Let be the set of linear and even-degree irreducible characters of a finite group G. In this paper, we prove that G has a normal Sylow 2-subgroup if for a positive integer m, which is the generalization of several recent results concerning the well-known Ito–Michler theorem.
The decomposition family of a family of graphs often helps us to determine the error term in the well-known Erdős–Stone–Simonovits theorem. We study the Turán number of families of nonbipartite graphs such that their decomposition families contain a matching and a star. To be precisely, we prove tight bounds on the Turán number of such families of graphs. Moreover, we find a graph which is a counterexample to an old conjecture of Erdős and Simonovits, while all previous counterexamples are families of graphs.
Censored data with functional predictors often emerge in many fields such as biology, neurosciences and so on. Many efforts on functional data analysis (FDA) have been made by statisticians to effectively handle such data. Apart from mean-based regression, quantile regression is also a frequently used technique to fit sample data. To combine the strengths of quantile regression and classical FDA models and to reveal the effect of the functional explanatory variable along with nonfunctional predictors on randomly censored responses, the focus of this paper is to investigate the semi-functional partial linear quantile regression model for data with right censored responses. An inverse-censoring-probability-weighted three-step estimation procedure is proposed to estimate parametric coefficients and the nonparametric regression operator in this model. Under some mild conditions, we also verify the asymptotic normality of estimators of regression coefficients and the convergence rate of the proposed estimator for the nonparametric component. A simulation study and a real data analysis are carried out to illustrate the finite sample performances of the estimators.
The Freidlin–Wentzell’s large deviation principle is established for a class of multi-scale stochastic models involving slow–fast components, where the slow component has general monotone drift and the fast component has dissipative drift driven by multiplicative noise. Our result is applicable to various slow–fast stochastic dynamical systems such as stochastic porous media equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. The weak convergence method and the technique of time discretization are used to establish the Laplace principle (equivalently, large deviation principle) under the multi-scale framework.
Model averaging has been considered to be a powerful tool for model-based prediction in the past decades. However, its application in ultra-high dimensional multi-categorical data is faced with challenges arising from the model uncertainty and heterogeneity. In this article, a novel two-step model averaging method is proposed for multi-categorical response when the number of covariates is ultra-high. First, a class of adaptive multinomial logistic regression candidate models are constructed where different covariates for each category are allowed to accommodate heterogeneity. Second, the optimal model weights is chosen by applying the Kullback–Leibler loss plus a penalty term. We show that the proposed model averaging estimator is asymptotically optimal by achieving the minimum Kullback–Leibler loss among all possible averaging estimators. Empirical evidences from simulation studies and a real data example demonstrate that the proposed model averaging method has superior performance to the state-of-the-art approaches.
Causal inference and missing data have attracted significant research interests in recent years, while the current literature usually focuses on only one of these two issues. In this paper, we develop two multiply robust methods to estimate the quantile treatment effect (QTE), in the context of missing data. Compared to the commonly used average treatment effect, QTE provides a more complete picture of the difference between the treatment and control groups. The first one is based on inverse probability weighting, the resulting QTE estimator is root-n consistent and asymptotic normal, as long as the class of candidate models of propensity scores contains the correct model and so does that for the probability of being observed. The second one is based on augmented inverse probability weighting, which further relaxes the restriction on the probability of being observed. Simulation studies are conducted to investigate the performance of the proposed method, and the motivated CHARLS data are analyzed, exhibiting different treatment effects at various quantile levels.
In this paper, we study continuous frames with symmetries from projective representations of compact groups. In particular, we study maximal spanning vectors in detail and we prove the existence of maximal spanning vectors for irreducible projective representations of compact abelian groups by a dimension counting method.
The optimal stopping problem for pricing Russian options in finance requires taking the supremum of the discounted reward function over all finite stopping times. We assume the logarithm of the asset price is a spectrally negative Markov additive process with finitely many regimes. The reward function is given by the exponential of the running supremum of the price process. Previous work on Russian optimal stopping problem suggests that the optimal stopping time would be an upcrossing time of the drawdown at a certain level for each regime. We derive explicit formulas for identifying the stopping levels and computing the corresponding value functions through a recursive algorithm. A numerical is provided for finding these stopping levels and their value functions.