arXiv subjects

Xiucai Ding

Publications and source records attributed to Xiucai Ding.

39 records · Page 3Linked to original sources

arXiv2019

Estimation and inference for precision matrices of non-stationary time series

In this paper, we consider the estimation and inference of precision matrices of a rich class of locally stationary and nonlinear time series assuming that only one realization of the time series is observed. Using a Cholesky decomposition technique, we show that the precision matrices can be directly estimated via a series of least squares linear regressions with smoothly time-varying coefficients. The method of sieves is utilized for the estimation and is shown to be efficient and optimally adaptive in terms of estimation accuracy and computational complexity. We establish an asymptotic theory for a class of ${\cal L}^2$ tests based on the nonparametric sieve estimators. The latter are used for testing whether the precision matrices are diagonal or banded. A high dimensional Gaussian approximation result is established for a wide class of quadratic form of non-stationary and nonlinear processes, which is of interest by itself.

math.ST↗

arXiv2017

Singular vector distribution of sample covariance matrices

We consider a class of sample covariance matrices of the form $Q=TXX^{*}T^*,$ where $X=(x_{ij})$ is an $M \times N$ rectangular matrix consisting of i.i.d entries and $T$ is a deterministic matrix satisfying $T^*T$ is diagonal. Assuming $M$ is comparable to $N$, we prove that the distribution of the components of the singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of $x_{ij}$ coincide with the Gaussian random variables. For the singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of $x_{ij}$ match with those of Gaussian random variables. Similar results have been proved for Wigner matrices by Knowles and Yin.

math.PR↗

arXiv2017

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal Q = TX(TX)^{*}$, where the sample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with mean zero and variance $N^{-1}$, and $T$ is an $M_1 \times M_2$ deterministic matrix satisfying $T^* T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal Q$ when $M:=\min\{M_1,M_2\}$ and $N$ tends to infinity with $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. Under mild assumptions of $T$, we prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal Q$ if and only if $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(| \sqrt{N} x_{ij}| \geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.

math.PR↗