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The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where $P$ is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of $P$. This provides some theoretical justification for the use of such estimators for cluster analysis in case that $P$ has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.

We present a space-time ultra-weak discontinuous Galerkin discretization of the linear Schr\"odinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal $h$-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

We unify Ryser's and Glynn's formulas for computing the permanent into a single framework. We then show via an orbital bound argument that the product rank of the permanent is asymptotically upper bounded by $ \frac{\exp\left(\pi\sqrt{\frac{2n}{3}}\right)}{4\sqrt{3}n} $.

In this paper, we develop an arbitrary-order locking-free enriched Galerkin method for the linear elasticity problem using the stress-displacement formulation in both two and three dimensions. The method is based on the mixed discontinuous Galerkin method in [30], but with a different stress approximation space that enriches the arbitrary order continuous Galerkin space with some piecewise symmetric-matrix valued polynomials. We prove that the method is well-posed and provide a parameter-robust error estimate, which confirms the locking-free property of the EG method. We present some numerical examples in two and three dimensions to demonstrate the effectiveness of the proposed method.

We present the new Orthogonal Polynomials Approximation Algorithm (OPAA), a parallelizable algorithm that solves two problems from a functional analytic approach: first, it finds a smooth functional estimate of a density function, whether it is normalized or not; second, the algorithm provides an estimate of the normalizing weight. In the context of Bayesian inference, OPAA provides an estimate of the posterior function as well as the normalizing weight, which is also known as the evidence. A core component of OPAA is a special transform of the square root of the joint distribution into a special functional space of our construct. Through this transform, the evidence is equated with the $L^2$ norm of the transformed function, squared. Hence, the evidence can be estimated by the sum of squares of the transform coefficients. The computations can be parallelized and completed in one pass. To compute the transform coefficients, OPAA proposes a new computational scheme leveraging Gauss--Hermite quadrature in higher dimensions. Not only does it avoid the potential high variance problem associated with random sampling methods, it also enables one to speed up the computation by parallelization, and significantly reduces the complexity by a vector decomposition.

We consider the general problem of Bayesian binary regression and we introduce a new class of distributions, the Perturbed Unified Skew Normal (pSUN, henceforth), which generalizes the Unified Skew-Normal (SUN) class. We show that the new class is conjugate to any binary regression model, provided that the link function may be expressed as a scale mixture of Gaussian densities. We discuss in detail the popular logit case, and we show that, when a logistic regression model is combined with a Gaussian prior, posterior summaries such as cumulants and normalizing constants can be easily obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general priors, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the p > n case, the proposed methodology shows better performances - both in terms of mixing and accuracy - compared to the existing methods. We illustrate the performance through several simulation studies and two data analyses.

We propose and analyze a finite element method for the Oseen eigenvalue problem. This problem is an extension of the Stokes eigenvalue problem, where the presence of the convective term leads to a non-symmetric problem and hence, to complex eigenvalues and eigenfunctions. With the aid of the compact operators theory, we prove that for inf-sup stable finite elements the convergence holds and hence, error estimates for the eigenvalues and eigenfunctions are derived. We also propose an a posteriori error estimator which results to be reliable and efficient. We report a series of numerical tests in two and three dimension in order to assess the performance of the method and the proposed estimator.

We investigate the so-called "MMSE conjecture" from Guo et al. (2011) which asserts that two distributions on the real line with the same entropy along the heat flow coincide up to translation and symmetry. Our approach follows the path breaking contribution Ledoux (1995) which gave algebraic representations of the derivatives of said entropy in terms of multivariate polynomials. The main contributions in this note are (i) we obtain the leading terms in the polynomials from Ledoux (1995), and (ii) we provide new conditions on the source distributions ensuring the MMSE conjecture holds. As illustrating examples, our findings cover the cases of uniform and Rademacher distributions, for which previous results in the literature were inapplicable.

In the realm of the Internet of Things (IoT), deploying deep learning models to process data generated or collected by IoT devices is a critical challenge. However, direct data transmission can cause network congestion and inefficient execution, given that IoT devices typically lack computation and communication capabilities. Centralized data processing in data centers is also no longer feasible due to concerns over data privacy and security. To address these challenges, we present an innovative Edge-assisted U-Shaped Split Federated Learning (EUSFL) framework, which harnesses the high-performance capabilities of edge servers to assist IoT devices in model training and optimization process. In this framework, we leverage Federated Learning (FL) to enable data holders to collaboratively train models without sharing their data, thereby enhancing data privacy protection by transmitting only model parameters. Additionally, inspired by Split Learning (SL), we split the neural network into three parts using U-shaped splitting for local training on IoT devices. By exploiting the greater computation capability of edge servers, our framework effectively reduces overall training time and allows IoT devices with varying capabilities to perform training tasks efficiently. Furthermore, we proposed a novel noise mechanism called LabelDP to ensure that data features and labels can securely resist reconstruction attacks, eliminating the risk of privacy leakage. Our theoretical analysis and experimental results demonstrate that EUSFL can be integrated with various aggregation algorithms, maintaining good performance across different computing capabilities of IoT devices, and significantly reducing training time and local computation overhead.

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump L\'evy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving L\'evy process, including the generalised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure L\'evy process, and to a Brownian-driven SDE in the case of the L\'evy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for L\'evy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM), and sequential Monte Carlo.