The spectrum of the normalized Laplacian matrix of a graph provides a lot of structural information of the graph, and it has applications in numerous areas and in different guises. In this paper, we completely characterize all connected graphs of order $n\ge 25$ with some normalized Laplacian eigenvalue $\rho \in \big (0,\,\frac{n-1}{n-2}\big )$ having multiplicity $n{-4}$.

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.

In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the $Q_g$ curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics.

Equivalence assessment via various indices such as relative risk has been widely studied in a matched-pair design with discrete or continuous endpoints over the past years. But existing studies mainly focus on the fully observed or missing at random endpoints. Nonignorable missing endpoints are commonly encountered in a matched-pair design. To this end, this paper proposes several novel methods to assess equivalence of two diagnostics via the difference between two correlated areas under ROC curves (AUCs) in a matched-pair design with nonignorable missing endpoints. An exponential tilting model is utilized to specify the nonignorable missing endpoint mechanism. Three nonparametric approaches and three semiparametric approaches are developed to estimate the difference between two correlated AUCs based on the kernel-regression imputation, inverse probability weighted (IPW), and augmented IPW methods. Under some regularity conditions, we show the consistency and asymptotic normality of the proposed estimators. Simulation studies are conducted to study the performance of the proposed estimators. Empirical results show that the proposed methods outperform the complete-case method. An example from clinical studies is illustrated by the proposed methodologies.

In the spatial autologistic model, the dependence parameter is often assumed to be a single value. To construct a spatial autologistic model with spatial heterogeneity, we introduce additional covariance in the dependence parameter, and the proposed model is suitable for the data with binary responses where the spatial dependency pattern varies with space. Both the maximum pseudo-likelihood (MPL) method for parameter estimation and the Bayesian information criterion (BIC) for model selection are provided. The exponential consistency between the maximum likelihood estimator and the maximum block independent likelihood estimator (MBILE) is proved for a particular case. Simulation results show that the MPL algorithm achieves satisfactory performance in most cases, and the BIC algorithm is more suitable for model selection. We illustrate the application of our proposed model by fitting the Bur Oak presence data within the driftless area in the midwestern USA.

Let $\sigma =\{\sigma _{i} \mid i\in I\}$ be some partition of the set of all primes and G a finite group. A subgroup A of G is $\sigma $-permutable in G provided G is $\sigma $-full; that is, G has a Hall $\sigma _{i}$-subgroup for all $i\in I$ and A permutes with all such Hall subgroups H of G; that is, $AH=HA$. Answering the Question 6.4 in Skiba (Probl Phys Math Tech 42(21):89–96, 2014), we get a description of finite $\sigma $-full groups G in which $\sigma $-permutability is a transitive relation.

Conditional quantile regression provides a useful statistical tool for modeling and inferring the relationship between the response and covariates in the heterogeneous data. In this paper, we develop a novel testing procedure for the ultrahigh-dimensional partially linear quantile regression model to investigate the significance of ultrahigh-dimensional interested covariates in the presence of ultrahigh-dimensional nuisance covariates. The proposed test statistic is an $L_2$-type statistic. We estimate the nonparametric component by some flexible machine learners to handle the complexity and ultrahigh dimensionality of considered models. We establish the asymptotic normality of the proposed test statistic under the null and local alternative hypotheses. A screening-based testing procedure is further provided to make our test more powerful in practice under the ultrahigh-dimensional regime. We evaluate the finite-sample performance of the proposed method via extensive simulation studies. A real application to a breast cancer dataset is presented to illustrate the proposed method.

The combined use of two drugs is a major treatment approach for complex diseases such as cancer and HIV due to its potential for efficacy at lower, less toxic doses and the need to reduce developmental time and cost. Experimental designs have been proposed in the literature to test whether there are synergistic or antagonistic actions between the combined drugs. The existing designs for synergy testing are primarily one-dimensional (1D), allocating the doses of one drug while keeping the dose of another, the mixing proportion, or the total dose of the two drugs fixed. This paper considers two-dimensional (2D) designs in which the doses of two drugs can be varied simultaneously over the entire dose region. Based on the premise that prior information about the single-drug experiments is already available, we propose a succinct dose-response model that encompasses a wide class of potential synergistic/antagonistic actions deviated from additivity. We show that the uniform design measure over the 2D dose region is optimal under the proposed model in the sense that it maximizes the minimum power in the F-test to detect drug synergy. Methods for sample size calculation and design generation for our 2D optimal design are given. We illustrate the use of the proposed design and demonstrate its advantages over the 1D optimal design via a combination study of two anticancer drugs.

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.

In the present paper, a scheme of path sampling is explored for stochastic diffusion processes. The core issue is the evaluation of the diffusion propagators (spatial–temporal Green functions) by solving the corresponding Kolmogorov forward equations with Dirac delta functions as initials. The technique can be further used in evaluating general functional of path integrals. The numerical experiments demonstrated that the simulation scheme based on this approach overwhelms the popular Euler scheme and Exact Algorithm in terms of accuracy and efficiency in fairly general settings. An example of likelihood inference for the diffusion driven Cox process is provided to show the scheme’s potential power in applications.