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.
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 {(Xi, Yi)}ni=1 be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér-type moderate deviation theorem for the general self-normalized sum ∑ni=1 ...Xi/(∑ni=1 Yi2)1/2, which unifies and extends the classical Cramér (Actual. Sci. Ind. 736 (1938) 5–23) theorem and the self-normalized Cramér-type moderate deviation theorems by Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) as well as the further refined version by Wang (J. Theoret. Probab. 24 (2011) 307–329). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramér-type moderate deviation theorems for one-dependent random variables, geometrically β-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramér-type moderate deviation theorems for self-normalized winsorized mean.
Applying the Benjamini and Hochberg (B-H) method to multiple Student's t tests is a popular technique for gene selection in microarray data analysis. Given the nonnormality 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 tn-1 or the bootstrap method to estimate the p-values. In this paper, we prove that when the population has the finite 4th moment and the dimension m and the sample size n satisfy log m = o(n⅓), the B-H method controls the false discovery rate (FDR) and the false discovery proportion (FDP) at a given level α asymptotically with p-values estimated from N(0, 1) or tn-1 distribution. However, a phase transition phenomenon occurs when logm ≥con⅓. In this case, the FDR and the FDP of the B-H method may be larger than α or even converge to one. In contrast, the bootstrap calibration is accurate for log m = o(n½) as long as the underlying distribution has the sub-Gaussian tails. However, such a light-tailed condition cannot generally be weakened. The simulation study shows that the bootstrap calibration is very conservative for the heavy tailed distributions. 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 its usual counterpart.
An exchangeable pair approach is commonly taken in the normal and nonnormal 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 nonnormal 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.
We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô's formula and Markov operator. As applications, we study ...the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler–Maruyama (EM) discretization for multi-dimensional Ornstein–Uhlenbeck stable process and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.
Stein's method has offered a completely novel way of evaluating the quality of normal approximations. This volume contains thorough coverage of the method's fundamentals. It includes a large number ...of recent developments in both theory and applications.
Exceptions to the generality of the stress-gradient hypothesis (SGH) may be reconciled by considering species-specific traits and stress tolerance strategies. Studies have tested stress tolerance and ...competitive ability in mediating interaction outcomes, but few have incorporated this to predict how species interactions shift between competition and facilitation along stress gradients. We used field surveys, salt tolerance and competition experiments to develop a predictive model interspecific interaction shifts across salinity stress gradients. Field survey and greenhouse tolerance tests revealed tradeoffs between stress tolerance and competitive ability. Modeling showed that along salinity gradients, (1) plant interactions shifted from competition to facilitation at high salinities within the physiological limits of salt-intolerant plants, (2) facilitation collapsed when salinity stress exceeded the physiological tolerance of salt-intolerant plants, and (3) neighbor removal experiments overestimate interspecific facilitation by including intraspecific effects. A community-level field experiment, suggested that (1) species interactions are competitive in benign and, facilitative in harsh condition, but fuzzy under medium environmental stress due to niche differences of species and weak stress amelioration, and (2) the SGH works on strong but not weak stress gradients, so SGH confusion arises when it is applied across questionable stress gradients. Our study clarifies how species interactions vary along stress gradients. Moving forward, focusing on SGH applications rather than exceptions on weak or nonexistent gradients would be most productive.
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 X, even when X and Y are independent. When the covariance matrix of 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 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.
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