We consider the problem of parameter estimation for the superposition of square-root diffusions. We first derive the explicit formulas for the moments and auto-covariances based on which we develop our moment estimators. We then establish a central limit theorem for the estimators with the explicit formulas for the asymptotic covariance matrix. Finally, we conduct numerical experiments to validate our method.

The electromagnetic scattering problem over a wide incident angle can be rapidly solved by introducing the compressive sensing theory into the method of moments, whose main computational complexity is comprised of two parts: a few calculations of matrix equations and the recovery of original induced currents. To further improve the method, a novel construction scheme of measurement matrix is proposed in this paper. With the help of the measurement matrix, one can obtain a sparse sensing matrix, and consequently the computational cost for recovery can be reduced by at least half. The scheme is described in detail, and the analysis of computational complexity and numerical experiments are provided to demonstrate the effectiveness.

Single-particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution three-dimensional (3D) structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3D reconstruction, known as Kam’s method, relies on the moments of the two-dimensional (2D) images. Inspired by Kam’s method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate the uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

In the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

This paper studies the parameter estimation for Ornstein–Uhlenbeck stochastic volatility models driven by Lévy processes. We propose computationally efficient estimators based on the method of moments that are robust to model misspecification. We develop an analytical framework that enables closed-form representation of model parameters in terms of the moments and autocorrelations of observed underlying processes. Under moderate assumptions, which are typically much weaker than those for likelihood methods, we prove large-sample behaviors for our proposed estimators, including strong consistency and asymptotic normality. Our estimators obtain the canonical square-root convergence rate and are shown through numerical experiments to outperform likelihood-based methods.

An integral equation-fast Fourier transform (IE-FFT) algorithm is applied to the electromagnetic solutions of the combined field integral equation (CFIE) for scattering problems by an arbitrary-shaped three-dimensional perfect electric conducting object. The IE-FFT with CFIE uses a Cartesian grid for known Green's function to considerably reduce memory storage and speed up CPU time for both matrix fill-in and matrix vector multiplication when used with a generalized minimal residual method. The uniform interpolation of the Green's function on an equally spaced Cartesian grid allows a global FFT for field interaction terms. However, the near interaction terms do not take care for the singularity of the Green's function and should be adequately corrected. The IE-FFT with CFIE does not always require a suitable preconditioner for electrically large problems. It is shown that the complexity of the IE-FFT with CFIE is found to be approximately O(N1.5) and O(N1.5log N) for memory and CPU time, respectively.

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$, and let $G(n)$ denote the number of subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdős–Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.

We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH model. We calculate higher-order moments and extend the first-jump approximation. These results are prerequisites for moment estimation and pseudo maximum likelihood estimation of the GJR-COGARCH model parameters, respectively, which we derive in detail.

A novel coplanar waveguide fed Industrial, Scientific, and Medical (ISM) band implantable crossed-type triangular slot antenna is proposed for biomedical applications. The antenna operates at the center frequency of 2450 MHz, which is in ISM band, to support GHz wideband communication for high-data rate implantable biomedical application. The size of the antenna is 78 mm3 (10 mm × 12 mm × 0.65 mm). The simulated and measured bandwidths are 7.9 and 8.2% at the resonant frequency of 2.45 GHz. The specific absorption rate distribution induced by the implantable antenna inside a human body tissue model is evaluated. The communication between the implanted antenna and external device is also examined. The proposed antenna has substantial merits such as miniaturization, lower return loss, better impedance matching, and high gain over other implanted antennas.

This paper has as its main theme the fitting in practice of the variance-gamma distribution, which allows for skewness, by moment methods. This fitting procedure allows for possible dependence of increments in log returns, while retaining their stationarity. It is intended as a step in a partial synthesis of some ideas of Madan, Carr and Chang (1998) and of Heyde (1999). Standard estimation and hypothesis-testing theory depends on a large sample of observations which are independently as well as identically distributed and consequently may give inappropriate conclusions in the presence of dependence.

The tapered (or generalized) Pareto distribution, also called the modified Gutenberg-Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.

A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d. The distribution of the number of intersections and approximate moments for large l are derived. The distribution is shown to converge weakly to an arc sine law as l/d→∞.