High-dimensional proportionality test of two covariance matrices and its application to gene expression data

With the development of modern science and technology, more and more high-dimensionaldata appear in the application fields. Since the high dimension can potentially increase the com-plexity of the covariance structure, comparing the covariance matrices among populations isstrongly motivated in high-dimensional data analysis. In this article, we consider the proportion-ality test of two high-dimensional covariance matrices, where the data dimension is potentiallymuch larger than the sample sizes, or even larger than the squares of the sample sizes. We devisea novel high-dimensional spatial rank test that has much-improved power than many exist-ing popular tests, especially for the data generated from some heavy-tailed distributions. Theasymptotic normality of the proposed test statistics is established under the family of ellipticallysymmetric distributions, which is a more general distribution family than the normal distribu-tion family, including numerous commonly used heavy-tailed distributions. Extensive numericalexperiments demonstrate the superiority of the proposed test in terms of both empirical sizeand power. Then, a real data analysis demonstrates the practicability of the proposed test forhigh-dimensional gene expression data.

Dimension Reduction for Detecting a Difference in Two High-Dimensional Mean Vectors

We consider the efficacy of a proposed linear-dimension-reduction method to potentially increase the powers of five hypothesis tests for the difference of two high-dimensional multivariate-normal population-mean vectors with the assumption of homoscedastic covariance matrices. We use Monte Carlo simulations to contrast the empirical powers of the five high-dimensional tests by using both the original data and dimension-reduced data. From the Monte Carlo simulations, we conclude that a test by Thulin [1], when performed with post-dimension-reduced data, yielded the best omnibus power for detecting a difference between two high-dimensional population-mean vectors. We also illustrate the utility of our dimension-reduction method real data consisting of genetic sequences of two groups of patients with Crohn’s disease and ulcerative colitis.

Large Dynamic Covariance Matrix Estimation with an Application to Portfolio Allocation:A Semiparametric Reproducing Kernel Hilbert Space Approach

The estimation of high dimensional covariance matrices is an interesting and important research topic for many empirical time series problems such as asset allocation. To solve this dimension dilemma, a factor structure has often been taken into account. This paper proposes a dynamic factor structure whose factor loadings are generated in reproducing kernel Hilbert space(RKHS), to capture the dynamic feature of the covariance matrix. A simulation study is carried out to demonstrate its performance. Four different conditional variance models are considered for checking the robustness of our method and solving the conditional heteroscedasticity in the empirical study. By exploring the performance among eight introduced model candidates and the market baseline, the empirical study from 2001 to 2017 shows that portfolio allocation based on this dynamic factor structure can significantly reduce the variance, i.e., the risk, of portfolio and thus outperform the market baseline and the ones based on the traditional factor model.

Selection of Fixed Effects in High-dimensional Generalized Linear Mixed Models

The selection of fixed effects is studied in high-dimensional generalized linear mixed models(HDGLMMs)without parametric distributional assumptions except for some moment conditions.The iterative-proxy-based penalized quasi-likelihood method(IPPQL)is proposed to select the important fixed effects where an iterative proxy matrix of the covariance matrix of the random effects is constructed and the penalized quasi-likelihood is adapted.We establish the model selection consistency with oracle properties even for dimensionality of non-polynomial(NP)order of sample size.Simulation studies show that the proposed procedure works well.Besides,a real data is also analyzed.

Influence of pump brightness,self-Kerr and cross-Kerr effects on the entanglement of partially degenerate triple-photon state

Recently,partially degenerate triple-photon states(TPS)generated by the third-order spontaneous parametric down-conversion have been observed in a superconducting cavity(2020,Phys.Rev.X 10,011011).Their non-Gaussian entanglement properties,characterized by a series of high-order covariance matrices,have also been theoretically revealed.Here,we use the non-Gaussian entanglement criterion proposed in(2021,Phy.Rev.Lett.127,150502)and the logarithmic negativity to study the effect of pump brightness,self-Kerr and cross-Kerr interactions on the entanglement of partially degenerate TPS(PDTPS).We find that the brighter the pump,the easier the entanglement of PDTPS leap to higher-order covariance matrices.Although both self-Kerr and cross-Kerr interactions induce nonlinear phase shifts and weaken the entanglement of PDTPS,cross-Kerr interactions can effectively raise the threshold of entanglement loaded on the third-order covariance matrix.These results can contribute to our understanding of the mechanism of the generation of unconditional non-Gaussian entanglement.