Baiqi Miao

On Sliced Inverse Regression With High-Dimensional Covariates

Sliced inverse regression is a promising method for the estimation of the central dimension-reduction subspace (CDR space) in semiparametric regression models. It is particularly useful in tackling cases with high-dimensional covariates. In this article we study the asymptotic behavior of the estimate of the CDR space with high-dimensional covariates, that is, when the dimension of the covariates goes to infinity as the sample size goes to infinity. Strong and weak convergence are obtained. We also suggest an estimation procedure of the Bayes information criterion type to ascertain the dimension of the CDR space and derive the consistency. A simulation study is conducted.

ON ASYMPTOTICS OF EIGENVECTORS OF LARGE SAMPLE COVARIANCE MATRIX

Let {X ij }, i, j = ..., be a double array of i.i.d. complex random variables with EX 11 = 0, E|X 11 | 2 = 1 and E|X 11 | 4 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix T n and X n is the n x N matrix of the upper-left comer of the double array. The matrix An can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T n , or as a multivariate F matrix if T n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of An, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if { X ij } and T n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when T n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

Asymptotic theory of least distances estimate in multivariate linear models

We consider the multivariate linear model where Yi is a p-vector random variable, Xi is a q X p matrix, βA0 is an unknown q-vector parameter and {si} is a sequence of iid p-vector random variable with median vector zero. The estimate βAn of βA0 such that is called the least distances (LD) estimator. It may be recalled that the least squares (LS) estimator is obtained by minimizing the sum of norm squares. In this paper, it is shown that the LD estimator is unique, consistent and has an asymptotic q-variate normal distribution with mean β0 and co variance matrix V which depends on the distribution of the error vectors {ei}. A consistent estimator of V is proposed which together with βAn enables an asymptotic inference on β0. In particular, tests of linear hypotheses on β0 analogous to those of analysis of variance in the GATJSS-MARKOFF linear model are developed. Explicit expressions are obtained in some cases for the asymptotic relative efficiency of the LD compared to the LS estimator.

Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices

In this paper, we improve known results on the convergence rates of spectral distributions of large-dimensional sample covariance matrices of size p × n. Using the Stieltjes transform, we first prove that the expected spectral distribution converges to the limiting Marcenko--Pastur distribution with the dimension sample size ratio y=yn=p/n at a rate of O(n- 1/2) if y keeps away from 0 and 1, under the assumption that the entries have a finite eighth moment. Furthermore, the rates for both the convergence in probability and the almost sure convergence are shown to be Op(n-2/5) and oa.s.(n-2/5+\eta), respectively, when y is away from 1. It is interesting that the rate in all senses is O(n-1/8) when y is close to 1.

Ontologies for crisis contagion management in financial institutions

What makes crisis management in financial institutions fairly unique and particularly complex is the accompanying crisis contagion or systemic risk. The subprime mortgage crisis currently happening in the USA is a typical example. In order to further deepen our understanding of how crisis contagion occurs and enhance information interchange and knowledge sharing among related entities, ontologies for crisis contagion management in financial institutions are proposed in this study. Three categories of ontologies, which include static ontology, dynamic ontology, and social ontology, are developed to deal with different perspectives in this domain. The three types of ontology are then united in the Ontology Web Language (OWL) and the Semantic Web Rules Languages (SWRL) framework, both of which are machine readable. Finally, the case of Long-Term Capital Management (LTCM) is offered to demonstrate how the proposed ontologies are used in financial institutions.

A note on the convergence rate of the spectral distributions of large random matrices

In this note we show that the spectral distribution of large dimensional Wigner matrix tends to the semicircular law with a convergence rate of Op(n-1/4). Similar results for sample covariance matrix are also given.

Optimal feature selection for sparse linear discriminant analysis and its applications in gene expression data

This work studies the theoretical rules of feature selection in linear discriminant analysis (LDA), and a new feature selection method is proposed for sparse linear discriminant analysis. An l1 minimization method is used to select the important features from which the LDA will be constructed. The asymptotic results of this proposed two-stage LDA (TLDA) are studied, demonstrating that TLDA is an optimal classification rule whose convergence rate is the best compared to existing methods. The experiments on simulated and real datasets are consistent with the theoretical results and show that TLDA performs favorably in comparison with current methods. Overall, TLDA uses a lower minimum number of features or genes than other approaches to achieve a better result with a reduced misclassification rate.

Knowledge level modeling for systemic risk management in financial institutions

The current subprime mortgage crisis is a typical case for systemic risk in financial institutions. In order to further our understanding and communication about systemic risk management (SRM) in financial institutions, this paper proposes a knowledge level model (KLM) for systemic risk management in financial institutions. There are two parts considered in the proposed KLM: ontologies and problem solving method (PSM). Ontologies are adopted to represent a knowledge base of KLM, which integrates top level ontology and domain level ontologies. And then the problem solving method is given to show the reasoning process of this knowledge. The symbol level of KLM is also discussed which integrates OWL, SWRL and JESS. Further, the discussion about Lehman Brothers Minibonds case in 2008 is provided to illustrate how proposed KLM is used in practice. With these, first, they will enhance the interchange of information and knowledge sharing for SRM within a financial institution. Second, they will assist knowledge base development for SRM design, for which a prototype of financial systemic risk management decision support system is given in this study. Third, they will support coordination among different institutions by using standardized vocabularies. And finally, from the design science perspective, the whole proposed framework could be meaningful to models in other domains.

Detection of change points using rank methods

In this paper, the detection and estimation of change points of local parameters are studied by means of localization procedures and rank statistics. These techniques are also applied to detection and estimation of the change points of scale parameters and that of location parameters of directional data.

Remarks on the Convergence Rate of the Spectral Distributions of Wigner Matrices

In this remark, we show that under the assumption of the finite fourth moment elements in the empirical spectral distribution of a large Wigner matrix converges to the Wigner semicircular law in probability of the order O(n−1/3).