For a one-dimensional simple symmetric random walk , an edge x (between points and x) is called a favorite edge at time n if its local time at n achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner (Comb Probab Comput 6:359–369, 1997), and Ding and Shen (Ann Probab 46:2545–2561, 2018), disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner (1997).
In this paper, we present new variants of both the de Casteljau subdivision algorithm for curves and Doo–Sabin subdivision algorithm for surfaces. Our subdivision schemes are built on nonlinear weighted averaging rules which are induced by monotonic functions. These averaging rules are used instead of midpoint averaging rule in the mentioned well-known subdivision algorithms. The analysis shows that the smoothness of the subdivision schemes for curves is inherited from the smoothness of the function which induces the averaging rule used in the refinement of the schemes. The results show that with our subdivision schemes, both convex surfaces and concave surfaces can be generated by the same scheme. This happens by only interchanging the weights of two adjacent points when computing the edge points in the subdivision refinement. This is an advantage since a designer can adjust the limit shape according to his interests.
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.
This paper studies the optimal investment problem with an option compensation scheme under rank-dependent expected utilities. Due to the presence of distortion functions and a nonconcave actual utility function, the conventional optimization tools like convex optimization and dynamic programming cannot be applied to this model. To address this challenge, a solution scheme for this nonconvex optimization problem and a procedure for fully solving this problem are proposed and demonstrated. We give explicit forms of optimal policies for a hyperbolic absolute risk-averse (HARA) manager assuming typical types of distortion functions. Numerical analysis illustrates how incentive fee rates and probability distortion influence the optimal investment policies of the fund manager. We find that under a not bad performance, (1) the increase of incentives reduces the asset volatility, (2) an increasing incentive fee rate results in a decreasing probability of bankruptcy, (3) a high risk-seeking degree leads to a high return and a high bankruptcy probability, and (4) a high risk-aversion degree reduces the bankruptcy probability and the asset volatility.
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.
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 -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.
Let be some partition of the set of all primes and G a finite group. A subgroup A of G is -permutable in G provided G is -full; that is, G has a Hall -subgroup for all and A permutes with all such Hall subgroups H of G; that is, . Answering the Question 6.4 in Skiba (Probl Phys Math Tech 42(21):89–96, 2014), we get a description of finite -full groups G in which -permutability is a transitive relation.
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.