In this article, we study a robust estimation method for a general class of integer-valued time series models. The conditional distribution of the process belongs to a broad class of distributions and unlike the classical autoregressive framework, the conditional mean of the process also depends on some exogenous covariates. We derive a robust inference procedure based on the minimum density power divergence. Under certain regularity conditions, we establish that the proposed estimator is consistent and asymptotically normal. In the case where the conditional distribution belongs to the exponential family, we provide sufficient conditions for the existence of a stationary and ergodic $\tau $-weakly dependent solution. Simulation experiments are conducted to illustrate the empirical performances of the estimator. An application to the number of transactions per minute for the stock Ericsson B is also provided.

In order to improve the efficiency of heart valve simulation, we proposed a fast isogeometric simulation approach for time-dependent heart valve simulation algorithm with the idea of Geometric-Independent Field approximation (GIFT for short). For the solution of the blood flow field problem in a heart valve, the fluid background mesh is first simplified, then a Bézier tetrahedral mesh is generated based on the simplified mesh to maintain geometric precision, and finally, the fluid velocity field and pressure are solved. In addition, the GIFT idea is used to represent the geometry of computational domain geometry and approximate the physical field solution with different basis function spaces to obtain the numerical solution with the same precision as before simplification. In the structural mechanics simulation of valve leaflets, NURBS surfaces are used to represent the geometric model. To avoid degeneration on geometric boundary, a single leaflet geometric patch is subdivided into four patches. The immersion geometry strategy is adopted in solving the deformation problem of cardiac valve leaflets to achieve high simulation precision, and the dynamic augmented Lagrangian algorithm is used to couple fluid–structure control equations. For the time discretization, the generalized $\alpha $ method is used to control high-frequency dissipation. Numerical examples and comparisons with previous methods are also presented. The proposed algorithm can reduce the computing costs by about 54.3%, which proves the effectiveness of the proposed method.

Multivariate regression models have been extensively studied in the literature and applied in practice. It is not unusual that some predictors may make the same nonnull contributions to all the elements of the response vector, especially when the number of predictors is very large. For convenience, we call the set of such predictors as the homogeneity set. In this paper, we consider a sparse high-dimensional multivariate generalized linear models with coexisting homogeneity and heterogeneity sets of predictors, which is very important to facilitate the understanding of the effects of different types of predictors as well as improvement on the estimation efficiency. We propose a novel adaptive regularized method by which we can easily identify the homogeneity set of predictors and investigate the asymptotic properties of the parameter estimation. More importantly, the proposed method yields a smaller variance for parameter estimation compared to the ones that do not consider the existence of a homogeneity set of predictors. We also provide a computational algorithm and present its theoretical justification. In addition, we perform extensive simulation studies and present real data examples to demonstrate the proposed method.

In this paper, we study the linear independence between the distribution of the number of prime factors of integers and that of the largest prime factors of integers. Under a restriction on the largest prime factors of integers, we will refine the Erdős–Kac Theorem and Loyd’s recent result on Bergelson and Richter’s dynamical generalizations of the Prime Number Theorem, respectively. At the end, we will show that the analogue of these results holds with respect to the Erdős–Pomerance Theorem as well.

Given integers m and f, let $S_n(m,f)$ be the set consisting of all integers e such that every n-vertex graph with e edges contains an m-vertex induced subgraph with f edges, and let $\sigma (m,f)=\limsup _{n\rightarrow \infty } |S_n(m,f)|/\left( {\begin{array}{c}n\\ 2\end{array}}\right) $. As a natural extension of an extremal problem of Erdős, this was investigated by Erdős, Füredi, Rothschild and Sós 20 years ago. Their main result indicates that integers in $S_n(m,f)$ are rare for most pairs (m, f), though they also found infinitely many pairs (m, f) whose $\sigma (m,f)$ is a fixed positive constant. Here we aim to provide some improvements on this study. Our first result shows that $\sigma (m,f)\le \frac{1}{2}$ holds for all but finitely many pairs (m, f) and the constant $\frac{1}{2}$ cannot be improved. This answers a question of Erdős et al. Our second result considers infinitely many pairs (m, f) of special forms, whose exact values of $\sigma (m,f)$ were conjectured by Erdős et al. We partially solve this conjecture (only leaving two open cases) by making progress on some constructions which are related to number theory. Our proofs are based on the research of Erdős et al. and involve different arguments in number theory. We also discuss some related problems.

For any finitely generated unital commutative associative algebra $\mathcal {R}$ over $\mathbb {C}$ and any complex finite-dimensional simple Lie algebra $\mathfrak {g}$ with a fixed Cartan subalgebra $\mathfrak {h}$, we classify all $\mathfrak {g}\otimes \mathcal {R}$-modules on $U(\mathfrak {h})$ such that $\mathfrak {h}$ as a subalgebra of $\mathfrak {g}\otimes \mathcal {R}$, acts on $U(\mathfrak {h})$ by the multiplication. We construct these modules explicitly and study their module structures.

Expected shortfall (ES), which conveys information regarding potential exceedances beyond the value-at-risk (VaR), is an important measure to characterize the properties of the tails of distribution. In this article, we study two two-step estimation procedures for ES regression with censored responses. Considering the potential dependence between the censoring variable and the covariates, two locally weighted estimation algorithms are proposed based on local Kaplan–Meier estimation and the joint elicitability of VaR and ES. The potential applications of this work are manifold, especially survival analysis, pharmacodynamic analysis, and sociological investigations. The resulting estimators are shown to be consistent. Extensive simulations demonstrate that the proposed method performs quite well in finite samples, with regard to estimation bias and mean squared errors. Last, the analysis of a real dataset illustrates the usefulness of our developed methodologies.

The online version contains supplementary material available at https://doi.org/10.1007/s40304-023-00357-3.

Data with missing values are often obtained using multivariate statistical analyses. It is crucial to study how to estimate parameters and test hypotheses using such data. There exists a step monotone incomplete sample as a simple model of data, which includes such missing values. In this study, we derive the asymptotic distribution of the estimator for the correlation matrix and propose a hypothesis testing method for it in a three-step monotone incomplete sample. Further, we investigate the accuracy of our results by numerical simulation.