This paper develops the theory of the kth power expectile estimation and considers its relevant hypothesis tests for coefficients of linear regression models. We prove that the asymptotic covariance matrix of kth power expectile regression converges to that of quantile regression as k converges to one and hence promise a moment estimator of asymptotic matrix of quantile regression. The kth power expectile regression is then utilized to test for homoskedasticity and conditional symmetry of the data. Detailed comparisons of the local power among the kth power expectile regression tests, the quantile regression test, and the expectile regression test have been provided. When the underlying distribution is not standard normal, results show that the optimal k are often larger than 1 and smaller than 2, which suggests the general kth power expectile regression is necessary. Finally, the methods are illustrated by a real example.

With the help of the local isomorphism between the Hurwitz space $M_{0;k-1,l-k-r+1,r-1}$ and the orbit space $\mathcal {M}_{k,k+r}(A_l)$, we will show the existence of a Frobenius manifold structure on the orbit space $\mathcal {M}_{k,k+r}(A_l)\setminus \Sigma _r$ of the extended affine Weyl group $\widetilde{W}^{(k,k+r)}(A_l)$ for $1\le k<k+r\le l$.

The gap between finite element analysis and computer-aided design derives the development of isogeometric analysis (IGA), which uses the same representation in the geometry and the analysis. However, the parameterization in IGA is non-trivial. Weighted extended B-splines (WEB) method replaces grid generation and parameterization with weight function construction (R-function or distance function). By using implicit spline representation, isogeometric analysis on implicit domains (IGAID) adopts the merits of the “isoparametric” in IGA and “weight function generation” in WEB. But the theoretical properties have not been fully studied yet. In this paper, we study the theoretical aspects of IGAID using tensor-product B-splines. Both the approximation and stability properties of IGAID are considered. By setting appropriate constraints on the weight function, we can derive the optimal approximation order and stability. Numerical examples show the effectiveness of the approach and validate the theoretical results.

Doubly robust (DR) methods that employ both the propensity score and outcome models are widely used to estimate the causal effect of a treatment and generally outperform those methods only using the propensity score or the outcome model. However, without appropriately chosen the working models, DR estimators may substantially lose efficiency. In this paper, based on the augmented inverse probability weighting procedure, we derive a new estimating equation for the causal effect by the strategy of combining estimating equations. The resulting estimator by solving the new estimating equation retains doubly robust and can improve the efficiency under the misspecification of conditional mean working model. We further show the large sample properties of the proposed estimator under some regularity conditions. Through simulation experiments and a real data analysis, we illustrate that the proposed method is competitive with its competitors, which is in line with those implied by the asymptotic theory.

In this paper, we determine the effect of the free multiplicative convolution on the pseudo-variance function of a Cauchy-Stieltjes kernel family. We then use the machinery of variance functions to establish some limit theorems related to this type of convolution and involving the free additive convolution and the Boolean additive convolution.

We construct a Floer type boundary operator for generalised Morse–Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $\mathbb {Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

Let $\Delta (n)$ denote the smallest positive integer m such that $a^3+a(1\leqslant a\leqslant n)$ are pairwise distinct modulo m. The purpose of this paper is to determine $\Delta (n)$ for all positive integers n.

We consider a kind of nonlinear systems on a locally finite graph $G=(V,E)$. We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solution which depends on the parameter $\lambda $ with some suitable assumptions on the potentials. Moreover, we pay attention to the concentration behavior of these solutions and prove that as $\lambda \rightarrow \infty $, these solutions converge to a ground state solution of a corresponding Dirichlet problem. Finally, we also provide some numerical experiments to illustrate our results.