ON THE ASYMPTOTIC BEHAVIOR OF THE INTERMEDIATE POINT IN THE HIGHER-ORDER LAGRANGE MEAN VALUE THEOREM

In 1982,Bernar Jacobson [2] proved that in the Mean Value Theorem forIntegrals,integral from 0 to x f(t)dt=f(c)(x-α),as x approaches α,the intermediate point capproaches the midpoint between a and x,i.e.(?)[(c-α)/(x-α)]=1/2.In the meantime,Alfonso G.Azpeitia established the analogous result for theTaylor Formula [1].Recently,Li Wenrong studied the asymptotic behavior ofthe intermediate points in the Cauchy Mean Value Theorem and the Generali-     

A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q(p) = inf {t : F(t)≥p}, 0≤p≤1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.     

ON THE BOREL SUBGROUPS OF LARGE TYPE

The author defines the large type Borel subgroups of a reductive algebraic group, which are used to discuss Langlands' L-groups and the Langlands classification of the admissible representations of reductive algebraic groups over R (see [1, 2, 5])and determine all of the Borel subgroups of large type for the classical semisimple Lie groups.     

ON LIMITING PROPERTIES OF BERNSTEIN-TROTTER TYPE OPERATOR

In this paper ,a class of Bernstein-Trotter type operator and its limiting properties are studied. By using both the limiting theorem and P. Le‘ vy continuity theorem on probability theory, a theorem of convergence is obtained. The result in [1] is included.     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

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ON GLOBAL SOLUTION FOR A CLASS OF SYSTEMS OF MULTI－DIMENSIONAL GENERALIZED ZAKHAROV TYPE EQUATION

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STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

ON THE NON-EXISTENCE OF L~2-EIGENVALUES OF MANIFOLDS OF COLUM TYPE

In[4],Li proved that Yau's conjecture“For non-compact connected completeRiemannian manifold M.M has no L~2-eigenvalues if its sectional curvature K_m≥0”holds if M can be represented as a Riemannian product M=R~1×N.Acturally,heproved(without the restriction K_m≥0)     

BAYESIAN PREDICTION FOR THE TWO-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON TYPE Ⅱ DOUBLY CENSORING

The two-parameter exponential distribution is proposed to be an underlying model, and prediction bounds for future observations are obtained by using Bayesian approach. Prediction intervals are derived for unobserved lifetimes in one-sample prediction and twosample prediction based on type Ⅱ doubly censored samples. A numerical example is given to illustrate the procedures,prediction intervals are investigated via Monte Carlo method,and the accuracy of prediction intervals is presented.     

Optimal convergence rates and asymptotic efficiency of point estimators under truncated distribution families

For regular and irregular truncated distribution families, the optimal convergence rates of consistent point estimators have been found and the corresponding asymptotic efficiencies established. Also, it has been justified that commonly used estimators are all efficient. The efficiencies here are compared to the efficiencies of asymptotically median unbiased estimators, providing a lot of counter estimator examples such that those estimators are efficient in the former sense, but not in the latter.     

非正则双边截断分布族点估计的中偏差

本文研究了某一类非正则双边截断分布族的参数估计,利用( X(1),X(n))的联合分布函数及应用Taylor渐近展开的方法,得到了它的未知参数(θ1,θ2)满足中偏差原理,且求出了其精确的速率函数表达式,它的表达式不同于一般的速率函数.     

Bootstrap for the conditional distribution function with truncated and censored data

We propose a resampling method for left truncated and right censored data with covariables to obtain a bootstrap version of the conditional distribution function estimator. We derive an almost sure representation for this bootstrapped estimator and, as a consequence, the consistency of the bootstrap is obtained. This bootstrap approximation represents an alternative to the normal asymptotic distribution and avoids the estimation of the complicated mean and variance parameters of the latter.     

The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases

In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.Research supported by University of Tsukuba Project Research.     

Edgeworth expansion of the Studentized product-limit estimator for truncated and censored data

Based on random left truncated and right censored data we investigate the one-term Edgeworth expansion for the Studentized product-limit estimator, and show that the Edgeworth expansion is close to the exact distribution of the Studentized product-limit estimator with a remainder of On(su-1/2).     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

左截断右删失数据光滑分位估计的渐近性质

文中提出了随机左截断右删失数据下的一种光滑分位估计,推导出此光滑估计的相合性和渐近正态性,同时获得了该估计的强弱Bahadur表示定理。     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

Estimation of parameters in the beta binomial model

This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970,Appl. Statist.,19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of ones has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.     

Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic PDEs Disturbed by Small Noise

Consider a parabolic stochastic partial differential equation perturbed by small noise observed on a time interval [0,T]. We construct the maximum likelihood estimators of the coefficients of the operators involved in these equations based on partial observations in the form of diffusion processes and show the asymptotic efficiency for loss functions with polynomial majorant as the variance goes to zero.