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稳健离散度矩阵与主成分的投影寻踪方法：基本理论与蒙特卡洛研究AI 解读

This article proposes and discusses a type of new robust estimators for covariance/correlation matrices and principal components via projection-pursuit techniques. The most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point. Besides, they are qualitatively robust and consistent at elliptic underlying distributions. The Monte Carlo study shows that the best of the new estimators compare favorably with other…

方向参数的最佳不变估计AI 解读

Let $X$ be an $n \times k$ random matrix whose coordinates are independently normally distributed with common variance $\sigma^2$ and means given by $EX = e\mu' + \theta\lambda',$ where $e$ is the vector in $R^n$ having all coordinates equal to $1, \theta \in R^n,$ and $\mu, \lambda \in R^k$ with $\sum^k_{j = 1} \lambda^2_j = 1.$ The problem is to estimate $\lambda$, say by $\hat{\lambda},$ with loss function $1 - (\lambda'\hat{\lambda})^2$ when $\mu, \theta,$ and $\sigma^2$…

主成分分析中的影响诊断AI 解读

In linear regression, the theoretical influence function and the various sample versions of it have an established place as diagnostic tools. These same functions are developed here to provide methods for the detection of influential observations in principal components analysis. The perturbation theory of real symmetric matrices unifies this development. Some interesting points of contrast with the regression case are noted and explained theoretically.

形状的典型成分与主成分AI 解读

Canonical components of shape, discriminant functions of shape and principal components of shape are defined from log shape vectors. Several results relate these components to the corresponding components of log measurement and to a particular size variable referred to as residual log size. The latter variable accounts for the variation which cannot be attributed to shape. Canonical components of shape are determined for Anderson's data for three species of iris, as well as…

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使用因子模型对二元响应进行量表评分AI 解读

SUMMARY A simple method for scaling a set of binary responses is proposed using the logit factor model of Bartholomew (1980). It is shown that individuals may be ranked on the basis of a linear combination, ∑αi x i of the responses (xi = 0 or 1) where the αi's are the appropriate factor loadings. This completely avoids the need to calculate the y-scores suggested in Bartholomew (1980) and hence the heavy numerical integration which that method involves.

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主成分的敏感性AI 解读

SUMMARY Simple analytical expressions are derived for the maximum changes in the coefficients of any principal component that are associated with a given (small) change in the variance of the component. This enables a sensitivity analysis to be conducted simultaneously with any principal component analysis, to investigate the stability of the derived components. This has implications for the interpretation of any set of components. A brief illustration is provided, and…

无符号方向数据的无分布区间估计方法AI 解读

A method of moments is suggested for large samples of unsigned directional data. The point estimators obtained coincide with the natural estimates obtained from maximum likelihood estimation if the data have a Bingham distribution. However, the calculations necessary for interval and regional estimation are considerably simpler. The statistical analysis is based upon the asymptotic sampling properties of principal components in nonnormal populations (Davis, 1977; Tyler,…

线性统计关系的估计AI 解读

This paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships, factor analysis, and simultaneous equations models. The emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators (under normality) in the different models. In the first two lectures the observable vector is decomposed into a "systematic part" and a random error; the systematic part…

因子分析的基础AI 解读

A new approach to factor analysis and related latent variable methods is proposed which is based on data reduction using the idea of Bayesian sufficiency. Considerations of symmetry, invariance and independence are used to determine an appropriate family of models. The results are expressed in terms of linear functions of the manifest variables after the manner of principal components analysis. The approach justifies some of the practices based on the normal theory factor…

无符号方向数据的分布无关区间估计方法AI 解读

A method of moments is suggested for large samples of unsigned directional data. The point estimators obtained coincide with the natural estimates obtained from maximum likelihood estimation if the data have a Bingham distribution. However, the calculations necessary for interval and regional estimation are considerably simpler. The statistical analysis is based upon the asymptotic sampling properties of principal components in nonnormal populations (Davis, 1977; Tyler,…

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…

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一般相对风险形式下 Cox 型回归模型的渐近分布理论AI 解读

The theory and application of the Cox (1972) failure time regression model has, almost without exception, assumed an exponential form for the dependence of the hazard function on regression variables. Other regression forms may be more natural or descriptive in some applications. For example, a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions. This note uses the counting process formulation of Andersen…

生命表分析中混合刺激的交互作用AI 解读

An interaction is a response to a mixture of stimuli that, based on the responses to the individual stimuli, seems unexpectedly large or small. The conclusions about the presence of interactions can depend upon the chosen parameterization. For dichotomous outcome variables, a simple probabilistic model of no interaction exists (Finney, 1952). In this model, individuals are at risk from challenges from the background and the separate stimuli, all acting independently. In this…

组成数据的主成分分析AI 解读

Compositional data, consisting of vectors of proportions, have proved difficult to handle statistically because of the awkward constraint that the components of each vector must sum to unity. Moreover such data sets frequently display marked curvature so that linear techniques such as standard principal component analysis are likely to prove inadequate. From a critical reexamination of previous approaches we evolve, through adaptation of recently introduced transformation…

生命表分析中混合刺激的交互效应AI 解读

An interaction is a response to a mixture of stimuli that, based on the responses to the individual stimuli, seems unexpectedly large or small. The conclusions about the presence of interactions can depend upon the chosen parameterization. For dichotomous outcome variables, a simple probabilistic model of no interaction exists (Finney, 1952). In this model, individuals are at risk from challenges from the background and the separate stimuli, all acting independently. In this…

股票收益风险结构变化的贝叶斯稳健检测AI 解读

In this article we provide a statistical procedure for the analysis of stock market prices that is robust toward departures from the normal distribution assumption and that can detect and evaluate a shift of parameters at an unknown time point. The method is an adaptation of a Bayesian inferential procedure developed by Box and Tiao that allows data to deviate moderately from the normal distribution model. It is applied to a set of U.S. stock market prices for 1971–1974. In…

都市工资率的单因子多元时间序列模型AI 解读

The paper formulates and estimates a single-factor multivariate time series model. The model is a dynamic generalization of the multiple indicator (or factor analysis) model. It is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the Kalman filter algorithm. The model is used to obtain estimates of the unobserved metropolitan wage rate for Los Angeles, based on observations of sectoral wages within the…

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.