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稳健协方差代入方法的注意事项AI 解读

The sample covariance matrix, which is well known to be highly nonrobust, plays a central role in many classical multivariate statistical methods. A popular way of making such multivariate methods more robust is to replace the sample covariance matrix with some robust scatter matrix. The aim of this paper is to point out that multivariate methods often require that certain properties of the covariance matrix hold also for the robust scatter matrix in order for the…

稳健函数型主成分：投影寻踪方法AI 解读

In many situations, data are recorded over a period of time and may be regarded as realizations of a stochastic process. In this paper, robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting. Our approach combines robust projection-pursuit with different smoothing methods. Consistency of the estimators are shown under mild assumptions. The performance of the classical and robust procedures are…

回归 MM 估计与约束 M 估计的最大偏差函数AI 解读

We derive the maximum bias functions of the MM-estimates and the constrained M-estimates or CM-estimates of regression and compare them to the maximum bias functions of the S-estimates and the τ-estimates of regression. In these comparisons, the CM-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM-estimate which has a smaller maximum bias function than a given S-estimate, that is,…

Tukey中位数的影响函数与最大偏差AI 解读

Tukey's median is among one of the earliest known high breakdown point multivariate location statistics. Aside from its breakdown point, though, little else appears to be known about its robustness properties. In this paper we investigate other aspects of Tukey's median, and in particular we derive and study its influence function and its maximum contamination bias function. When judged by these other robustness criteria, Tukey's median again proves to be highly robust.

约束M估计与回降型M估计的正则性和唯一性AI 解读

Constrained M-estimates of multivariate location and scatter are found by finding the global minimum of an objective function subject to a constraint. They are related to redescending M-estimates of multivariate location and scatter since any stationary point of the objective function corresponds to such an M-estimate. Unfortunately, even for the population form of the estimator, that is, the constrained M-functional, the objective function may have multiple stationary…

类别时间序列的谱分析：赋值与谱包络AI 解读

Many studies produce categorical time series in which harmonic analysis is of interest. Although there exist time domain approaches for the analysis of categorical time series such as Markov chains or link function based regression models, there is apparently little statistical theory or methodology for analyzing qualitative-valued time series in the frequency domain. The purpose of this paper is to initiate the development of a general framework for the frequency domain…

类别时间序列的谱分析：数值赋值与谱包络AI 解读

Many studies produce categorical time series in which harmonic analysis is of interest. Although there exist time domain approaches for the analysis of categorical time series such as Markov chains or link function based regression models, there is apparently little statistical theory or methodology for analyzing qualitative-valued time series in the frequency domain. The purpose of this paper is to initiate the development of a general framework for the frequency domain…

Wielandt不等式及其在随机对称矩阵特征值渐近分布中的应用AI 解读

A relatively obscure eigenvalue due to Wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix. The asymptotic distributions are obtained under a fairly general setting. An application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given.

多元散布的分布无关 M 估计量AI 解读

The existence and uniqueness of a limiting form of a Huber-type $M$-estimator of multivariate scatter is established under certain conditions on the observed sample. These conditions hold with probability one when sampling randomly from a continuous multivariate distribution. The existence of the estimator is proven by showing that it is the limiting point of a specific algorithm. Hence, the proof is constructive. For continuous populations, the estimator of multivariate…

球面角中心高斯分布的统计分析AI 解读

The angular central Gaussian distribution is an alternative to the Bingham distribution for modeling antipodal symmetric directional data. In this paper the statistical theory for the angular central Gaussian model is presented. Some topics treated are maximum likelihood estimation of the parameters, testing for uniformity and circularity, and principal components analysis. Comparisons to methods based upon the sample second moments are made via an example.

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《散布矩阵的稳健性与效率性质》更正AI 解读

The notation used in defining the likelihood ratio statistics, ln,H,9(⁠Λ^n⁠), on p. 416, line 3 from bottom, should have the argument Λ^n deleted. For finite sample sizes, the likelihood ratio statistic is not necessarily strictly a function of the maximum likelihood estimator Λ^n⁠. The results of §4 remain valid.

主成分向量的一类渐近检验AI 解读

In this paper, the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered. A class of invariant asymptotic tests based on the sample covariance matrix is derived. Tests in this class are shown to be consistent and their local power functions are given. The arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of…

总体重根局部备择下主成分特征根的渐近分布AI 解读

The asymptotic distribution for the principal component roots under local alternatives to multiple population roots is derived. The asymptotic theory assumes the estimate of the population covariance or scatter matrix to be asymptotically normal and to possess certain invariance properties. These assumptions are satisfied for the affine-invariant $M$-estimates of scatter for an elliptical distribution. The local alternative framework is used in deriving a local power…

散布矩阵的稳健性与效率性质AI 解读

The robustness and efficiency properties of likelihood ratio tests for functions of the population covariance matrix are studied. An alternative class of tests based upon affine-invariant M-estimates of scatter is proposed whenever the function of the covariance matrix is invariant under a common scale change. For such inferences, it is shown that a single scalar-valued index of efficiency is sufficient.

径向估计与球形性检验AI 解读

The first two moments for random symmetric matrices with radial distributions are given. This result enables one to generalize asymptotic procedures based on the sample covariance matrix to procedures based on a class of estimates for a positive-definite matrix parameter. This class includes the maximum likelihood estimates for the scatter matrix of an elliptical population, and affine-invariant robust estimates of a scatter matrix. Application of the results to the test for…

特征向量的渐近推断AI 解读

Asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors. These asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest. Applications of the general results to principal components analysis and canonical variate analysis are given.