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缺失数据下多元位置与散布的稳健估计AI 解读

Two main issues regarding data quality are data contamination (outliers) and data completion (missing data). These two problems have attracted much attention and research but surprisingly, they are seldom considered together. Popular robust methods such as S-estimators of multivariate location and scatter offer protection against outliers but cannot deal with missing data, except for the obviously inefficient approach of deleting all incomplete cases. We generalize the…

多变量数据中的异常值传播AI 解读

We investigate the performance of robust estimates of multivariate location under nonstandard data contamination models such as componentwise outliers (i.e., contamination in each variable is independent from the other variables). This model brings up a possible new source of statistical error that we call “propagation of outliers.” This source of error is unusual in the sense that it is generated by the data processing itself and takes place after the data has been…

中位数的稳健非参数推断AI 解读

We consider the problem of constructing robust nonparametric confidence intervals and tests of hypothesis for the median when the data distribution is unknown and the data may contain a small fraction of contamination. We propose a modification of the sign test (and its associated confidence interval) which attains the nominal significance level (probability coverage) for any distribution in the contamination neighborhood of a continuous distribution. We also define some…

尺度未知时稳健位置估计的一致渐近理论AI 解读

Most asymptotic results for robust estimates rely on regularity conditions that are difficult to verify in practice. Moreover, these results apply to fixed distribution functions. In the robustness context the distribution of the data remains largely unspecified and hence results that hold uniformly over a set of possible distribution functions are of theoretical and practical interest. Also, it is desirable to be able to determine the size of the set of distribution…

稳健回归估计的自助法AI 解读

We introduce a new computer-intensive method to estimate the distribution of robust regression estimates. The basic idea behind our method is to bootstrap a reweighted representation of the estimates. To obtain a bootstrap method that is asymptotically correct, we include the auxiliary scale estimate in our reweighted representation of the estimates. Our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations. The…

非椭圆回归变量下稳健回归的最大偏差曲线AI 解读

Maximum bias curves for some regression estimates were previously derived assuming that (i) the intercept term is known and/or (ii) the regressors have an elliptical distribution. We present a single method to obtain the maximum bias curves for a large class of regression estimates. Our results are derived under very mild conditions and, in particular, do not require the restrictive assumptions (i) and (ii) above. Using these results it is shown that the maximum bias curves…

位置参数的最优稳健 M 估计AI 解读

We find optimal robust estimates for the location parameter of n independent measurements from a common distribution F that belongs to a contamination neighborhood of a normal distribution. We follow an asymptotic minimax approach similar to Huber's but work with full neighborhoods of the central parametric model including nonsymmetric distributions. Our optimal estimates minimize monotone functions of the estimate's asymptotic variance and bias, which include asymptotic…

一类局部与全局稳健的回归估计量AI 解读

We present a new class of regression estimates called generalized τ estimates. These estimates are defined by minimizing the τ scale of the weighted residuals, with weights that penalize high-leverage observations. Like the τ estimates, the generalized τ estimates utilize for their definition two loss functions, ρ1 and ρ2, which together with the weights can be chosen to achieve simultaneously high breakdown point, finite gross error sensitivity, and high efficiency. We…

一类局部与全局稳健的回归估计量AI 解读

We present a new class of regression estimates called generalized τ estimates. These estimates are defined by minimizing the τ scale of the weighted residuals, with weights that penalize high-leverage observations. Like the τ estimates, the generalized τ estimates utilize for their definition two loss functions, ρ1 and ρ2, which together with the weights can be chosen to achieve simultaneously high breakdown point, finite gross error sensitivity, and high efficiency. We…

最小α分位数估计的极小极大偏差性质AI 解读

A natural measure of the degree of robustness of an estimate $\mathbf{T}$ is the maximum asymptotic bias $B_\mathbf{T}(\varepsilon)$ over an $\varepsilon$-contamination neighborhood. Martin, Yohai and Zamar have shown that the class of least $\alpha$-quantile regression estimates is minimax bias in the class of $M$-estimates, that is, they minimize $B_\mathbf{T}(\varepsilon)$, with $\alpha$ depending on $\varepsilon$. In this paper we generalize this result, proving that the…

尺度参数的偏差稳健估计AI 解读

In this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location-scale family. We define the class of $M$-estimates of scale with general location, and show that under certain regularity assumptions, these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution, and with respect…

效率约束下的位置参数偏差稳健估计AI 解读

In 1964, P. Huber established the following minimax bias robustness result for estimating the location $\mu$ in the $\varepsilon$-contamination family $F(x)=(1-\varepsilon)\phi[(x-\mu)/s]+\varepsilon H(x)$, where $\phi$ is the standard normal distribution and H is an arbitrary distribution function: The median minimizes the maximum asymptotic bias among all translation equivariant estimates of location. However, the median efficiency of $2/\pi$ at the Gaussian model may be…

正交回归中的抗偏估计AI 解读

Orthogonal regression $M$-estimates are considered from a bias robust point of view. Their maximum bias over epsilon-contamination neighborhoods is characterized, and maximum bias curves are computed. The most bias robust orthogonal regression $M$-estimate is derived and shown to be a "mode type" estimate; for instance, in the two-dimensional case this estimate can be computed by locating a strip of fixed width covering the maximum number of data points. It will be shown…

正值随机变量尺度的渐近极小极大偏差稳健 M 估计AI 解读

Asymptotically min—max bias robust M-estimates of scale are obtained for positive random variables with ε-contaminated distributions, using any one of a broad class of loss functions of positive and negative bias. Any such estimate is a scaled order statistic, with the order statistic and scaling constant determined by ε, the nominal distribution, and the loss function. Some calculations for specific cases show that for a large range of ε the min—max order statistic solution…

摘要暂缺

变量含测量误差模型中的稳健估计AI 解读

Orthogonal regression analogues of M estimates of regression, called here orthogonal regression M estimates, are defined. These estimates are shown to be consistent at elliptical errors-in-variables models and robust, if the corresponding loss function is bounded. The orthogonal regression analogues of regression scale estimates, called here orthogonal regression S estimates, are considered as well. In particular, they provide a robust estimate for the scale of the…

通过最小化高效尺度构造高崩溃点回归估计AI 解读

A new class of robust estimates, τ estimates, is introduced. The estimates have simultaneously the following properties: (a) they are qualitatively robust, (b) their breakdown point is .5, and (c) they are highly efficient for regression models with normal errors. They are defined by minimizing a new scale estimate, τ, applied to the residuals. Asymptotically, a τ estimate is equivalent to an M estimate with a ψ function given by a weighted average of two ψ functions, one…