Precise Asymptotics in Complete Moment Convergence of Parameter Estimator in the Gaussian Autoregressive Process

In this article, we study the precise asymptotic behaviors of the least-squares estimator in the Gaussian autoregressive process. Two kinds of complete moment convergence of this estimator can be obtained by the methods of deviation inequalities for this estimator and nonuniform Berry-Esseen bound for martingales.     

Rates of Convergence for a Bayesian Level Set Estimation

Abstract.  We are interested in estimating level sets using a Bayesian non-parametric approach, from an independent and identically distributed sample drawn from an unknown distribution. Under fairly general conditions on the prior, we provide an upper bound on the rate of convergence of the Bayesian level set estimate, via the rate at which the posterior distribution concentrates around the true level set. We then consider, as an application, the log-spline prior in the two-dimensional unit cube. Assuming that the true distribution belongs to a class of Hölder, we provide an upper bound on the rate of convergence of the Bayesian level set estimates. We compare our results with existing rates of convergence in the frequentist non-parametric literature: the Bayesian level set estimator proves to be competitive and is also easy to compute, which is of no small importance. A simulation study is given as an illustration.     

Convergence results for sequences of quadratic forms

Let (XI,)be a sequence of independent random variables, and let Q_n= where for each N,(an:,k)is a doubly indexed sequence of weights. The convergence and the rate of convergence of the sequence of quadratic forms {Q_n} are studied. These quadratic forms are linear sums of dependent variables; however, their convergence properties are similar to those of linear sums of independent variables provided the variables have finite rth absolute moments with 0 < r 2.while the rate of convergence has not been obtained for r< 2, it is shown to be different from that of linear sums.     

On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.     

Rates of Posterior Convergence for iid Observations

By means of the Hausdorff α-entropy introduced by Xing and Ranneby (2009 Xing , Y. , Ranneby , B. ( 2009 ). Sufficient conditions for Bayesian consistency . J. Statist. Plann. Inference. 139 : 2479 – 2489 . [Google Scholar]), we give two theorems on rates of in-probability convergence of posterior distributions. The result is applied in study of the Bernstein polynomial priors.     

On the convergence of moving average processes under dependent conditions

This paper considers a moving average process for a sequence of negatively associated random variables. It discusses the complete convergence of such a moving average process under suitable conditions. These results generalize and complement earlier results on independent random variables. Also, a conjecture for the case of a sequence of independent and identically distributed random variables is resolved and its moment condition weakened.     

幂平均数若干性质之证明

幂平均数作为算术平均数、几何平均数、调和平均数等的一般形式,它与各具体形式之间必存在一定的关系。在目前的一些教材中,这种逻辑关系的描述是不正确的。又由于这些教材对所作的描述并未给予证明,因此,错误被隐藏起来。本文重新描述这种逻辑关系,并给予数学证明。     

On Empirical Processes for Ergodic Diffusions and Rates of Convergence of M-estimators

Abstract. This paper contributes to the development of empirical process theory for ergodic diffusions. We prove an entropy‐type maximal inequality for the increments of the empirical process of an ergodic diffusion. The inequality is used to study the rate of convergence of M‐estimators.     

零息收益率曲线期限结构变化的主成分分析

文章认为运用主成分分析方法能极大地简化对利率曲线变化的分析,便于准确了解利率曲线结构变动的模式。同时运用主成分分析方法能为金融机构计算投资组合资产VaR提供切实可行的方法,为金融机构风险管理提供依据。     

Entropy-Based Moment Selection in the Presence of Weak Identification

Hall et al. (2007 Hall , A. R. , Inoue , A. , Jana , K. , Shin , C. (2007). Information in generalized method of moments estimation and entropy based moment selection. Journal of Econometrics 138:488–512.[Crossref] , [Google Scholar]) propose a method for moment selection based on an information criterion that is a function of the entropy of the limiting distribution of the Generalized Method of Moments (GMM) estimator. They establish the consistency of the method subject to certain conditions that include the identification of the parameter vector by at least one of the moment conditions being considered. In this article, we examine the limiting behavior of this moment selection method when the parameter vector is weakly identified by all the moment conditions being considered. It is shown that the selected moment condition is random and hence not consistent in any meaningful sense. As a result, we propose a two-step procedure for moment selection in which identification is first tested using a statistic proposed by Stock and Yogo (2003 Stock , J. H. , Yogo , M. ( 2003 ). Testing for weak instruments in linear IV regression . Discussion paper, Kennedy School of Government, Harvard University, Cambridge, MA . [Google Scholar]) and then only if this statistic indicates identification does the researcher proceed to the second step in which the aforementioned information criterion is used to select moments. The properties of this two-step procedure are contrasted with those of strategies based on either using all available moments or using the information criterion without the identification pre-test. The performances of these strategies are compared via an evaluation of the finite sample behavior of various methods for inference about the parameter vector. The inference methods considered are based on the Wald statistic, Anderson and Rubin's (1949 Anderson , T. W. , Rubin , H. ( 1949 ). Estimation of the parameters of a single equation in a complete system of stochastic equations . Annals of Mathematical Statistics 20 : 46 – 63 .[Crossref] , [Google Scholar]) statistic, Kleibergen (2002 Kleibergen , F. ( 2002 ). Pivotal statistics for testing structural parameters in instrumenatl variables regression . Econometrica 70 : 1781 – 1803 .[Crossref], [Web of Science ®] , [Google Scholar]) K statistic, and combinations thereof in which the choice is based on the outcome of the test for weak identification.     

中国国债利率期限结构实证研究

文章利用三次样条函数构造出上交所和深交所的国债利率期限结构,并通过对两市利率期限结构曲线进行比较与分析,得出这样的结论:中国国债利率期限结构虽然呈现出向右上方倾斜的正常态势,但是还存在着曲线趋于平坦、国债品种单一化等问题。     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

The authors discuss the convergence for weighted sums of pairwise negatively quadrant dependent (NQD) random variables and obtain some new results which extend and improve the result of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar]. In addition, we relax some restrictions of the conditions in their result. Some new methods are used in this article which differ from that of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar].     

中国公共教育消费区域差异及收敛性研究

基于全国31省市(区)相关的公共教育消费数据,对中国八大经济区的公共教育消费进行区域间和区域内的差异分析,研究发现:中国公共教育消费总体差异呈现"W"型趋势,并且区域内差异和区域间差异交替成为公共教育消费差异的主导部分,区域内差异表现出显著的收敛性;相对于人口数而言,经济发展对公共教育消费具有更大的影响;全国公共教育消费收敛性检验显示,公共教育消费具有明显的δ-收敛,但β-收敛不明显。     

Asymptotic properties for the moment estimators in Rayleigh distribution with two parameters

ABSTRACT

We consider the asymptotic properties for the moment estimators in Rayleigh distribution with two parameters. The law of the iterated logarithm for the estimators can be obtained. Moreover, we can give a simple proof of the asymptotic normality which has been obtained by Li and Li (2012) Li, Y.W., Li, M.H. (2012). Moment estimation of the parameters in Rayleigh distribution with two parameters. Commun. Stat.-Theor. Methods 41:2643–2660.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

Estimation of Parameters of the Ornstein-Uhlenbeck Type Processes with Continuum of Moment Conditions

In this article, we develop a new method of parametric estimation for process of Ornstein-Uhlenbeck type. The proposed method is based on GMM with a continuum of moment conditions (CGMM), which is introduced in Carrasco and Florens (2002 Carrasco, M., Florens, J.P. (2002). Efficient GMM Estimation Using the Empirical Characteristic Function. Working paper, CREST. Paris. [Google Scholar]), and the estimator is called the CGMM estimator. We show that this CGMM estimator has consistency and asymptotic normality. Simulation studies evidence that the proposed method performs quite well in small-sample cases.