In this paper, we consider a random matrix of the form D-n circle dot X-n, a variance profile random matrix with variance profile D-n circle dot D-n where circle dot denotes the Hadamard product of the two matrices, D-n is a deterministic matrix and X-n is a random matrix. We call D-n circle dot X-n a missing data matrix of X-n when the entries of D-n are either 0 or 1. This framework is commonly used in various applied fields, such as biology, neuroscience and network data analysis. We study the convergence and asymptotic freeness of missing data matrices of iid, elliptic and covariance random matrices. Specifically, it is known that independent iid, elliptic and covariance matrices converge to freely independent circular, elliptic and Mar & ccaron;enko-Pastur variables, respectively. In this paper, we provide the necessary and sufficient conditions on deterministic matrices D-n for which these results to hold true for independent missing data matrices of these three types of random matrices.
This paper investigates the spectral properties of spatial-sign covariance matrices, a self-normalized version of sample covariance matrices, for data from alpha-regularly varying populations with general covariance structures. By exploiting the elegant properties of self-normalized random variables, we establish the limiting spectral distribution and a central limit theorem for linear spectral statistics. We demonstrate that the Marcenko-Pastur equation holds under the condition alpha >= 2, while the central limit theorem for linear spectral statistics is valid for alpha > 4, which are shown to be nearly the weakest possible conditions for spatial-sign covariance matrices from heavy-tailed data in the presence of dependence.
Conformal prediction (CP) offers a principled framework for quantifying predictive uncertainty with finite-sample coverage guarantees. However, when the calibration data are limited, the coverage of CP sets can deviate substantially from the nominal target. This paper introduces Enhanced Conformal Prediction (ECP), a new framework that incorporates abundant but potentially corrupted auxiliary data to recalibrate prediction sets by left-shifting the score thresholds derived from the clean calibration set, provably reducing set size while preserving finite-sample coverage guarantees. Further theoretical analysis demonstrates that ECP achieves a higher breakdown point than existing methods. Extensive experiments confirm the robustness and efficiency of ECP across a variety of settings.
In this paper, we study the spectral properties of a class of random matrices of the form Sn- = n(-1)(X 1X2 & lowast;- X 2X1 & lowast;), where X-k = Sigma(1/2)(k)Z( k), Z(k)'s are independent p x n complex-valued random matrices, and Sigma(k) are p x p positive semi-definite matrices that commute and are independent of the Z(k)'s for k = 1, 2. We assume that Z(k)'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix S(n)(- )will be referred to as a random commutator matrix associated with the samples X-1 and X-2. We show that, when the dimension p and sample size n increase simultaneously, so that p/n -> c is an element of (0,infinity), there exists a limiting spectral distribution (LSD) for S-n(-), supported on the imaginary axis, under the assumptions that the joint spectral distribution of Sigma(1), Sigma(2) converges weakly. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a system of Mar & ccaron;enko-Pastur-type functional equations. Moreover, we show that the companion matrix Sn+ = n(-1)(X X-1(2)& lowast; + X 2X1 & lowast;), under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.
In this paper, we use dimensional reduction technique to study the central limit theory (CLT) of Hotelling's T-2 statistic. Specifically, we use a matrix denoted by U-pxq, to map q-dimensional sample vectors to a p-dimensional subspace, where q >= p or q >> p. Under the condition of p/n -> 0 as (p,n) ->infinity, we obtain the CLT of Hotelling's T-2 statistic. Moreover, the selection of U-pxq is also considered, which demonstrates good performance in the simulation studies.
