Asymptotic Joint Normality of Outdegrees of Nodes in Random Recursive Trees

We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k.     

Asymptotic Normality of the Recursive M-estimators of the Scale Parameters

In this paper, the limit distributions of the recursive M-estimators of scatter parameters in a multivariate linear model setting are studied. Under some mild conditions, the asymptotic normality of the recursive M-esimtators is established. Some Monte Carlo simulation results are presented to illustrate the performance of the recursive M-estimators.     

回归函数基于分割的改良递推估计的渐近正态性

提出了回归函数基于分割的改良递推估计,mn(x)=sum (IAn(x) (Xi)YiI(|Yi|) from i=1 to n≤bi)/sum (IAn(x)(Xi)) from i=1 to n,并在较为简洁的条件下证明了该估计量的渐近正态性.     

NA随机变量的递归密度核估计的渐近正态性

设{Xn,n≥1}为同分布的NA样本序列，其未知概率密度函数为f(x),基于样本X1,…,Xn，用递归密度核估计fn(x)=1/n∑j=1 n 1/hj K(x-Xj/hj)对f(x)进行估计。本文研究了在一定条件下，fn(x)的渐近正态性。     

k-U统计量的渐进正态性

称 为k-U统计量,其中g(x1,…,xm)是一对称函数,k为小于等于n的自然数,k可能依赖于n.这一表达形式是一类统计量,在k=n时, Unm,n就是U-统计量.本文证明了Unm,k的渐进正态性.     

Asymptotic Normality of Autoregressive Processes

In the present paper, the form of iterated limits of the moderate deviation principle for dependent variables is considered and as an application, the moderate deviation principle of m-dependent random variables is obtained.     

多元t分布的一致渐进正态性

讨论了多元t分布一致渐进正态性的条件,利用不等式法估计了二者之间的Kullback-Leibler距离,给出了Kullback-Leibler距离表达式,最后对多元t分布一致渐进正态性进行了模拟验证.     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

一个半参数回归模型的渐近正态性

考虑半多数回归模型yi＝xiβ＋g（xi）＋εi，lin，这里xi是具有已知方差σ的独立同分布随机样本，εi是具有零均值和有限方基σ2的独立同分布随机误差．β，g和εi的分布密度是未知的．本文作者构造了一个具有更小渐近方差的β的一个渐近正态估计．     

Asymptotic Normality of a Combined Regression Estimator

In this paper, we propose a combined regression estimator by using a parametric estimator and a nonparametric estimator of the regression function. The asymptotic distribution of this estimator is obtained for cases where the parametric regression model is correct, incorrect, and approximately correct. These distributional results imply that the combined estimator is superior to the kernel estimator in the sense that it can never do worse than the kernel estimator in terms of convergence rate and it has the same convergence rate as the parametric estimator in the case where the parametric model is correct. Unlike the parametric estimator, the combined estimator is robust to model misspecification. In addition, we also establish the asymptotic distribution of the estimator of the weight given to the parametric estimator in constructing the combined estimator. This can be used to construct consistent tests for the parametric regression model used to form the combined estimator.     

Local Asymptotic Normality in Extreme Value Index Estimation

This paper deals with the estimation of the extreme value index in local extreme value models. We establish local asymptotic normality (LAN) under certain extreme value alternatives. It turns out that the central sequence occurring in the LAN expansion of the likelihood process is up to a rescaling procedure the Hill estimator. The central sequence plays a crucial role for the construction of asymptotic optimal statistical procedures. In particular, the Hill estimator is asymptotically minimax.     

Asymptotic Normality of Nonparametric Estimate for Zero-Utility Premiums

Acta Mathematicae Applicatae Sinica, English Series - Zero-utility principle is one of the main premium pricing principles, which has been widely used in insurance practice. In this paper, the...     

Asymptotic Normality of Kernel Density Estimators under Dependence

In this paper, we study the kernel methods for density estimation of stationary samples under generalized conditions, which unify both the linear and -mixing processes discussed in the literature and also adapt to the non-linear or/and non--mixing processes. Under general, mild conditions, the kernel density estimators are shown to be asymptotically normal. Some specific theorems are derived within various contexts, and their applications and relationship with the relevant references are considered. It is interesting that the conditions on the bandwidth may be very simple, even in the generalized context. The stationary sequences discussed cover a large number of (linear or nonlinear) time series and econometric models (such as the ARMA processes with ARCH errors).     

矩阵Г分布的一致渐近正态性

相对于两个密度函数之间的Kullback-Leibler距离,本文获得了矩阵Γ分布一致渐近正态分布的条件,由于矩阵Γ分布包含了Wishart分布,因此我们也指出了 Wishart分布一致渐近正态分布的条件．     

加权和线性过程的渐近正态性

本文建立了α-混合序列情形的加权和平稳线性过程的渐近正态性.获得的结论基于最少的权条件.所得结论将Abadir等[Econometric Theory,2014,30(1):252-284]中的结论推广至α-混合序列情形.     

部分线性变系数模型中估计的渐进正态性

作为部分线性模型与变系数模型的推广,部分线性变系数模型是一类应用非常广泛的模型,本文基于Profile最小二乘方法给出了模型中参数分量与非参数分量的估计,并在异方差情形下证明了这些估计的渐进正态性.     

Asymptotic Normality of Cross-correlogram Estimates of the Response Function

The problem of estimation of an unknown response function of a time-invariant continuous linear system is considered. Discrete-time sample input–output cross-correlograms are taken as estimates of the response function. The inputs are supposed to be zero-mean stationary Gaussian processes close, in some sense, to a white noise. Both asymptotic normality of finite-dimensional distributions of the estimates and their asymptotic normality in spaces of continuous functions are studied. Our basic tool is a new integral representation for cumulants of the estimate as a finite sum of integrals involving cyclic products of kernels. Some inequalities for these integrals are obtained and their asymptotic behaviour is studied.     

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

部分线性模型中估计的渐近正态性

考虑回归模型其中是未知函数，（ｘ＿ｉ，ｔ＿ｉ，ｕ＿ｉ）是固定非随机设计点列，β是待估参数，ｅ＿ｉ是随机误差。基于ｇ（·）及ｆ（·）的一类非参数估计（包括常见的核估计和近邻估计），我们构造了β的加权最小二乘估计，并证得了最小二乘估计和加权最小二乘估计的渐近正态性。