Dimension estimation in sufficient dimension reduction: A unifying approach

Sufficient Dimension Reduction (SDR) in regression comprises the estimation of the dimension of the smallest (central) dimension reduction subspace and its basis elements. For SDR methods based on a kernel matrix, such as SIR and SAVE, the dimension estimation is equivalent to the estimation of the rank of a random matrix which is the sample based estimate of the kernel. A test for the rank of a random matrix amounts to testing how many of its eigen or singular values are equal to zero. We propose two tests based on the smallest eigen or singular values of the estimated matrix: an asymptotic weighted chi-square test and a Wald-type asymptotic chi-square test. We also provide an asymptotic chi-square test for assessing whether elements of the left singular vectors of the random matrix are zero. These methods together constitute a unified approach for all SDR methods based on a kernel matrix that covers estimation of the central subspace and its dimension, as well as assessment of variable contribution to the lower-dimensional predictor projections with variable selection, a special case. A small power simulation study shows that the proposed and existing tests, specific to each SDR method, perform similarly with respect to power and achievement of the nominal level. Also, the importance of the choice of the number of slices as a tuning parameter is further exhibited.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.     

Singular random matrix decompositions: Jacobians

For a singular random matrix X, we find the Jacobians associated to the following decompositions: QR, Polar, Singular Value (SVD), L′U, L′DM and modified QR (QDR). Similarly, for the cross-product matrix S=X′X we find the Jacobians of the Spectral, Cholesky's, L′DL and symmetric nonnegative definite square root decompositions.     

Transplantation and Jacobians of Sunada isospectral Riemann surfaces

We consider relations among the Jacobians of isospectral compact Riemann surfaces constructed using Sunada's theorem. We use a simple algebraic formulation of “transplantation” of holomorphic 1-forms and singular 1-cycles to obtain two main results. First, we obtain a geometric proof of a result of Prasad and Rajan that Sunada isospectral Riemann surfaces have isogenous Jacobians. Second, we determine a relationship (weaker than isogeny) that holds among the Jacobians of Sunada isospectral Riemann surfaces when the Jacobians’ extra structure as principally polarized abelian varieties is taken into account. We also show all Sunada isospectral manifolds have isomorphic real cohomology algebras. Finally, we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral Riemann surfaces constructed by Brooks and Tse.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Image classification based on Markov random field models with Jeffreys divergence

This paper considers image classification based on a Markov random field (MRF), where the random field proposed here adopts Jeffreys divergence between category-specific probability densities. The classification method based on the proposed MRF is shown to be an extension of Switzer's soothing method, which is applied in remote sensing and geospatial communities. Furthermore, the exact error rates due to the proposed and Switzer's methods are obtained under the simple setup, and several properties are derived. Our method is applied to a benchmark data set of image classification, and exhibits a good performance in comparison with conventional methods.     

Conditional information criteria for selecting variables in linear mixed models

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, here we use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: marginal AIC and conditional AIC. In this paper, we derive three versions of conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the proposed conditional AIC’s are superior to the marginal and conditional AIC’s proposed in the literature in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.     

Random volumes under a general matrix-variate model

The convex hull generated by p linearly independent points in Euclidean n-space, n?p will almost surely determine a p-simplex and the corresponding p-parallelotope. The volume of this p-parallelotope is where the rows of the p×n,n?p matrix of rank p represent the p linearly independent points. If the points are random points in some sense then v becomes a random volume. The distribution of this random volume v when the matrix X has a very general real rectangular matrix-variate density is the topic of this paper. The complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on Jacobians of matrix transformations and functions of matrix argument. Apart from the distribution of v under this general model, arbitrary moments of v, connection to the likelihood ratio statistic or λ-criterion for testing hypotheses on the parameters of multivariate normal distributions, connections to Mellin-Barnes integrals and Meijer’s G-function, connection to the concept of generalized variance, various structural decompositions of v and special cases are also examined here.     

Rank test for heteroscedastic functional data

In this paper, we consider (mid-)rank based inferences for testing hypotheses in a fully nonparametric marginal model for heteroscedastic functional data that contain a large number of within subject measurements from possibly only a limited number of subjects. The effects of several crossed factors and their interactions with time are considered. The results are obtained by establishing asymptotic equivalence between the rank statistics and their asymptotic rank transforms. The inference holds under the assumption ofα-mixing without moment assumptions. As a result, the proposed tests are applicable to data from heavy-tailed or skewed distributions, including both continuous and ordered categorical responses. Simulation results and a real application confirm that the (mid-)rank procedures provide both robustness and increased power over the methods based on original observations for non-normally distributed data.     

Gauss inequalities on ordered linear spaces

Markov inequalities on ordered linear spaces are tightened through the α-unimodality of the corresponding measures. Modality indices are studied for various induced measures, including the singular values of a random matrix and the periodogram of a time series. These tools support a detailed study of linear inference and the ordering of random matrices, to include fixed and random designs and probability bounds on their comparative efficiencies. Other applications include probability bounds on quadratic forms and of order statistics on Rⁿ, on periodograms in the analysis of time series, and on run-length distributions in multivariate statistical process control. Connections to other topics in applied probability and statistics are noted.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

Inference for the invariance of canonical analysis under linear transformations

Using a recent result about the invariance problem in linear canonical analysis (LCA), we introduce a criterion by means of which one can see if this invariance holds when the related random vectors are transformed by linear maps. Then, the estimation of this criterion is considered as well as the problem of testing for invariance of LCA. Particularly, a new test for additional information in canonical analysis is proposed and simulations are used to gain understanding of the finite sample performance of this test and to compare it with the likelihood ratio test.     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

Reconstruction of vector fields and their singularities from ray transforms

In this paper, numerical methods for reconstruction of the singular support of a vector field from its known longitudinal and (or) transverse ray transforms are proposed. Apart from a modification for the Vainberg operator, we use integral operators of angular moments and back projections as well as differential operators of tensor analysis for solving the problem. Results of numerical simulation for reconstructing discontinuous vector fields and with discontinuities in the derivatives are presented. Visualization of their singular support is shown.     

The diffraction of plane elastic unsteady waves by a delaminated inclusion in the case of smooth contact in the delamination region

A solution of the problem of the diffraction of unsteady elastic waves by a thin strip-like delaminated rigid inclusion in an unbounded elastic medium under conditions of planer strain is proposed. We have in mind an inclusion, one side of which is completely bonded with the medium while, the other side is delaminated and conditions of smooth contact are satisfied on it. The method of solution is based on the use of discontinuous solutions of the Lamé equations of motion under conditions of planer strain, which have been constructed earlier in the space of Laplace transforms. As a result, the problem reduces to solving a system of three singular integral equations for the transforms of the unknown discontinuities. The inverse transforms are found by a numerical method, based on the replacement of a Mellin integral by a Fourier series.     

On the Motive of Moduli Spaces of Rank Two Vector Bundles over a Curve

We study the motive of the moduli spaces of rank two vector bundles on a curve. In the smooth case we obtain the Hodge numbers, intermediate Jacobians and number of points over a finite field as corollaries. In the singular case our computations yield the Poincaré–Hodge polynomial of Seshadri's smooth model.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

A novel trace test for the mean parameters in a multivariate growth curve model

A trace test for the mean parameters of the growth curve model is proposed. It is constructed using the restricted maximum likelihood followed by an estimated likelihood ratio approach. The statistic reduces to the Lawley-Hotelling trace test for the Multivariate Analysis of Variance (MANOVA) models. Our test statistic is, therefore, a natural extension of the classical trace test to GMANOVA models. We show that the distribution of the test under the null hypothesis does not depend on the unknown covariance matrix Σ. We also show that the distributions under the null and alternative hypotheses can be represented as sums of weighted central and non-central chi-square random variables, respectively. Under the null hypothesis, the Satterthwaite approximation is used to get an approximate critical point. A novel Satterthwaite type approximation is proposed to obtain an approximate power. A simulation study is performed to evaluate the performance of our proposed test and numerical examples are provided as illustrations.     

Pseudo-inverse multivariate/matrix-variate distributions

The Moore-Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore-Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore-Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudo-inverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.     

The simplex dispersion ordering and its application to the evaluation of human corneal endothelia

A multivariate dispersion ordering based on random simplices is proposed in this paper. Given a R^d-valued random vector, we consider two random simplices determined by the convex hulls of two independent random samples of sizes d+1 of the vector. By means of the stochastic comparison of the Hausdorff distances between such simplices, a multivariate dispersion ordering is introduced. Main properties of the new ordering are studied. Relationships with other dispersion orderings are considered, placing emphasis on the univariate version. Some statistical tests for the new order are proposed. An application of such ordering to the clinical evaluation of human corneal endothelia is provided. Different analyses are included using an image database of human corneal endothelia.