二阶常微分方程边值问题

本文通过对非线性常微分方程两边边值问题：ｘ〃＝ｆ（ｔ,ｘ,ｘ’）,ｘ（０）＝Ａ,ｘ（１）＝Ｂ的解的先验估计,并利用Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ度理论讨论了解的存在性及唯一性。     

Likelihood Factorizations for Mixed Discrete and Continuous Variables

Some general remarks are made about likelihood factorizations, distinguishing parameter-based factorizations and concentration-graph factorizations. Two parametric families of distributions for mixed discrete and continuous variables are discussed. Conditions on graphs are given for the circumstances under which their joint analysis can be split into separate analyses, each involving a reduced set of component variables and parameters. The result shows marked differences between the two families although both involve the same necessary condition on prime graphs. This condition is both necessary and sufficient for simplified estimation in Gaussian and for discrete log linear models.     

Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarev's ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n^−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 (k?3) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.     

Combining Multivariate Bioassays: Accurate Inference Using Small Sample Asymptotics

For several independent multivariate bioassays performed at different laboratories or locations, the problem of testing the homogeneity of the relative potencies is addressed, assuming the usual slope‐ratio or parallel line assay model. When the homogeneity hypothesis holds, interval estimation of the common relative potency is also addressed. These problems have been investigated in the literature using likelihood‐based methods, under the assumption of a common covariance matrix across the different studies. This assumption is relaxed in this investigation. Numerical results show that the usual likelihood‐based procedures are inaccurate for both of the above problems, in terms of providing inflated type I error probabilities for the homogeneity test, and providing coverage probabilities below the nominal level for the interval estimation of the common relative potency, unless the sample sizes are large, as expected. Correction based on small sample asymptotics is investigated in this article, and this provides significantly more accurate results in the small sample scenario. The results are also illustrated with examples.     

On the Mean Value Theorem for Estimating Functions

Abstract

Feng et al. revealed that the usual mean value theorem (MVT) should not be applied directly to a vector-valued function (e.g., the score function or a general estimating function under a multiparametric model). This note shows that the application of the Cramer–Wold’s device to a corrected version of the MVT is sufficient to obtain standard asymptotics for the estimators attained from vector-valued estimating functions.     

Demystifying the Integrated Tail Probability Expectation Formula

ABSTRACT

Calculating the expected values of different types of random variables is a central topic in mathematical statistics. Targeted toward students and instructors in both introductory probability and statistics courses and graduate-level measure-theoretic probability courses, this pedagogical note casts light on a general expectation formula stated in terms of distribution and survival functions of random variables and discusses its educational merits. Often consigned to an end-of-chapter exercise in mathematical statistics textbooks with minimal discussion and presented under superfluous technical assumptions, this unconventional expectation formula provides an invaluable opportunity for students to appreciate the geometric meaning of expectations, which is overlooked in most undergraduate and graduate curricula, and serves as an efficient tool for the calculation of expected values that could be much more laborious by traditional means. For students’ benefit, this formula deserves a thorough in-class treatment in conjunction with the teaching of expectations. Besides clarifying some commonly held misconceptions and showing the pedagogical value of the expectation formula, this note offers guidance for instructors on teaching the formula taking the background of the target student group into account.     

律师拒证权的法经济学分析 ——以证据资源的稀缺性为视角

在刑事诉讼对抗制体系下,控辩双方对证据资源的需求和证据的供给之间发生了紧张关系,使证据成为一种稀缺资源.对于证人证言这样一种公共物品,法律通过设定真相作证义务满足了刑事诉讼对证据的需求.而对于律师基于职业关系所掌握的被告人的秘密信息,如果严格贯彻真相义务要求律师作证,则会产生类似于经济学意义上的外部影响问题.对于外部影响的解决,科斯定理给出了基本思路,即明确被告人对其秘密信息的排他性产权,并设立律师拒证权予以保护.     

A quadratic lower bound for colourful simplicial depth

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in ℝ ^d is contained in at least ⌊(d+2)²/4⌋ simplices with one vertex from each set. Both authors were supported by NSERC Discovery grants. Additionally, T. Stephen was supported by DFG FG-468 and the Dynamical Systems research focus at the University of Magdeburg.     

Generalized anticipated backward stochastic differential equations driven by Brownian motion and continuous increasing process*

This article mainly considers a new class of anticipated BSDEs driven by Brownian motion and continuous increasing process, which are called generalized anticipated backward stochastic differential equations(GABSDEs). We first give the form of GABSDEs. Then, existence and uniqueness result for GABSDEs is established as well as a comparison theorem is obtained under the certain assumptions. At last, we give an application about the duality between SDDEs and GABSDEs.