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单元格层面的最小协方差行列式估计量AI 解读

The usual Minimum Covariance Determinant (MCD) estimator of a covariance matrix is robust against casewise outliers. These are cases (that is, rows of the data matrix) that behave differently from the majority of cases, raising suspicion that they might belong to a different population. On the other hand, cellwise outliers are individual cells in the data matrix. When a row contains one or more outlying cells, the other cells in the same row still contain useful information…

回归深度AI 解读

In this article we introduce a notion of depth in the regression setting. It provides the “rank” of any line (plane), rather than ranks of observations or residuals. In simple regression we can compute the depth of any line by a fast algorithm. For any bivariate dataset Z n of size n there exists a line with depth at least n/3. The largest depth in Z n can be used as a measure of linearity versus convexity. In both simple and multiple regression we introduce the deepest…

广义 S 估计量AI 解读

In this article we introduce a new type of positive-breakdown regression method, called a generalized S-estimator (or GS-estimator), based on the minimization of a generalized M-estimator of residual scale. We compare the class of GS-estimators with the usual S-estimators, including least median of squares. It turns out that GS-estimators attain a much higher efficiency than S-estimators, at the cost of a slightly increased worst-case bias. We investigate the breakdown…

重复中位数斜率估计量的渐近性质AI 解读

The influence function is determined for (twice) repeated median estimators with arbitrary kernel functions, and more generally in the case where the two medians are replaced by a general class of estimators. Asymptotic normality is then established for the repeated median estimator of the slope parameter in simple linear regression. In this case the influence function is bounded. For bivariate Gaussian data the efficiency becomes $4/\pi^2 \approx 40.5{\tt \%},$ which is the…

中位绝对偏差的替代尺度估计量AI 解读

In robust estimation one frequently needs an initial or auxiliary estimate of scale. For this one usually takes the median absolute deviation MAD n = 1.4826 med, {|xi − med j x j |}, because it has a simple explicit formula, needs little computation time, and is very robust as witnessed by its bounded influence function and its 50% breakdown point. But there is still room for improvement in two areas: the fact that MAD n is aimed at symmetric distributions and its low (37%)…

多元位置与协方差矩阵仿射等变估计量的崩溃点AI 解读

Finite-sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices. The role of various equivariance properties is illustrated. The breakdown point is related to a measure of performance based on large deviations probabilities. Finally, we show that one-step reweighting preserves the breakdown point.

揭示被掩蔽的多元异常值与杠杆点AI 解读

Detecting outliers in a multivariate point cloud is not trivial, especially when there are several outliers. The classical identification method does not always find them, because it is based on the sample mean and covariance matrix, which are themselves affected by the outliers. That is how the outliers get masked. To avoid the masking effect, we propose to compute distances based on very robust estimates of location and covariance. These robust distances are better suited…

Remedian：适用于大型数据集的稳健汇总方法AI 解读

It is often assumed that to compute a robust estimator on n data values one needs at least n storage elements (contrary to the sample average, that may be calculated with an updating mechanism). This is one of the main reasons why robust estimators are seldom used for large data sets and why they are not included in most statistical packages. We introduce a new estimator that takes up little storage space, investigate its statistical properties, and provide an example on…

Remedian：适用于大型数据集的稳健平均方法AI 解读

It is often assumed that to compute a robust estimator on n data values one needs at least n storage elements (contrary to the sample average, that may be calculated with an updating mechanism). This is one of the main reasons why robust estimators are seldom used for large data sets and why they are not included in most statistical packages. We introduce a new estimator that takes up little storage space, investigate its statistical properties, and provide an example on…

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最小中位数平方回归AI 解读

Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of nearly 50% of contamination in the data. In the special case of simple…

方差变化曲线与最优再下降型 M 估计量AI 解读

We define the change-of-variance curve (CVC) of location M-estimators in order to investigate the infinitesimal stability of the asymptotic variance. We also construct the so-called hyperbolic tangent estimators, proving their existence and performing certain numerical computations of their defining constants. Their introduction is motivated by a theorem that shows they are the optimally robust redescending M-estimators in the sense of the CVC.

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