Necessary Conditions of L_1-Convergenoe of Kernel Regression Estimators

Let (X_1,Y_1),\cdots,(X_n,Y_n) be iid. and R^d *R-valued samples of (X,Y). The kernel estimator of the regression function m(x)\triangleq E(Y|X=x) (if it exists), with kernel K, is denoted by $\[{m_n}(x) = \sum\limits_{i = 1}^n {{Y_i}K(\frac{{{X_i} - x}}{{{h_n}}})/\sum\limits_{j = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} } \]$ Many authors discussed the convergence of m_n(x) in various senses, under the conditions h_n\rightarrow 0 and nh_u^d\rightarrow \infinity asn\rightarrow \infinity. Are these conditions necessary? This paper gives an affirmative answer to this bprolemuithe case of L_1-conversence, when K satisfies (1.3) and E(|Y|log^+|Y|)<\infinity.     

Strong Convergence of the Kernel Estimates of Nonparametric Regression Functions

Let (X, Y), (X_1, Y_1),\cdots, (X_n, Y_n) be i. i. d. random vectors taking values in R_d\times R with E(|Y|)<\infinity, To estimate the regression function m(x)=E(Y|X=x), we use the kernel estimate $m_n(x)=[\sum\limits_{i = 1}^n {K(\frac{{{X_i} - x}}{{{h_n}}}){Y_i}/} \sum\limits_{i = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} \]$ where K(x) is a kernel function and h_n a window width. In this paper, we establish the strong consistency of m_n(x) when E(|Y|^p)<\infinity for some p>l or E{exp(t|Y|^\lambda)}<\infinity for some \lambda>0 and t>0. It is remakable that other conditions imposed here are independent of the distribution of (X, Y).     

Higher Commutators of Pseudo-Differential Operator

In this paper the following result is established: For a_i,f\in \phi(R^K),i=1,\cdots,n and $T(a,f)(x)=w(x,D)()[\prod\limits_{i = 1}^n {{P_{{m_i}}}({a_i},x, \cdot )f( \cdot )} \]$ It holds that $||T(a,f)||_q\leq C||f||_p_0[\prod\limits_{i = 1}^n {||{\nabla ^{{m_i}}}|{|_{{p_i}}}} \]$ where a=(a_1,\cdots,a_n), q^-1=p^-1_0+[\sum\limits_{i = 1}^n {p_i^{ - 1} \in (0,1),\forall i,{p_i} \in (1,\infty )} \] or \forall i,p_i=\infinity,p_0\in (1,\infinity), for an integer m_i\geq 0, $P_m_m(a_i,x,y)=a_i(x)-[\sum\limits_{|\beta | < {m_i}} {\frac{{a_i^{(\beta )}(y)}}{{\beta !}}} {(x - y)^\beta }\]$ w(x,\xi) is a classical symbol of order |m|, m=(m_1,\cdots, m_n), |m|=m_1+\cdots+m_n, m_i are nonnegative integers. Besides, a representation theorem is given. The methods used here closely follow those developed by Coifman, R. and Meyer, Y. in [5] and by Cohen, J. in [3].     

STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA

Let X_1,…,X_n be a sequence of independent identically distributed random variableswith distribution function F and density function f.The X_are censored on the right byY_i,where the Y_i are i.i.d.r.v.s with distribution function G and also independent of theX_i.One only observesLet S=1-F be survival function and S be the Kaplan-Meier estimator,i.e.,where Z_are the order statistics of Z_i and δ_((i))are the corresponping censoring indicatorfunctions.Define the density estimator of X_i by where =1-and h_n(>0)↓0.     

THE OPTIMAL RATE OF CONVERGENCE OF ERROR FOR kNN MEDIAN REGRESSION ESTIMATES

Let(X,Y),(X_1,Y_1),…,(X_n,Y_n)be iid.random vectors,where Y is one-dimensional.It is desired to estimate the conditional median(X)of Y,by use of Z_n={(X_i,Y_i),i=1,…,n}and X.Denote by(X,Z_n)the kNN estimate of(X),and putH_(nk)(Z_n)=E{|(X,Z_n)-(X)||Z_n},the conditional mean absolute error.This articalestablishes the optimal convergence rate of P(H_(nk_n)(Z_n)>ε),under fairly generalassumptions on(X,Y)and k_n,which tends to ∞ in some suitable way.     

ON ADMISSffilLITY OF VARIANCE COMPONENTS ESTIMATES

Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.     

A NECESSARY AND SUFFICIENT CONDITION FOR CONVERGENCE OF ERROR PROBABILITY ESTIMATES IN K-NN DISCRIMINATION

Let(X,θ)be R~d×{1,…,s}valued random vector,(X_j,θ_j),j=1,…,n,be its observedvalues, be the K-nearest neighbor estimate of θ_j,R~((K)) be the limit of error probabilityand be the error probability estimate.In this paper it is shown thatA_ε>0, constants α>0,c<∞ such thatif add only if there is no unregular atom of(X,θ)defined below and the various conver-gences R_(nk)→R~(k) are equivalent.     

On the uniform complete convergence of estimates for multivariate density functions and regression curves

Let (X ₁,Y ₁),...(X _n ,Y _n ) be a random sample from the (k+1)-dimensional multivariate density functionf ^*(x,y). Estimates of thek-dimensional density functionf(x)=∫f ^*(x,y)dy of the form $$\hat f_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n W \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered whereW(x) is a bounded, nonnegative weight function andb ₁ (n),...,b _k (n) and bandwidth sequences depending on the sample size and tending to 0 asn→∞. For the regression function $$m(x) = E(Y|X = x) = \frac{{h(x)}}{{f(x)}}$$ whereh(x)=∫y(f) ^* (x, y)dy , estimates of the form $$\hat h_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n {Y_i W} \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered. In particular, unform consistency of the estimates is obtained by showing that \(||\hat f_n (x) - f(x)||_\infty \) and \(||\hat m_n (x) - m(x)||_\infty \) converge completely to zero for a large class of “good” weight functions and under mild conditions on the bandwidth sequencesb _k (n)'s.     

Oscillatory and Asymptotic Behavior of First Order Functional Differential Equations

In this paper the author discusses the following first order functional differential equations: $x''(t)+[\int_a^b {p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (1)$ $x''(t)+[\int_a^b {f(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (2)$ Some sufficient conditions of oscillation and nonoscillation are obtained, and two asymptotic properties and their criteria are given. These criteria are better than those in [1, 2], and can be used to the following equations: $x''(t)+[\sum\limits_{i = 1}^n {{p_i}(t)x[{g_i}(t)] = 0} \] (3)$ $x''(t)+[\sum\limits_{i = 1}^n {{f_i}(t)x[{g_i}(t)] = 0} \] (4)$     

ON THE DIOPHANUNE EQUATION $\[\sum\limits_{i = 0}^N {\frac{{{x_i}}}{{{d_i}}}} \equiv 0\]$ (mod 1) AND ITS APPLICATIONS

The number $\[A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\[\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\[{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\[N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\[{F_q}\]$ of the equation $$\[\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\[a_i^''s\]$ belong to $\[F_q^*\]$. the multiplication group of $\[F_q^{[1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\[A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\[A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\[{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\[A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$     

Banach空间中有限个增生算子公共零点的带误差项的迭代逼近

令E为实一致凸Banach空间,满足Opial条件或其范数是Frechet可微的.令为增生算子,满足值域条件且为非空闭凸子集且满足 .将引入新的带误差项的迭代算法并证明迭代序列弱收敛于{Ai}ki=1的公共零点.     

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

Banach空间中有限个极大单调算子公共零点的迭代格式

令E为实光滑、一致凸Banach空间,E~*为其对偶空间．令A_i,B_i (?) E×E~*,i= 1,2,…,m,为极大单调算子且(?)(A_i~(-1)0∩B_i~(-1)0)≠φ．引入新的迭代算法,并利用Lyapunov泛函,Q_r算子与广义投影算子等技巧,证明迭代序列弱收敛于极大单调算子A_i,B_i,i= 1,2,…,m的公共零点的结论．     

ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

Let X_1,…,X_n be iid samples drawn from an m-dimensional population with a probabilitydensity f,belonging to the family C_(ka),i.e.the family of all densities whose partialderivatives of order k are bounded by a.It is desired to estimate the value of f at somepredetermined point a,for example a=0.Farrell obtained some results concerning the bestpossible convergence rates for all estimator sequence,from which it follows,for example,thatthere exists no estimator sequence{γ_n(0)=γ_n(X_1,…,X_n,0)}such that(?)E_f[γ_n(0)-f(0)]~2=o(n~(-2k/(2k m))).This article pursues this problem further and proves that there existsno estimator sequence{γ_n(0)}such thatn~(-k/(2k m))(γ_n(0)-f(0))(?)0,for each f∈C_(ka),where(?)denotes convergence in probability.     

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

两类广泛递归序列的动力特性

We investigate the dynamics of two extensive classes of recursive sequences:xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j/c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1 and xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1/c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j We prove that their unique positive equilibrium x = 1 is globally asymptotically stable.And a new access is presented to study the theory of recursive sequences.     

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.     

Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

In this paper, we concern the divergence Kohn-Laplace equation$$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left( {X_j^*({a^{ij}}{X_i}u) + Y_j^*({b^{ij}}{Y_i}u)} \right)} } + Tu = f - \sum\limits_{i = 1}^n {\left( {X_i^*{f^i} + Y_i^*{g^i}} \right)}$$ with bounded coefficients on the Heisenberg group ${{\mathbb{H}}^n}$, where ${X_1}, \cdots, {X_n},{Y_1}, \cdots, {Y_n}$ and $T$ are real smooth vector fields defined in a bounded region $\Omega \subset {\mathbb{H}^n}$. The local maximum principle of weak solutions to the equation is established. The oscillation properties of the weak solutions are studied and then the Hölder regularity and weak Harnack inequality of the weak solutions are proved.     

A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relations■hold as x→∞,where the random weightsθ_1,···,θ_(n )are bounded away both from 0 and from∞with no dependency assumptions,independent of the primary random variables X_1,···,X_(n )which have a certain kind of dependence structure and follow non-identically subexponential distributions.In particular,the asymptotic relations remain true whenX_1,···,X_(n )jointly follow a pairwise Sarmanov distribution.