新题征展(129)

A 题组新编1.(1)设x∈R+,e表示自然对数的底,求证:函数y=(1+1/x)s,y=(1+1/x)(x+1)分别单调递增、递减,且(1+1/x)x＜e＜(1+1/x)(x+1);(2)已知数列{an}满足2Sn=nan,其中Sn是{an}的前n项和,a2=1,求证:3/2≤(1+1/(2an+1))n＜√e.2.已知a1C0n+ a2C1n+a3C2n+…+an+1Cnn=n·2n对任意的正整数n恒成立.(1)若a1,a2,a3,…,an+1成等差数列,求出该数列的通项公式;(2)若a1是已知数,求数列a1,a2,a3,…,an+1的通项公式.     

方差分量组合的非负二次估计的可容许性

设有方差分量模型Y=X_β+U_(1ε1)+…+U_(NεN),其中XU_i已知,ε_1,…,ε_1相互独立。Eε_(if)=0,Eε_(if)~2=σ~2,Eε_(if)~3=0.Eε_(if)~4=3σ_i~4,这里(ε_(i1),…,ε_(in_i)εi。(β,σ~2)∈R~n×Ω为未知参数。Ω={(σ_1~2,…,σ_N~2):0≠sum from i=1 to n σ_i~2U_iU'_i≥0}。本文给出了Y'AY是sum from i=1 to n f_iσ_i~2在损失(Y'AY-sum from i=1 to N f_iσ_i~2)~2下在类{Y'BY:B≥0}中可容许估计的一个充分条件。同时也给出了Y'AY+l'Y+a是sum from i=1 to N f_iσ_i~2的可容许估计(在类{Y'BY+m'Y+b}中)的一个充要条件。研究了非负二次估计与局部最优估计之间的关系。     

正态—正态模式下,结构串联系统可靠度的置信下界

设X_1,…,X_niid.～N(μ,σ~2),Y～N(0,1),Y与X_1,…,X_n独立,Z_1,…,Z_m为N(μ,σ~2)的iid.样本,σ~2已知,μ未知,本文证明R=P_r{X_1≥Y,…,X_n≥Y}的MVUE存在的充要条件是m≥nσ~2/n+σ~2,得到R的MVUE和UMA置信下界。     

实二次城 Q(■)类数的可除性

设 d 是无平方因子正整数,h(d)是实二次域 Q(d~(1/2))的类数.本文证明了:如果 da~2=1+4k~(2n),a、k、n 是正整数,k>1,n>1,n 的奇素因子 p和 k 的素因子 q 都适合 gcd(p,(q-1)q)=1,而且 2k~n+ad~(1/2)是 Pell 方程u′~2-dv′~2=-1 的基本解,则除了(a,d,k,n)=(5,41,2,4) 以及 n=2,k=P_mP_(m+1) 或者 2Q_mQ_(m+1) 以外,h(d)=0(modn),这里 m 是正整数,P_m=1/2((1+2~(1/2))~m+(1-2~(1/2))~m),Q_m=1/22~(1/2)((1+2~(1/2))~m-(1-2~(1/2))~m).由此可推得:对于任何正整数 n,存在无限多个实二次域,可使 n 整除其类数.     

实二次城 Q(■)类数的可除性

设 d 是无平方因子正整数,h(d)是实二次域 Q(d~(1/2))的类数.本文证明了:如果 da~2=1+4k~(2n),a、k、n 是正整数,k>1,n>1,n 的奇素因子 p和 k 的素因子 q 都适合 gcd(p,(q-1)q)=1,而且 2k~n+ad~(1/2)是 Pell 方程u′~2-dv′~2=-1 的基本解,则除了(a,d,k,n)=(5,41,2,4) 以及 n=2,k=P_mP_(m+1) 或者 2Q_mQ_(m+1) 以外,h(d)=0(modn),这里 m 是正整数,P_m=1/2((1+2~(1/2))~m+(1-2~(1/2))~m),Q_m=1/22~(1/2)((1+2~(1/2))~m-(1-2~(1/2))~m).由此可推得:对于任何正整数 n,存在无限多个实二次域,可使 n 整除其类数.     

课外练习

高一年级1.已知:函数f(x)=ax2+(2a-1)x+1在x∈[-3/2,2]上的最大值为3,求实数a的值. (山东汶上县一中(272500)张宪铸)2.解关于x的不等式|loga(ax2)|<|logax|+2. (江苏张家港职业教育中心校 (215600) 周文国)3.已知数列{an}是等差数列,且a1=2,a1+a2+a3=12. (1)求数列{an}的通项公式; (2)令bn=anxn(x∈R),求数列{bn}前n项 和的公式. (江苏如东县掘港钲北街274号 (226400) 童枚     

关于误差方差估计渐近正态收敛速度的上,下界

考虑线性回归模型 Y_■=x_4~′β+e_■ i=1,2,…设误差序列■,i≥1满足条件:e_■ i≥1 i.i.d.,Ee_1=0,Ee_1~2=σ~2>0,∞>Var e_1~2=τ~2>0。记■_n~2=1/(n-r){sum from j=1 to n e■-sum from k=1 to r (sum from j=1 to n a_(akj)■_j)~2} δ(n)=τ~(-2)E(■_1~2-σ~2)~2I_((|■-σ~2|≥■τ)+τ~(-3)n~(1/2)|E(■_1~2-σ~2)~3I_((|■_1~2-σ~2|<(nτ)~(1/2))+τ~(-4)n~(-1)E■_1~2-σ~2)~4I_((|■-σ~2|0使得■|P(■_n~2-σ~2)/(Var■_n~2)~(1/2))≤x)-Φ(x)|≤C(δ(n)+n~(-1/2)) ■|P(■_n~2-σ~2)/(Var■_n~2)~(1/2))≤x)-Φ(x)|+n~(-1/2)≥C_1δ(n)。     

虚二次域Q((1-4k~n)~(1/2))类数整除性的例外情况

设d是无平方因子正整数,h_K是虚二次域K=Q((-d)~(1/2))的类数.又设d满足1+da~2=4k~n,其中a,k,n是适合k1,n2的正整数.运用初等数论方法给出了数组(d,a,k,n)可使n|h_K成立的必要条件.     

一类“数学问题”的统一证法

文[1]给出了不等式:已知x,y,z∈R+,m∈N+.求证:x/mx+y+z+y/x+my+z+z/x+y+mz≤3/m+2. 文[2]给出了不等式:已知xi＞0(i=1,2,…n),k＜1,求证: n∑i=1 xi/x1+x2+…+xi-1+kxi+xi+1+…+xn≥n/n+k-1. 文[3]给出了不等式:设ai＞0(i=1,2,3,…,n),p∈R,q＞0,且n∑i=1ai=A,Si=pai+q(A一ai)＞0(i=1,2,…,n),求证:     

关于一个数论函数的导数及应用

Kanemitsu教授给出了欧拉求和函数的推广公式Lu(x,a)=0n相似文献   

长沙市规模以下零售业抽样调查方案

本文介绍了一个根据具体城市情况制定的零售贸易餐饮业的抽样调查方案.以长沙市规模以下零售业、餐饮业为总体,用区域抽样代替一般的名录抽样,并采用细致的分层解决商业单位分布不均匀问题.文中也给出了与抽样方案配套的总体和域目标量的估计及相应的方差估计公式.     

具有随机右删失随机变量分位数的经验似然推断

本文研究了具有随机右删失随机变量分位数的置信域的构造.利用经验似然和截尾值估算相结合的方法,给出了分位数的对数经验似然比统计量,在较少的条件下证明了该统计量的极限分布为自由度为1的x~2分布.使得完全数据下的分位数的经验似然推断方法应用到非完全数据中.     

Convergence Properties of Two-Stage Stochastic Programming

This paper considers a procedure of two-stage stochastic programming in which the performance function to be optimized is replaced by its empirical mean. This procedure converts a stochastic optimization problem into a deterministic one for which many methods are available. Another strength of the method is that there is essentially no requirement on the distribution of the random variables involved. Exponential convergence for the probability of deviation of the empirical optimum from the true optimum is established using large deviation techniques. Explicit bounds on the convergence rates are obtained for the case of quadratic performance functions. Finally, numerical results are presented for the famous news vendor problem, which lends experimental evidence supporting exponential convergence.     

Penalized Empirical Likelihood Estimation for Partially Linear Single Index Panel Model with Fixed Effects

This paper constructs a penalized empirical likelihood estimation method via quadratic inference function method, filter method and empirical likelihood estimation method. Under some regular conditions, we derived the large sample properties of estimators and show that the proposed empirical likelihood ratio is asymptotically to chi-square distribution. Furthermore, the infinite sample performance of the proposed method is evaluated by Monte Carlo simulation and real data analysis.     

Specification tests for the response distribution in generalized linear models

Goodness-of-fit tests are proposed for the case of independent observations coming from the same family of distributions but with different parameters. The most popular related context is that of generalized linear models (GLMs) where the mean of the distribution varies with regressors. In the proposed procedures, and based on suitable estimators of the parameters involved, the data are transformed to normality. Then any test for normality for i.i.d. data may be applied. The method suggested is in full generality as it may be applied to arbitrary laws with continuous or discrete distribution functions, provided that an efficient method of estimation exists for the parameters. We investigate by Monte Carlo the relative performance of classical tests based on the empirical distribution function, in comparison to a corresponding test which instead of the empirical distribution function, utilizes the empirical characteristic function. Standard measures of goodness-of-fit often used in the context of GLM are also included in the comparison. The paper concludes with several real-data examples.     

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??This paper constructs a penalized empirical likelihood estimation method via quadratic inference function method, filter method and empirical likelihood estimation method. Under some regular conditions, we derived the large sample properties of estimators and show that the proposed empirical likelihood ratio is asymptotically to chi-square distribution. Furthermore, the infinite sample performance of the proposed method is evaluated by Monte Carlo simulation and real data analysis.     

Empirical Likelihood Semiparametric Regression Analysis under Random Censorship

This paper considers large sample inference for the regression parameter in a partly linear model for right censored data. We introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. A Monte Carlo method is proposed to approximate the limiting distribution. This enables one to make empirical likelihood-based inference for the regression parameter. We also develop an adjusted empirical likelihood method which only appeals to standard chi-square tables. Finite sample performance of the proposed methods is illustrated in a simulation study.     

截断情况下线性回归模型响应变量均值的经验似然

本文讨论了在带有截断情况的线性回归模型中 ,响应变量均值的估计问题 .将经验似然的方法应用到带有截断情况的回归模型中 ,在估计响应变量的均值时构造了调整的经验似然统计量 ,证明了在一定的条件下 ,该统计量渐近服从 χ2 分布 ,给出了均值的置信区间 ,并与正态下得到的结果进行了比较 ,模拟的结果说明了经验似然的优良性 .     

The Efficiency Distribution Approach in Data Envelopment Analysis: An Application

A method is developed here for characterizing the empirical distribution of the efficient units in data envelopment analysis. Two empirical applications illustrate the various uses of the distribution approach. One involves the cost frontier which exhibits increasing returns to scale and the other involves a dynamic production frontier, where technological change causes a shift of the production frontier over time.