Consider $k$ independent random samples from $p$-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of $k$ covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when $k$ and $p$ are fixed integers. Jiang and Yang~\cite{JY13} and Jiang and Qi~\cite{JQ15} have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality $p$ goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either $p$ or $k$ goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for $p$ and $k$. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.
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The covariance of two random variables measures the average joint deviations from their respective means. We generalise this well-known measure by replacing the means with other statistical functionals such as quantiles, expectiles, or thresholds. Deviations from these functionals are defined via generalised errors, often induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation is constructed. Replacing the common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding normalisation, we obtain attainability of the entire interval $[-1, 1]$ for any given marginals. We uncover favourable properties of these new dependence measures. The families of quantile and threshold correlations give rise to function-valued distributional correlations, exhibiting the entire dependence structure. They lead to tail correlations, which should arguably supersede the coefficients of tail dependence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The applicability and usefulness of our new dependence measures is illustrated on demographic data from the Panel Study of Income Dynamics.
This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension $d$; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself. As a corollary, we propose a simple stopping rule (applicable to any optimization method) which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.
We consider the problem of comparison-sorting an $n$-permutation $S$ that avoids some $k$-permutation $\pi$. Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak prove that when $S$ is sorted by inserting the elements into the GreedyFuture binary search tree, the running time is linear in the extremal function $\mathrm{Ex}(P_\pi\otimes \text{hat},n)$. This is the maximum number of 1s in an $n\times n$ 0-1 matrix avoiding $P_\pi \otimes \text{hat}$, where $P_\pi$ is the $k\times k$ permutation matrix of $\pi$, $\otimes$ the Kronecker product, and $\text{hat} = \left(\begin{array}{ccc}&\bullet&\\\bullet&&\bullet\end{array}\right)$. The same time bound can be achieved by sorting $S$ with Kozma and Saranurak's SmoothHeap. In this paper we give nearly tight upper and lower bounds on the density of $P_\pi\otimes\text{hat}$-free matrices in terms of the inverse-Ackermann function $\alpha(n)$. \[ \mathrm{Ex}(P_\pi\otimes \text{hat},n) = \left\{\begin{array}{ll} \Omega(n\cdot 2^{\alpha(n)}), & \mbox{for most $\pi$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))\alpha(n)}), & \mbox{for all $\pi$.} \end{array}\right. \] As a consequence, sorting $\pi$-free sequences can be performed in $O(n2^{(1+o(1))\alpha(n)})$ time. For many corollaries of the dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix theory. Our analysis may be useful in analyzing other classes of access sequences on binary search trees.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years. While it seems to defy the conventional wisdom that overfitting leads to poor prediction, recent research reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This renders the Ridgeless interpolator a theoretically tractable proxy that offers useful insights into the mechanisms of modern machine learning methods. This paper takes a different perspective that aims at understanding the precise stochastic behavior of the Ridgeless interpolator as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which plays the role of the prescribed implicit regularization in the context of prediction risk. Our distributional characterizations hold for general random designs and extend uniformly to positively regularized Ridge estimators. As a demonstration of the analytic power of these characterizations, we derive approximate formulae for a general class of weighted $\ell_q$ risks for Ridge(less) estimators that were previously available only for $\ell_2$. Our theory also provides certain further conceptual reconciliation with the conventional wisdom: given any data covariance, a certain amount of regularization in Ridge regression remains beneficial for `most' signals across various statistical tasks including prediction, estimation and inference, as long as the noise level is non-trivial. Surprisingly, optimal tuning can be achieved simultaneously for all the designated statistical tasks by a single generalized or $k$-fold cross-validation scheme, despite being designed specifically for tuning prediction risk.
Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. In this paper, we propose a non-classical parametrization for density estimation using sample moments, which does not require the choice of such functions. The parametrization is induced by the squared Hellinger distance, and the solution of it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and can be obtained by convex optimization. Statistical properties of the density estimator, together with an asymptotic error upper bound are proposed for the estimator by power moments. Applications of the proposed density estimator in signal processing tasks are given. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. To the best of our knowledge, the proposed estimator is the first one in the literature for which the power moments up to an arbitrary even order exactly match the sample moments, while the true density is not assumed to fall within specific function classes.
In this paper, we study the problems of detection and recovery of hidden submatrices with elevated means inside a large Gaussian random matrix. We consider two different structures for the planted submatrices. In the first model, the planted matrices are disjoint, and their row and column indices can be arbitrary. Inspired by scientific applications, the second model restricts the row and column indices to be consecutive. In the detection problem, under the null hypothesis, the observed matrix is a realization of independent and identically distributed standard normal entries. Under the alternative, there exists a set of hidden submatrices with elevated means inside the same standard normal matrix. Recovery refers to the task of locating the hidden submatrices. For both problems, and for both models, we characterize the statistical and computational barriers by deriving information-theoretic lower bounds, designing and analyzing algorithms matching those bounds, and proving computational lower bounds based on the low-degree polynomials conjecture. In particular, we show that the space of the model parameters (i.e., number of planted submatrices, their dimensions, and elevated mean) can be partitioned into three regions: the impossible regime, where all algorithms fail; the hard regime, where while detection or recovery are statistically possible, we give some evidence that polynomial-time algorithm do not exist; and finally the easy regime, where polynomial-time algorithms exist.
The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson's equation and the linear elasticity equations.
Entangled states shared among distant nodes are frequently used in quantum network applications. When quantum resources are abundant, entangled states can be continuously distributed across the network, allowing nodes to consume them whenever necessary. This continuous distribution of entanglement enables quantum network applications to operate continuously while being regularly supplied with entangled states. Here, we focus on the steady-state performance analysis of protocols for continuous distribution of entanglement. We propose the virtual neighborhood size and the virtual node degree as performance metrics. We utilize the concept of Pareto optimality to formulate a multi-objective optimization problem to maximize the performance. As an example, we solve the problem for a quantum network with a tree topology. One of the main conclusions from our analysis is that the entanglement consumption rate has a greater impact on the protocol performance than the fidelity requirements. The metrics that we establish in this manuscript can be utilized to assess the feasibility of entanglement distribution protocols for large-scale quantum networks.
In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalised shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For the high-frequency data contaminated by microstructure noise, we introduce a localised pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario, and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine the kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.
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