Factor analysis as a tool for data analysis

The use of factor analysis in analyzing real data is influenced not only by mathematical models but also by the objectives of the research at hand, the amount of data to be analyzed and the departures of the data from the model. Factor analysis is a process performed in several steps, including data screening and assessment of assumptions necessary for the model as well as the actual computations—the new analyst may need assistance in deter-mining the initial method of extraction, how many factors to request, the method of rotation and how to interpret the factors— these steps are discussed with reference to figures containing computer output for a real problem. The importance of auxiliary information and graphical displays to aid the statistician during the factor analysis process is stressed.     

Regressor space outliers in ridge regression

A simple analytical expression is derived for leverage in ridge regression. Leverage is shown to be a monotonically decreasing function of the value of the ridge parameter. This reduction in leverage is greatest for those observations lying substantially in the direction of the minor principal axes. Thus, ridge estimation copes with outliers in regressor space by downweighting their influence. A brief illustration is provided.     

Some results on commutation matrices,with statistical applications

The commutation matrix P _mn changes the order of multiplication of a Kronecker matrix product. The vec operator stacks columns of a matrix one under another in a single column. It is possible to express the vec of a Kronecker matrix product in terms of a Kronecker product of vecs of matrices. The commutation matrix plays an important role here. “Super-vec-operators” like vec A ? vec A vec ( A ? A ), and vec{( A ? A ) P _nn} are very convenient. Several of their properties are being studied. Both the traditional commutation matrix and vec operator and the newer concepts developed from these are applied to multivariate statistical and related problems.     

Applied Nonparametric Instrumental Variables Estimation

Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used to identify and estimate these parameters. However, linear and other finite‐dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

带电粒子在磁化等离子体中辐射模拟研究

研究了带电粒子沿磁化等离子体中匀速运动产生辐射的机理。从理论上推导了辐射场强的解析表达式和特征值形式,并通过对特征值讨论得出了辐射条件。采用数值计算得到了磁化等离子体中Cherenkov辐射特征值分布,发现磁环等离子体回旋频率和等离子体频率对辐射场特征值有直接的影响。通过计算机模拟研究了在该介质中辐射特性及辐射规律,结果表明磁化等离子体中引导磁场强度影响辐射角、辐射强度,并观察到了尾场辐射。     

Fulton＇s猜想的另一种证法

讨论了带有耦合边界条件的自Sturm-Liouville问题．作为已有解决下标问题简单方法的应用,证明了Fulton’s猜想是正确的．     

Estimation of eigenvalues of the scale matrix of the multivariate f distribution

Let F have the multivariate F distribution with a scale matrix Δ. In this paper, the problem of estimating the eigenvalues of the scale matrix Δ is considered. New class of estimators are obtained which dominate the best linear estimator of the form cF. Simulation study is also carried out to compare the performance of these estimators.     

Proof of phenomenon B

In their paper “Some application, properties and conjectures for higher order cumulants of a Markovian stepping-stone model”, Zheng and Matis make use of a conjecture known as “Phenomenon B” regarding the eigenvalues of a tridiagonal matrix having a certain form. While the conjecture was shown to hold for particular cases, it was not proven in general. To supplement the Zheng and Matis paper, a formal proof of Phenomenon B is given here.