一类混合回归模型中估计的收敛速度

考虑回归模型y_i=x_iβ+g(t_i)+e_4,i=1,2,…,n,其中 g(·)是未知光滑函数,β是未知待估参数,e_4是随机误差.设{(x_4,t_4,y_4),1≤i≤n}是 i.i.d.子样.本文首先给出了 g(·)的一类近邻估计n(·),然后基于模型 y_4=x_4β+(t_4)+e_4得到了β的最小二乘估计.在适当条件下,证得了及 g(·)的最终估计(·)的强弱收敛速度.     

误差为相依序列时的非参数回归模型

本文对非参数多元回归模型Y~(j)(x_(in))=g(x_(in))+e~(j)(x_(in))中误差序列{e~(j)(x_(in)),j≥1}为m_0—相依序列和局部广义高斯序列时,给出未知函数g(x)的估计量的相合性。     

在K—NN判别和预测中条件风险函数的估计量的强收敛性

设(θ,x),(θ_1,x_1),……,(θ_n,x_n)是独立同分布的随机向量,θ_(j 1)~((k))(θ_(k 1)~((k)))是(θ_1,x_1),……,(θ_i,x_i)和x_(j 1)对θ_(j 1)的K—NN判别(预测)值。本文用来作为L_n~((k)))的估计,并讨论了其强收敛性,即在很一般的条件下,证明了:其中L_u~((k))(L_u~((k)))是K—NN判别(预测)的条件风险函数,■     

纵向数据变系数回归模型小波估计的渐近性质

考虑纵向数据下的变系数回归模型y_(ij)=x_(ij)~Tθ(t_(ij))+e_(ij)i=1,2,…,n j=1,2,…,m.利用小波光滑和加权最小二乘方法,分别研究了模型中未知参数θ(·)的小波估计θ(·)和误差方差σ~2的小波估计σ~2,在适当的条件下,证明了θ的强相合性,强相合速度,并得到θ和σ~2的渐近正态性.     

等距节点上(0,3)插值五次样条

§1.引言 设f(x)是定义在[0,1]上的连续函数,n是自然数。记h=1/n, f_v~((r))=f~((r))(vh),v=0,1,…,n;r=0,1,…,5, f_(v 1/2)~((r))=f~((r))((v 1/2)h),v=0,1,…,n-1;r=0,1,…,5, ω_r(j)=max |f~((r))(x_1)-f~((r))(x_2)|,r=0,1,…,6. |x_1-x_2|≤h 0≤x_1,x_2≤1又设s(x)是[0,1]上满足(i)s(x)∈C~3[0,1],(ii)在[vh,(v 1)h]上s(x)∈∏_5,v=0,1,…,n-1的五次样条.它们的全体记为?_(n5)~((3)) .     

纵向数据下部分线性EV模型的渐近性质

研究了纵向数据下部分线性EV函数关系模型.应用一般非参数权函数法和广义最小二乘法给出了未知参数β,误差方差σ2以及未知函数g(·)的估计.在一般的条件下,证明了β,σ2估计的渐近正态性,同时也给出了未知函数g(·)估计的收敛速度,其结果是独立数据情形下相应结果的推广.     

部分线性模型中参数估计的Bootstrap逼近

考虑回归模型 :yi=xiβ+g(ti) +σiei,1≤ i≤ n.其中 σ2i=f(ui) ,(xi,ti,ui)是固定非随机设计点列 ,f (· )和 g(· )是未知函数 ,β是待估参数 ,ei 是随机误差 .对文 [1 ]给出的基于 g(· )及 f(· )的一类非参数估计的β的最小二乘估计β^ n和加权最小二乘估计βn,本文通过重抽样的方法构造了 β^n 和 βn 的 Bootstrap统计量 β^ *n 和 β*n .证明了在给定原样本的条件下 ,n (β^ *n -β^ n)和 n (β*n -β^ n)分别与 n (β^ n-β)和 n (βn-β)有相同的渐近分布 .     

在Banach空间中推广的Roper-Suffridge算子(Ⅲ)

假定X是具有范数‖·‖的复Banach空间,n是一个满足dim X≥n≥2的正整数.本文考虑由下式定义的推广的Roper-Suffridge算子Φ_(n,β_22γ_2,…,β_(n+1),γ_(n+1))(f):(?)其中x∈Ω_(p1,p2,…,pn+1),β_1=1,γ_1=0和(?)这里p_j1(j=1,2,…,n+1),线性无关族{x_1,x_2,…,x_n}(?)X与{x_1~*,x_2~*,…,x_n~*}(?) X~*满足x_j~*(x_j)=‖x_j‖=1(j=1,2,…,n)和x_j~*(x_k)=0(j≠k),我们选取幂函数的单值分支满足(f(ξ)/ξ)~(β_j)|ξ=0=1和(f′(ξ))~(γ_j)|ξ=0=1,j=2,…,n+1.本文将证明:对某些合适的常数β_j,γ_j,算子Φ_(n,β_2,γ_2,…,β_(n+1),γ_(n+1))(f)在Ω_(p_1,p_2,…,p_(n+1))上保持α阶的殆β型螺形映照和α阶的β型螺形映照.     

AANA误差下半参数回归模型中估计量的相合性

考虑如下半参数回归模型y_i=x_iβ+g(t_i)+σ_ie_i,i=1,2,…,n,n≥1,其中σ_i~2=f(u_i),(x_i,t_i,u_i)为已知设计点列.在适当的条件下,当误差为AANA变量时,本文研究了未知参数β和未知函数g的最小二乘估计与加权最小二乘估计的相合性,特别是p(p1)阶矩相合性和完全相合性,所得结果推广了误差为NA变量的相应结论.     

ON UNIQUENESS OF SOLUTION TO POINTWISE LANDAU PROBLEM ON THE FINITE INTERVAL

1. Introduction Let W_∞~((r)) (β) = {f| f∈W_∞~((r)) [-1,1], ||f||_(C[-1,1]) β, ||f~((r))||_∞ 1}.In this paper, we will consider the following Landau problem:λf~((k))(ξ) + μf~((k-1)) (ξ) →inf, f∈W_∞~((r)) (β), (1.1)where ξ∈[-1,1], 1(?)k(?)r-1, and λ, μ real and not all zero, (if k=1,suppose λ≠0 in addition ). A. Pinkus studied it first. To begin with, we introduce some fundamental definitions anddenotions. The perfect spline f, which satisfies || f~((r))||_∞ = 1 andhas n knots and n+r+1 points of equioscillation in [-1,1], isdenoted by x_(nr), which is refered as Tchebyshev perfect spline. And     

核实数据下误差为鞅差序列的部分线性模型的估计及性质

Consider the partly linear model Y = xβ + g(t) + e where the explanatory x is erroneously measured,and both t and the response Y are measured exactly,the random error e is a martingale difference sequence.Let x be a surrogate variable observed instead of the true x in the primary survey data.Assume that in addition to the primary data set containing N observations of {(Y_j,x_j,t_j)_(j=n+1)~(n+N),the independent validation data containing n observations of {(x_j,x_j,t_j)_(j=1)~n} is available.In this paper,a semiparametric method with the primary data is employed to obtain the estimator of β and g(·) based on the least squares criterion with the help of validation data.The proposed estimators are proved to be strongly consistent.Finite sample behavior of the estimators is investigated via simulations too.     

纵向数据下半参数回归模型估计的渐近性质

本文考虑纵向数据下半参数回归模型: $y_{ij}=x_{ij}'\beta+g(t_{ij})+e_ij},\;i=1,\cdots,m,\;j=1,\cdots,n_i$. 基于最小二乘法和一般的非参数权函数方法给出了模型中参数$\beta$和回归函数$g(\cdot)$的估计, 并在适当条件下证明了$\beta$估计量的渐近正态性和$g(\cdot)$估计量的最优收敛速度\bd 模拟结果表明我们的估计方法在有限样本情形有良好的效果     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma \to \Gamma$, $\sigma_{\varphi}:X^{\Gamma} \to X^{\Gamma}$with $\sigma_{\varphi}((x_{\alpha})_{\alpha \in \Gamma})=(x_{\varphi(\alpha)})_{\alpha \in \Gamma}$ (for $(x_{\alpha})_{\alpha \in \Gamma} \in X^{\Gamma}$) is a generalized shift. In this text for $\mathcal{S} = \{\sigma_{\varphi}:\varphi \in \Gamma^{\Gamma}\}$ and $\mathcal{H}=\{\sigma_{\varphi}:\Gamma \xrightarrow{\varphi} \Gamma$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$. Regarding proximal relation we prove: $$P(\mathcal{S}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \exists \beta \in \Gamma (x_{\beta} = y_{\beta})\}$$and $P(\mathcal{H}, X^{\Gamma} ) \subseteq \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\beta \in \Gamma : x_{\beta} = y_{\beta}\}$ is infinite$\}$ $\cup\{($ $x,x) : x \in \mathcal{X}\}$. Moreover, for infinite $\Gamma$, both transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$ are regionally proximal, i.e., $Q(\mathcal{S}, X^{\Gamma}) = Q(\mathcal{H}, X^{\Gamma} ) = X^{\Gamma} \times X^{\Gamma}$, also for sydetically proximal relation we have $L(\mathcal{H}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\gamma ∈ \Gamma :$ $x_{\gamma} \neq y_{\gamma}\}$ is finite$\}$.     

The Consistency for the Estimators of Semiparametric Regression Model with Dependent Samples

For the semiparametric regression model:Y^(j)(x_in,t_in)=t_inβ+g(x_in)+e^(j)(x_in),1≤j≤k,1≤i≤n,where t_in∈R and x(in)∈Rpare known to be nonrandom,g is an unknown continuous function on a compact set A in R^p,e^j(x_in)are m-extended negatively dependent random errors with mean zero,Y^(j)(x_in,t_in)represent the j-th response variables which are observable at points xin,tin.In this paper,we study the strong consistency,complete consistency and r-th(r>1)mean consistency for the estimatorsβ_k,nand g__k,nofβand g,respectively.The results obtained in this paper markedly improve and extend the corresponding ones for independent random variables,negatively associated random variables and other mixing random variables.Moreover,we carry out a numerical simulation for our main results.     

2×2阶上三角型算子矩阵的Moore-Penrose谱

设$H_{1}$和$H_{2}$是无穷维可分Hilbert空间. 用$M_{C}$表示$H_{1}\oplusH_{2}$上的2$\times$2阶上三角型算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\\end{array}\right)$. 对给定的算子$A\in{\mathcal{B}}(H_{1})$和$B\in{\mathcal{B}}(H_{2})$,描述了集合$\bigcap\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$与$\bigcup\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$,其中$\sigma_{M}(\cdot)$表示Moore-Penrose谱.     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.     

On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation

In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.     

限制零点个数的亚纯函数的正规定则

设k,n(≥k+1)是两个正整数,a(≠0),b是两个有穷复数,F为区域D内的一族亚纯函数.如果对于任意的f∈F,f的零点重级大于等于k+1,并且在D内满足f+a[L(f)]~n-b至多有n-k-1个判别的零点,那么F在D内正规·这里L(f)=f~((k))(z)+a_1f~((k-1))(z)+…+a_(k-1)f'(z)+a_kf(z),其中a_1(z),a_2(z),…,a_k(z)是区域D上的全纯函数.     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0<λ<+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l<+∞）has p(1≤p<+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0<p<+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1<θ2<…<θp<2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l<+∞）has a finite nonzero deficient value.