Test for bandedness of high-dimensional covariance matrices and bandwidth estimation

Motivated by the latest effort to employ banded matrices to estimate a high-dimensional covariance $\Sigma$, we propose a test for $\Sigma$ being banded with possible diverging bandwidth. The test is adaptive to the "large $p$, small $n$" situations without assuming a specific parametric distribution for the data. We also formulate a consistent estimator for the bandwidth of a banded high-dimensional covariance matrix. The properties of the test and the bandwidth estimator are investigated by theoretical evaluations and simulation studies, as well as an empirical analysis on a protein mass spectroscopy data.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1002 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two sample tests for high-dimensional covariance matrices

We propose two tests for the equality of covariance matrices between two high-dimensional populations. One test is on the whole variance--covariance matrices, and the other is on off-diagonal sub-matrices, which define the covariance between two nonoverlapping segments of the high-dimensional random vectors. The tests are applicable (i) when the data dimension is much larger than the sample sizes, namely the "large $p$, small $n$" situations and (ii) without assuming parametric distributions for the two populations. These two aspects surpass the capability of the conventional likelihood ratio test. The proposed tests can be used to test on covariances associated with gene ontology terms.Comment: Published in at http://dx.doi.org/10.1214/12-AOS993 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the approximate maximum likelihood estimation for diffusion processes

The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. A\"{\i}t-Sahalia [J. Finance 54 (1999) 1361--1395; Econometrica 70 (2002) 223--262] proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on A\"{\i}t-Sahalia's [Econometrica 70 (2002) 223--262; Ann. Statist. 36 (2008) 906--937] proposal and analysis on the AMLE, we establish the consistency and convergence rate of the AMLE, which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations. We find conditions under which the AMLE has the same asymptotic distribution as that of the full MLE. A first order approximation to the Fisher information matrix is proposed.Comment: Published in at http://dx.doi.org/10.1214/11-AOS922 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tests for High Dimensional Generalized Linear Models

We consider testing regression coefficients in high dimensional generalized linear models. An investigation of the test of Goeman et al. (2011) is conducted, which reveals that if the inverse of the link function is unbounded, the high dimensionality in the covariates can impose adverse impacts on the power of the test. We propose a test formation which can avoid the adverse impact of the high dimensionality. When the inverse of the link function is bounded such as the logistic or probit regression, the proposed test is as good as Goeman et al. (2011)'s test. The proposed tests provide p-values for testing significance for gene-sets as demonstrated in a case study on an acute lymphoblastic leukemia dataset.Comment: The research paper was stole by someone last November and illegally submitted to arXiv by a person named gong zi jiang nan. We have asked arXiv to withdraw the unfinished paper [arXiv:1311.4043] and it was removed last December. We have collected enough evidences to identify the person and Peking University has begun to investigate the plagiarize

High dimensional generalized empirical likelihood for moment restrictions with dependent data

This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over-identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting

Thermal effects on bipartite and multipartite correlations in fiber coupled cavity arrays

We investigate the thermal influence of fibers on the dynamics of bipartite and multipartite correlations in fiber coupled cavity arrays where each cavity is resonantly coupled to a two-level atom. The atom-cavity systems connected by fibers can be considered as polaritonic qubits. We first derive a master equation to describe the evolution of the atom-cavity systems. The bipartite (multipartite) correlations is measured by concurrence and discord (spin squeezing). Then, we solve the master equation numerically and study the thermal effects on the concurrence, discord, and spin squeezing of qubits. On the one hand, at zero temperature, there are steady-state bipartite and multipartite correlations. One the other hand, the thermal fluctuations of a fiber may blockade the generation of entanglement of two qubits connected directly by the fiber while the discord can be generated and stored for a long time. This thermal-induced blockade effects of bipartite correlations may be useful for quantum information processing. The bipartite correlations of a longer chain of qubits is more robust than a shorter one in the presence of thermal fluctuations

Simultaneous Testing of Mean and Variance Structures in Nonlinear Time Series Models

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness of fit. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives which are different from the null hypothesis at an optimal rate.Bootstrap, empirical likelihood, goodness{of{t test, kernel estimation, least squares empirical likelihood, rate-optimal test

A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression

We propose an empirical likelihood test that is able to test the goodness of fit of a class of parametric and semi-parametric multiresponse regression models. The class includes as special cases fully parametric models; semi-parametric models, like the multiindex and the partially linear models; and models with shape constraints. Another feature of the test is that it allows both the response variable and the covariate be multivariate, which means that multiple regression curves can be tested simultaneously. The test also allows the presence of infinite-dimensional nuisance functions in the model to be tested. It is shown that the empirical likelihood test statistic is asymptotically normally distributed under certain mild conditions and permits a wild bootstrap calibration. Despite the large size of the class of models to be considered, the empirical likelihood test enjoys good power properties against departures from a hypothesized model within the class.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ208 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Nonparametric Estimation of Expected Shortfall

The paper evaluates the properties of nonparametric estimators of the expected shortfall, an increasingly popular risk measure in financial risk management. It is found that the existing kernel estimator based on a single bandwidth does not offer variance reduction, which is surprising considering that kernel smoothing reduces the variance of estimators for the value at risk and the distribution function. We reformulate the kernel estimator such that two different bandwidths are employed in the kernel smoothing for the value at risk and the shortfall itself. We demonstrate by both theoretical analysis and simulation studies that the new kernel estimator achieves a variance reduction. The paper also covers the practical issues of bandwidth selection and standard error estimation