria-Feser (2002) propose different solutions. In this paper we actually concentrate on robust estimators with missing data, in particular we propose the use of faster algorithms for their computation and compare them through extensive simulations in terms of their robustness properties when data are contaminated and also in terms of the speed of two different algorithms used to compute the robust estimators.…”

“…Robust estimation of covariance matrices when some of the data at hand are missing is an important problem. It has been studied by Little and Smith (1987) and more recently by Cheng and Victoria-Feser (2002). The latter propose the use of high breakdown estimators and so-called hybrid algorithms (see, e.g., Woodruff and Rocke, 1994).…”