arXiv subjects

Jianfeng Yao

Publications and source records attributed to Jianfeng Yao.

At least 55 records · Page 3Linked to original sources

arXiv2019

Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model

This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson-Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral processes, corresponding to the extreme and linear spectral statistics respectively.

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arXiv2018

On testing for high-dimensional white noise

Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (upto the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, non trivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.

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arXiv2018

Joint CLT for eigenvalue statistics from several dependent large dimensional sample covariance matrices with application

Let $\mathbf{X}_n=(x_{ij})$ be a $k \times n$ data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously $R$ sample covariance matrices $\mathbf{B}_{nr}=\frac1n \mathbf{Q}_r \mathbf{X}_n \mathbf{X}_n^*\mathbf{Q}_r^\top,~1\le r\le R$, where the $\mathbf{Q}_{r}$'s are nonrandom real matrices with common dimensions $p\times k~(k\geq p)$. Assuming that both the dimension $p$ and the sample size $n$ grow to infinity, the limiting distributions of the eigenvalues of the matrices $\{\mathbf{B}_{nr}\}$ are identified, and as the main result of the paper, we establish a joint central limit theorem for linear spectral statistics of the $R$ matrices $\{\mathbf{B}_{nr}\}$. Next, this new CLT is applied to the problem of testing a high dimensional white noise in time series modelling. In experiments the derived test has a controlled size and is significantly faster than the classical permutation test, though it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix ($R=1$).

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arXiv2017

Testing for Heteroscedasticity in High-dimensional Regressions

Testing heteroscedasticity of the errors is a major challenge in high-dimensional regressions where the number of covariates is large compared to the sample size. Traditional procedures such as the White and the Breusch-Pagan tests typically suffer from low sizes and powers. This paper proposes two new test procedures based on standard OLS residuals. Using the theory of random Haar orthogonal matrices, the asymptotic normality of both test statistics is obtained under the null when the degree of freedom tends to infinity. This encompasses both the classical low-dimensional setting where the number of variables is fixed while the sample size tends to infinity, and the proportional high-dimensional setting where these dimensions grow to infinity proportionally. These procedures thus offer a wide coverage of dimensions in applications. To our best knowledge, this is the first procedures in the literature for testing heteroscedasticity which are valid for medium and high-dimensional regressions. The superiority of our proposed tests over the existing methods are demonstrated by extensive simulations and by several real data analyses as well.

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arXiv2017

CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data

This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $\mathbf{B}_n=n^{-1}\sum_{j=1}^{n}\mathbf{Q}\mathbf{x}_j\mathbf{x}_j^{*}\mathbf{Q}^{*}$ where $\mathbf{Q}$ is a nonrandom matrix of dimension $p\times k$, and $\{\mathbf{x}_j\}$ is a sequence of independent $k$-dimensional random vector with independent entries, under the assumption that $p/n\to y>0$. A key novelty here is that the dimension $k\ge p$ can be arbitrary, possibly infinity. This new model of sample covariance matrices $\mathbf{B}_n$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $k=p$ and $\mathbf{Q}=\mathbf{T}_n^{1/2}$ for some positive definite Hermitian matrix $\mathbf{T}_n$. Also with $k=\infty$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (2004). Applications of this new CLT are proposed for testing the structure of a high-dimensional covariance matrix. The derived tests are then used to analyse a large fMRI data set regarding its temporary correlation structure.

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arXiv2017

On structure testing for component covariance matrices of a high-dimensional mixture

By studying the family of $p$-dimensional scale mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix {\em does not converge} to the celebrated Marčenko-Pastur law. A different and new limit is found and characterized. The reasons of failure of the Marčenko-Pastur limit in this situation are found to be a strong dependence between the $p$-coordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analize the traditional John's type test we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. It is shown that the John's test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used ICL and BIC criteria in detecting non spherical component covariance matrices of a high-dimensional mixture.

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arXiv2017

On a spiked model for large volatility matrix estimation from noisy high-frequency data

Recently, inference about high-dimensional integrated covariance matrices (ICVs) based on noisy high-frequency data has emerged as a challenging problem. In the literature, a pre-averaging estimator (PA-RCov) is proposed to deal with the microstructure noise. Using the large-dimensional random matrix theory, it has been established that the eigenvalue distribution of the PA-RCov matrix is intimately linked to that of the ICV through the Marcenko-Pastur equation. Consequently, the spectrum of the ICV can be inferred from that of the PA-RCov. However, extensive data analyses demonstrate that the spectrum of the PA-RCov is spiked, that is, a few large eigenvalues (spikes) stay away from the others which form a rather continuous distribution with a density function (bulk). Therefore, any inference on the ICVs must take into account this spiked structure. As a methodological contribution, we propose a spiked model for the ICVs where spikes can be inferred from those of the available PA-RCov matrices. The consistency of the inference procedure is established and is checked by extensive simulation studies. In addition, we apply our method to the real data from the US and Hong Kong markets. It is found that our model clearly outperforms the existing one in predicting the existence of spikes and in mimicking the empirical PA-RCov matrices.

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arXiv2016

Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a $N\times n$ matrix $Σ_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \[\operatorname {Trace}f \bigl(Σ_nΣ_n^*\bigr)=\sum_{i=1}^Nf(λ_i),\qquad (λ_i)\ eigenvalues\ of\ Σ_nΣ_n^*,\] are shown to be Gaussian, in the regime where both dimensions of matrix $Σ_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to $ \vert \mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and $κ=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert ^2-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_n$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer-Sjöstrand's formula. The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.

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arXiv2016

On the Surprising Explanatory Power of Higher Realized Moments in Practice

Realized moments of higher order computed from intraday returns are introduced in recent years. The literature indicates that realized skewness is an important factor in explaining future asset returns. However, the literature mainly focuses on the whole market and on the monthly or weekly scale. In this paper, we conduct an extensive empirical analysis to investigate the forecasting abilities of realized skewness and realized kurtosis towards individual stock's future return and variance in the daily scale. It is found that realized kurtosis possesses significant forecasting power for the stock's future variance. In the meanwhile, realized skewness is lack of explanatory power for the future daily return for individual stocks with a short horizon, in contrast with the existing literature.

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arXiv2016

Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

This paper focuses on the prominent sphericity test when the dimension $p$ is much lager than sample size $n$. The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty, n\rightarrow\infty$ is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.

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arXiv2015

Identifying the number of factors from singular values of a large sample auto-covariance matrix

Identifying the number of factors in a high-dimensional factor model has attracted much attention in recent years and a general solution to the problem is still lacking. A promising ratio estimator based on the singular values of the lagged autocovariance matrix has been recently proposed in the literature and is shown to have a good performance under some specific assumption on the strength of the factors. Inspired by this ratio estimator and as a first main contribution, this paper proposes a complete theory of such sample singular values for both the factor part and the noise part under the large-dimensional scheme where the dimension and the sample size proportionally grow to infinity. In particular, we provide the exact description of the phase transition phenomenon that determines whether a factor is strong enough to be detected with the observed sample singular values. Based on these findings and as a second main contribution of the paper, we propose a new estimator of the number of factors which is strongly consistent for the detection of all significant factors (which are the only theoretically detectable ones). In particular, factors are assumed to have the minimum strength above the phase transition boundary which is of the order of a constant; they are thus not required to grow to infinity together with the dimension (as assumed in most of the existing papers on high-dimensional factor models). Empirical Monte-Carlo study as well as the analysis of stock returns data attest a very good performance of the proposed estimator. In all the tested cases, the new estimator largely outperforms the existing estimator using the same ratios of singular values.

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arXiv2015

Testing the independence of two random vectors where only one dimension is large

For testing the independence of two vectors with respective dimensions $p_1$ and $p_2$, the existing literature in high-dimensional statistics all assume that both dimensions $p_1$ and $p_2$ grow to infinity with the sample size. However, as evidenced in the RNA-sequencing data analysis discussed in the paper, it happens frequently that one of the dimension is quite small and the other quite large compared to the sample size. In this paper, we address this new asymptotic framework for the independence test. A new test procedure is introduced and its asymptotic normality is established when the vectors are normal distributed. A Mote-Carlo study demonstrates the consistency of the procedure and exhibits its superiority over some existing high-dimensional procedures. Applied to the RNA-sequencing data mentioned above, we obtain very convincing results on pairwise independence/dependence of gene isoform expressions as attested by prior knowledge established in that field. Lastly, Monte-Carlo experiments show that the procedure is robust against the normality assumption on the population vectors.

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arXiv2015

Extreme eigenvalues of large-dimensional spiked Fisher matrices with application

Consider two $p$-variate populations, not necessarily Gaussian, with covariance matrices $Σ_1$ and $Σ_2$, respectively, and let $S_1$ and $S_2$ be the sample covariances matrices from samples of the populations with degrees of freedom $T$ and $n$, respectively. When the difference $Δ$ between $Σ_1$ and $Σ_2$ is of small rank compared to $p,T$ and $n$, the Fisher matrix $F=S_2^{-1}S_1$ is called a {\em spiked Fisher matrix}. When $p,T$ and $n$ grow to infinity proportionally, we establish a phase transition for the extreme eigenvalues of $F$: when the eigenvalues of $Δ$ ({\em spikes}) are above (or under) a critical value, the associated extreme eigenvalues of the Fisher matrix will converge to some point outside the support of the global limit (LSD) of other eigenvalues; otherwise, they will converge to the edge points of the LSD. Furthermore, we derive central limit theorems for these extreme eigenvalues of the spiked Fisher matrix. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in $Δ$ are {\em simple}. Numerical examples are provided to demonstrate the finite sample performance of the results. In addition to classical applications of a Fisher matrix in high-dimensional data analysis, we propose a new method for the detection of signals allowing an arbitrary covariance structure of the noise. Simulation experiments are conducted to illustrate the performance of this detector.

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arXiv2015

Forecasting High-Dimensional Realized Volatility Matrices Using A Factor Model

Modeling and forecasting covariance matrices of asset returns play a crucial role in finance. The availability of high frequency intraday data enables the modeling of the realized covariance matrix directly. However, most models in the literature suffer from the curse of dimensionality. To solve the problem, we propose a factor model with a diagonal CAW model for the factor realized covariance matrices. Asymptotic theory is derived for the estimated parameters. In an extensive empirical analysis, we find that the number of parameters can be reduced significantly. Furthermore, the proposed model maintains a comparable performance with a benchmark vector autoregressive model.

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arXiv2015

On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T=\sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. This paper investigates the limiting behavior of the singular values of $X_T$ under the so-called {\em ultra-dimensional regime} where $p\to\infty$ and $T\to\infty$ in a related way such that $p/T\to 0$. First, we show that the singular value distribution of $X_T$ after a suitable normalization converges to a nonrandom limit $G$ (quarter law) under the forth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of $G$. Both results are derived using the moment method.

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arXiv2015

On Two Simple and Effective Procedures for High Dimensional Classification of General Populations

In this paper, we generalize two criteria, the determinant-based and trace-based criteria proposed by Saranadasa (1993), to general populations for high dimensional classification. These two criteria compare some distances between a new observation and several different known groups. The determinant-based criterion performs well for correlated variables by integrating the covariance structure and is competitive to many other existing rules. The criterion however requires the measurement dimension be smaller than the sample size. The trace-based criterion in contrast, is an independence rule and effective in the "large dimension-small sample size" scenario. An appealing property of these two criteria is that their implementation is straightforward and there is no need for preliminary variable selection or use of turning parameters. Their asymptotic misclassification probabilities are derived using the theory of large dimensional random matrices. Their competitive performances are illustrated by intensive Monte Carlo experiments and a real data analysis.

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arXiv2014

Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models

In this paper, we derive a joint central limit theorem for random vector whose components are function of random sesquilinear forms. This result is a natural extension of the existing central limit theory on random quadratic forms. We also provide applications in random matrix theory related to large-dimensional spiked population models. For the first application, we find the joint distribution of grouped extreme sample eigenvalues correspond to the spikes. And for the second application, under the assumption that the population covariance matrix is diagonal with $k$ (fixed) simple spikes, we derive the asymptotic joint distribution of the extreme sample eigenvalue and its corresponding sample eigenvector projection.

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arXiv2014

Moment approach for singular values distribution of a large auto-covariance matrix

Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T= \sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. Since $X_T$ is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of $X_TX^T_T$. Therefore, the purpose of this paper is to investigate the limiting behaviors of the eigenvalues of $X_TX^T_T$ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit $F$. Second, we establish the convergence of its largest eigenvalue to the right edge of $F$. Both results are derived using moment methods.

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