On Covariance Estimators of Factor Loadings in Factor Analysis

We report a matrix expression for the covariance matrix of MLEs of factor loadings in factor analysis, We then derive the analytical formula for covariance matrix of the covariance estimators of MLEs of factor loadings by obtaining the matrix of partial derivatives, which maps the differential of sample covariance matrix (in vector form) into the differential of the covariance estimators

Asymptotic Distribution of Restricted Canonical Correlations and Relevant Resampling Methods

As restricted canonical correlation with a nonnegativity condition on the coefficients depend only on the covariance matrix, their sample counterparts can be obtained from the sample covariance matrix. For such estimators, asymptotic normality results are established, and the role of resampling methods in this context is critically examined. The effectiveness of the usual jackknife and bootstrap methods is studied analytically, and the findings are supplemented by numerical studies

Quasi U-statistics of innite order and applications to the subgroup decomposition of some diversity measures

In several applications, information is drawn from quali- tative variables. In such cases, measures of central tendency and dis- persion may be highly inappropriate. Variability for categorical data can be correctly quantied by the so-called diversity measures. These measures can be modied to quantify heterogeneity between groups (or subpopulations). Pinheiro et al. (2005) shows that Hamming distance can be employed in such way and the resulting estimator of hetero- geneity between populations will be asymptotically normal under mild regularity conditions. Pinheiro et al. (2009) proposes a class of weighted U-statistics based on degenerate kernels of degree 2, called quasi U-statistics, with the property of asymptotic normality under suitable conditions. This is generalized to kernels of degree m by Pinheiro et al. (2011). In this work we generalize this class to an innite order degenerate kernel. We then use this powerful tools and the reverse martingale nature of U-statistics to study the asymptotic behavior of a collection of trans- formed classic diversity measures. We are able to estimate them in a common framework instead of the usual individualized estimation procedures. MSC 2000: primary - 62G10; secondary - 62G20, 92D20.In several applications, information is drawn from quali- tative variables. In such cases, measures of central tendency and dis- persion may be highly inappropriate. Variability for categorical data can be correctly quantied by the so-called diversity measures. These measures can be modied to quantify heterogeneity between groups (or subpopulations). Pinheiro et al. (2005) shows that Hamming distance can be employed in such way and the resulting estimator of hetero- geneity between populations will be asymptotically normal under mild regularity conditions. Pinheiro et al. (2009) proposes a class of weighted U-statistics based on degenerate kernels of degree 2, called quasi U-statistics, with the property of asymptotic normality under suitable conditions. This is generalized to kernels of degree m by Pinheiro et al. (2011). In this work we generalize this class to an innite order degenerate kernel. We then use this powerful tools and the reverse martingale nature of U-statistics to study the asymptotic behavior of a collection of trans- formed classic diversity measures. We are able to estimate them in a common framework instead of the usual individualized estimation procedures. MSC 2000: primary - 62G10; secondary - 62G20, 92D20

Spherical uniformity and some characterizations of the Cauchy distribution

AbstractGiven a fixed line L (in Rn) and a uniform distribution of points (c) on the unit sphere, L(tc), the point of intersection of L and the hyperplane P · c = 0, leads to a mapping Xn : Rn → R, which is shown to have a Cauchy distribution

Second Order Hadamard Differentiability in Statistical Applications

A formulation of the second-order Hadamard differentiability of (extended) statistical functionals and some related theoretical results are established. These results are applied to derive the limiting distributions of a class of generalized Cramér-von Mises type test statistics, which include some proposed new ones for the tests of goodness of fit in the 3-sample problems, the tests in linear regression models, and the tests of bivariate independence, as special cases

Locally best rotation-invariant rank tests for modal location

For a general class of unipolar, rotationally symmetric distributions on the multi-dimensional unit spherical surface, a characterization of locally best rotation-invariant test statistics is exploited in the construction of locally best rotation-invariant rank tests for modal location. Allied statistical distributional problems are appraised, and in the light of these assessments, asymptotic relative efficiency of a class of rotation-invariant rank tests (with respect to some of their parametric counterparts) is studied. Finite sample permutational distributional perspectives are also appraised

Asymptotically optimal tests for parametric functions against ordered functional alternatives

There are hypothesis testing problems for (nonlinear) functions of parameters against functional ordered alternatives for which a reduction to a conventional order-restricted hypothesis testing problem may not be feasible. While such problems can be handled in an asymptotic setup, among the available choices, it is shown that the union-intersection principle may have certain advantages over the likelihood principle or its ramifications. An application to a genomic model is also considered

On inadmissibility of Hotelling T2-tests for restricted alternatives

For multinormal distributions, testing against a global shift alternative, the Hotelling T2-test is uniformly most powerful invariant, and hence admissible. For testing against restricted alternatives this feature may no longer be true. It is shown that whenever the dispersion matrix is an M-matrix, Hotelling's T2-test is inadmissible, though some union-intersection tests may not be so