arXiv subjects
Publications and source records attributed to Qi-Man Shao.
We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained.
Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application.
Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^α\rfloor, 0< α< 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdös-Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramér-type moderate deviations via Stein's method.
Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm.
Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{θ\in Θ} E_θ[\cdot]$ for a given set of linear expectations $\{E_θ: θ\in Θ\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.
Let $(ξ_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^nξ_i $ and $[ S]_n=\sum_{i=1}^n ξ_i^2.$ We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.
An exchangeable pair approach is commonly taken in the normal and non-normal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein's method, a new Berry-Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and non-normal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie-Weiss model, mean field Heisenberg model and colored graph model.
A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student's statistic and autoregressive process are also discussed.
Let ${\bf R}$ be the Pearson correlation matrix of $m$ normal random variables. The Rao's score test for the independence hypothesis $H_0 : {\bf R} = {\bf I}_m$, where ${\bf I}_m$ is the identity matrix of dimension $m$, was first considered by Schott (2005) in the high dimensional setting. In this paper, we study the asymptotic minimax power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Rao's score test is rate-optimal for detecting the dependency signal $\|{\bf R} - {\bf I}_m\|_F$ of order $\sqrt{m/n}$, where $\|\cdot\|_F$ is the matrix Frobenius norm.
Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries from these data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions about the exogeneity of the covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given a certain number of predictors, namely, the distribution of the correlation of a response variable $Y$ with the best $s$ linear combinations of $p$ covariates $\mathbf{X}$, even when $\mathbf{X}$ and $Y$ are independent. When the covariance matrix of $\mathbf{X}$ possesses the restricted eigenvalue property, we derive such distributions for both a finite $s$ and a diverging $s$, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of $\mathbf{X}$. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where the residuals are from regularized fits. Our approach is then used to construct the upper confidence limit for the maximum spurious correlation and to test the exogeneity of the covariates. The former provides a baseline for guarding against false discoveries and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated with both numerical examples and real data analysis.
Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample $t$-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.
Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.
We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider two types of self-normalization: the big-block-small-block scheme and the interlacing or equal-block scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and construction of simultaneous confidence intervals for high-dimensional time series mean vectors.
Let $\mathbf {x}_1,\ldots,\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\to\infty$ and $\log p=o(n^β)$ for some $0<β\leq1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}L_n\to2$ in probability if and only if $Ee^{t_0|x_{11}|^α}<\infty$ for some $t_0>0$, where $α$ satisfies $β=α/(4-α)$. Asymptotic distributions of $L_n$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the population are $m$ dependent. The proofs are based on self-normalized moderate deviations, the Stein-Chen method and a newly developed randomized concentration inequality.
Applying Benjamini and Hochberg (B-H) method to multiple Student's $t$ tests is a popular technique in gene selection in microarray data analysis. Because of the non-normality of the population, the true p-values of the hypothesis tests are typically unknown. Hence, it is common to use the standard normal distribution N(0,1), Student's $t$ distribution $t_{n-1}$ or the bootstrap method to estimate the p-values. In this paper, we first study N(0,1) and $t_{n-1}$ calibrations. We prove that, when the population has the finite 4-th moment and the dimension $m$ and the sample size $n$ satisfy $\log m=o(n^{1/3})$, B-H method controls the false discovery rate (FDR) at a given level $α$ asymptotically with p-values estimated from N(0,1) or $t_{n-1}$ distribution. However, a phase transition phenomenon occurs when $\log m\geq c_{0}n^{1/3}$. In this case, the FDR of B-H method may be larger than $α$ or even tends to one. In contrast, the bootstrap calibration is accurate for $\log m=o(n^{1/2})$ as long as the underlying distribution has the sub-Gaussian tails. However, such light tailed condition can not be weakened in general. The simulation study shows that for the heavy tailed distributions, the bootstrap calibration is very conservative. In order to solve this problem, a regularized bootstrap correction is proposed and is shown to be robust to the tails of the distributions. The simulation study shows that the regularized bootstrap method performs better than the usual bootstrap method.
Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\sum_{i=1}^nX_i$ and $V^2_n=\sum_{i=1}^nX^2_i$. A Cramér type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-Φ(x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.
Poisson approximation using Stein's method has been extensively studied in the literature. The main focus has been on bounding the total variation distance. This paper is a first attempt on moderate deviations in Poisson approximation for right-tail probabilities of sums of dependent indicators. We obtain results under certain general conditions for local dependence as well as for size-bias coupling. These results are then applied to independent indicators, 2-runs, and the matching problem.