A Diagnostic Plot for Estimating the Tail Index of a Distribution

The problem of estimating the tail index in heavy-tailed distributions is very important in many applications. We propose a new graphical method that deals with this problem by selecting an appropriate number of upper order statistics. We also investigate the method's theoretical properties are investigated. Several real datasets are analyzed using this new procedure and a simulation study is carried out to examine its performance in small, moderate and large samples. The results suggest that the new procedure overcomes many of the shortcomings present in some of the most common techniques—for example, the Hill and Zipf plots—used in the estimation of the tail index, and it performs very competitively when compared with other adaptive threshold procedures based on the asymptotic mean squared error of the Hill estimator.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

Quasi-Profile Log Likelihoods for Unbiased Estimating Functions

This paper presents a new quasi-profile loglikelihood with the standard kind of distributional limit behaviour, for inference about an arbitrary one-dimensional parameter of interest, based on unbiased estimating functions. The new function is obtained by requiring the corresponding quasi-profile score function to have bias and information bias of order O(1). We illustrate the use of the proposed pseudo-likelihood with an application to robust inference in linear models.     

调和单叶函数族的一些结果

借用著名的Ruscheweyh导数,引入了一类单叶保向调和函数族. 通过建立极值理论,得到了关联该族的最优系数边界、最优增长定理和最优偏差定理. 同时,给出了该族与先前已有调和函数族之间的转换半径. 最后,讨论了基于该族的修正哈达玛乘积结果.     

含有解析函数与亚纯函数的一些不等式

指出文献 [2 ]和文献 [4]中的错误 ,并且得到含有单位圆盘内接解析函数与亚纯函数的几个不等式 .     

On Time-Reversibility and Estimating Functions for Markov Processes

The time-reversibility of a Markov process implies a particular structure of the score function. It is explored which martingale estimating functions and other unbiased estimating functions have a similar structure. This leads to an estimating function with a semiparametric efficiency property. Also relations to martingale estimating functions based on eigenfunctions of the infinitesimal generator are found.     

Some Generalizations of the Notion of Length

Two numerical characteristics of a nonrectifiable arc generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums , where are the lengths of segments of a polygonal line inscribed in the curve and is a given function. On the other hand, the length of is the norm of the functional in the space ; its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve implies its generalized analytic rectifiability.     

The Open-Faced Sandwich Adjustment for MCMC Using Estimating Functions

A situation frequently arises where working with the likelihood function is problematic. This can happen for several reasons—perhaps the likelihood is prohibitively computationally expensive, perhaps it lacks some robustness property, or perhaps it is simply not known for the model under consideration. In these cases, it is often possible to specify alternative functions of the parameters and the data that can be maximized to obtain asymptotically normal estimates. However, these scenarios present obvious problems if one is interested in applying Bayesian techniques. This article describes open-faced sandwich adjustment, a way to incorporate a wide class of nonlikelihood objective functions within Bayesian-like models to obtain asymptotically valid parameter estimates and inference via MCMC. Two simulation examples show that the method provides accurate frequentist uncertainty estimates. The open-faced sandwich adjustment is applied to a Poisson spatio-temporal model to analyze an ornithology dataset from the citizen science initiative eBird. An online supplement contains an appendix with additional figures, tables, and discussion, as well as R code.     

Some Generating Functions for the Generalized Bessel Polynomials

The authors begin by presenting a systematic (historical) account of some linear generating functions for the generalized Bessel polynomials. It is then shown how these linear generating functions can be applied with a view to obtaining various new families of bilinear, bilateral, or mixed multilateral generating functions for the generalized Bessel polynomials. Some interesting generalizations of these classes of generating functions, involving hypergeometric functions, are also considered.     

函数一致连续的比较判别法

在一般教材上对无穷区间上的函数,通常都采用定义的方法判别其一致连续性,对于复杂的函数,判别其是否一致连续一般来说常常比较困难.本文给出了判别无穷区间上函数一致连续性的一种比较判别法.     

解析函数的p叶性条件

本文指出 Ｍ．Ｊａｈａｎｇｉｒｉ的评论［Ｍａｔｈｅｍａｔｉｃａｌ Ｒｅｖｉｅｗｓ　９８ｅ：３００２０］错误，并且导出解析函数ｐ叶星形性与ｐ叶凸性的某些充分条件．     

Some Generalized Problems for Polyharmonic Functions

In the first section of this paper, some non-local boundary value problem for the polyharmonic equation in the plane is considered. This problem consists in determining solution of the polyharmonic equation satisfying some special non-local-type boundary condition on two curves. The existence theorem is proved. In the second section, an example for the case of the biharmonic equation is considered. In the third section, some non-local, non-linear problem of Riquier type is examined. The Riquier-type problem consists in determining the polyharmonic function in the plane whose value together with its successive Laplacians are prescribed on the boundary. The existence theorem is proved and an example for the case of the biharmonic equation is considered.     

On the Equivalence of Some Conditions for Convex Functions

We study classes of convex functions on (1, ∞] that tend to zero at infinity. Relations between different elements of these classes are determined. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 427–431, March, 2005.     

N-Width for Some Classes of Periodic Functions

This paper considers the Kolmogorov width,the Gelfand width and the Linear width of the periodic smooth functions with boundary conditions in Orlicz space.Furth...     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

加权Bergman空间中函数的几个严格不等式——纪念李国平院士吴新谋教授诞辰100周年

该文给出加权Bergman空间中函数的几个严格不等式.     

若干函数不等式探析

针对近几年大学生高等数学竞赛和硕士研究生入学考试中出现的若干有关函数不等式的试题,探讨分析其命题背景,并推广深化相关结果.