关于ord_a(b~n±1)的上界

设ａ ,ｂ 是互素的正整数,ａ ＞ １,ｂ＞ １ ．本文给出了ｏｒｄａ（ ｂｎ ±１） 的最好的上界：设ν是使得ａα｜ｂν±１ 　及　ｇｃｄ（ａ ,ｂυ±１ａα） ＝１成立的最小正整数,而α≥２ 是一个整数．令ｃａ ,ｂ ＝ ａα／υ,则对于任何正整数ｎ,都有ｏｒｄａ（ｂｎ±１） ≤ｌｏｇ（ Ｃａ ,ｂｎ）ｌｏｇａ ．     

The Effect of Treatment Compliance in a Placebo-controlled Trial: Regression with Unpaired Data

The causal effect of a treatment is estimated at different levels of treatment compliance, in a placebo-controlled trial on the reduction of blood pressure. The structural nested mean model makes no direct assumptions on selected treatment compliance levels and placebo prognosis but relies on the randomization assumption and a parametric form for causal effects. It can be seen as a regression model for unpaired data, where pre- and post-randomization covariables are treated differently. The causal parameters are found as solutions to estimating equations involving estimated placebo response and treatment compliance based on base-line covariates for all subjects. Our example considers a linear effect of the percentage of prescribed dose taken on achieved diastolic blood pressure reduction. We propose an exploration of structural model checks. In the example, this reveals an interaction between the causal effect of active dose taken and the base-line body weight of the patient.     

Persistence and existence of stationary measures for a logistic growth model with predation

We consider a stochastic logistic growth model with a predation term, and a diffusive stochastic part with a power-type coefficient. We provide criteria for the persistence of the population and for the existence and uniqueness of a stationary measure. Furthermore, we perform a detailed study of the densities of the stationary measures resorting to the forward Kolmogorov equation. We compile our results in a stochastic bifurcation diagram, drawing comparisons with the corresponding deterministic model.     

Efficient Computation of Reduced Regression Models

We consider settings where it is of interest to fit and assess regression submodels that arise as various explanatory variables are excluded from a larger regression model. The larger model is referred to as the full model; the submodels are the reduced models. We show that a computationally efficient approximation to the regression estimates under any reduced model can be obtained from a simple weighted least squares (WLS) approach based on the estimated regression parameters and covariance matrix from the full model. This WLS approach can be considered an extension to unbiased estimating equations of a first-order Taylor series approach proposed by Lawless and Singhal. Using data from the 2010 Nationwide Inpatient Sample (NIS), a 20% weighted, stratified, cluster sample of approximately 8 million hospital stays from approximately 1000 hospitals, we illustrate the WLS approach when fitting interval censored regression models to estimate the effect of type of surgery (robotic versus nonrobotic surgery) on hospital length-of-stay while adjusting for three sets of covariates: patient-level characteristics, hospital characteristics, and zip-code level characteristics. Ordinarily, standard fitting of the reduced models to the NIS data takes approximately 10 hours; using the proposed WLS approach, the reduced models take seconds to fit.     

Stability in distribution of Markov-modulated stochastic differential delay equations with reflection

In this paper, we discuss Markov-modulated stochastic differential delay equations with reflection. The aim of this paper is to extend the stability criterion in distribution as in [Systems and Control Letters Vol 50 (2003) 195–207] to the equations with reflection. Interesting examples are provided to demonstrate not only our theory, but also the importance of Markov-modulating.     

On the Characterization of Copulas by Differential Equations

We study the semigroup action induced by univariate conditioning of copulas. Based on this, we give a new characterization of bivariate copulas in terms of flows generated by solutions of ordinary differential equations with not necessary continuous right side. Several applications, related to concordance ordering of copulas, illustrate the usefulness of this result.     

具有正负项系数的一阶中立型时滞微分方程的振动性

利用时滞微分方程与时滞微分不等式之间的一种等价关系,得到了具有正负项系数的一阶中立型时滞微分方程：d/dt[x（t）-C（t）x（t-r）]＋P（t）x（t-τ）-Q（t）x（t-δ）=0一切解振动的充分条件。结果推广了当Q（t）≡q〉0时的主要结论。     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

Robust statistical inference: weighted likelihoods or usual m-estimation?

We review the weighted likelihood estimating equations methodology introduced by Markatou, Basu and Lindsay (1995). and Basu, Markatou and Lindsay (1995) and compare it, in the case of symmetric and asymmetric contamination, with Huber's M-estimators of location. The simulation study shows that the weighted likelihood estimating equations estimator is at least as competitive as Huber's M-estimators in the case of symmetric contamination. In the case of asymmetric contamination it may be superior than Huber's M-estimators