Cramer-type Moderate Deviations Research Articles

Self-normalized Cramér-type Moderate Deviations for Functionals of Markov Chain

Let xn, n ≥ 0} be a Markov chain with a countable state space S and let f(·) be a measurable function from S to ℝ and consider the functionals of the Markov chain yn:= f(xn). We construct a new type of self-normalized sums based on the random-block scheme and establish a Cramer-type moderate deviations for self-normalized sums of functionals of the Markov chain.

Asymptotic behaviors of semidefinite programming with a covariance perturbation

In this paper, we study asymptotic behaviors of semidefinite programming with a covariance perturbation. We obtain some moderate deviations, Cramer-type moderate deviations and a law of the iterated logarithm of estimates of the respective optimal value and optimal solutions when the covariance matrix is estimated by its sample covariance. As an example, we also apply the main results to the Minimum Trace factor Analysis.

Moderate deviations and nonparametric inference for monotone functions

This paper considers self-normalized limits and moderate deviations of nonparametric maximum likelihood estimators for monotone functions. We obtain their self-normalized Cramer-type moderate deviations and limit distribution theorems for the nonparametric maximum likelihood estimator in the current status model and the Grenander-type estimator. As applications of the results, we present a new procedure to construct asymptotical confidence intervals and asymptotical rejection regions of hypothesis testing for monotone functions. The theoretical results can guarantee that the new test has the probability of type II error tending to 0 exponentially. Simulation studies also show that the new nonparametric test works well for the most commonly used parametric survival functions such as exponential and Weibull survival distributions.

Detecting rare and faint signals via thresholding maximum likelihood estimators

Motivated by the analysis of RNA sequencing (RNA-seq) data for genes differentially expressed across multiple conditions, we consider detecting rare and faint signals in high-dimensional response variables. We address the signal detection problem under a general framework, which includes generalized linear models for count-valued responses as special cases. We propose a test statistic that carries out a multi-level thresholding on maximum likelihood estimators (MLEs) of the signals, based on a new Cramer-type moderate deviation result for multidimensional MLEs. Based on the multi-level thresholding test, a multiple testing procedure is proposed for signal identification. Numerical simulations and a case study on maize RNA-seq data are conducted to demonstrate the effectiveness of the proposed approaches on signal detection and identification.

Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory

In this paper, we consider the asymptotic behavior of Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramer type moderate deviation principle (MDP) for Xfn(n) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramer type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case (which means fn has a particular form f(x)I(x≥θn) where θn is close to the right edge of equilibrium measure and f is a smooth function), Xfn(n) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively.

Asymptotic behavior for log-determinants of several non-Hermitian random matrices

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A-1B; where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramer type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

Self-normalized Cramér-type moderate deviations under dependence

We establish a Cramer-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 three types of self-normalization: the equal-block scheme, the big-block-small-block scheme and the interlacing 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 ultra-high dimensional time series mean vectors.

From Stein identities to moderate deviations

Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.

Cramér type moderate deviations for Studentized U-statistics

Let T n be a Studentized U-statistic. It is proved that a Cramer type moderate deviation P (T n ≥ x )/(1 − Φ(x )) → 1 holds uniformly in x ∈ [0, o (n 1/6 )) when the kernel satisfies some regular conditions.