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Concentration inequalities form an essential toolkit in the study of high-dimensional statistical methods. Most of the relevant statistics literature in this regard is, however, based on the assumptions of sub-Gaussian or sub-exponential random variables/vectors. In this paper, we first bring together, through a unified exposition, various probabilistic inequalities for sums of independent random variables under much more general exponential type (namely sub-Weibull) tail assumptions. These results extract a part sub-Gaussian tail behavior of the sum in finite samples, matching the asymptotics governed by the central limit theorem, and are compactly represented in terms of a new Orlicz quasi-norm—the Generalized Bernstein–Orlicz norm—that typifies such kind of tail behaviors. We illustrate the usefulness of these inequalities through the analysis of four fundamental problems in high-dimensional statistics. In the first two problems, we study the rate of convergence of the sample covariance matrix in terms of the maximum elementwise norm and the maximum |$k$| -sub-matrix operator norm that are key quantities of interest in bootstrap procedures and high-dimensional structured covariance matrix estimation, as well as in high-dimensional and post-selection inference. The third example concerns the restricted eigenvalue condition, required in high-dimensional linear regression, which we verify for all sub-Weibull random vectors through a unified analysis, and also prove a more general result related to restricted strong convexity in the process. In the final example, we consider the Lasso estimator for linear regression and establish its rate of convergence to be generally |$\sqrt{k\log p/n}$|⁠ , for |$k$| -sparse signals, under much weaker than usual tail assumptions (on the errors as well as the covariates), while also allowing for misspecified models and both fixed and random design. To our knowledge, these are the first such results for Lasso obtained in this generality. The common feature in all our results over all the examples is that the convergence rates under most exponential tails match the usual (optimal) ones obtained under sub-Gaussian assumptions. Finally, we also establish some complementary results on analogous tail bounds for the suprema of empirical processes indexed by sub-Weibull variables. All our results are finite samples. [ABSTRACT FROM AUTHOR]

A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distributionfree estimators shares a shrinkage-type form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed-form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable signal-to-interference-plus-noise ratio (SINR) both for spatial and Doppler processing scenarios assuming different data statistical characterizations. The results show that interesting SINR improvementswith respect to some counterparts available in the open literature can be achieved especially in training starved regimes. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of estimating the disturbance covariance matrix for radar signal processing applications, when a limited number of training data is present. We determine the maximum likelihood (ML) estimator of the covariance matrix starting from a set of secondary data, assuming a special covariance structure (i.e., the sum of a positive semi-definite matrix plus a term proportional to the identity), and a condition number upper-bound constraint. We show that the formulated constrained optimization problem falls within the class of MAXDET problems and develop an efficient procedure for its solution in closed form. Remarkably, the computational complexity of the algorithm is of the same order as the eigenvalue decomposition of the sample covariance matrix. At the analysis stage, we assess the performance of the proposed algorithm in terms of achievable signal-to-interference-plus-noise ratio (SINR) both for a spatial and a Doppler processing. The results show that interesting SINR improvements, with respect to some existing covariance matrix estimation techniques, can be achieved. [ABSTRACT FROM AUTHOR]