Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least-squares approach

In this paper, we characterize and construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. An efficient estimator equals the empirical estimator minus a correction term and provides significant improvements over the empirical estimator. We construct an efficient estimator by estimating the correction term. For this we use the least-squares principle and an estimated orthonormal basis for the Hilbert space of square-integrable functions under the unknown equal marginal distribution. Simulations confirm the asymptotic behavior of this estimator in moderate sample sizes and the considerable theoretical gains over the empirical estimator.     

Frontier estimation with local polynomials and high power-transformed data

We present a new method for estimating the frontier of a sample. The estimator is based on a local polynomial regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth goes to zero. We give conditions on these two parameters for obtaining almost complete convergence. The asymptotic conditional bias and variance of the estimator are provided and its good performance is illustrated for some finite sample situations.     

Estimators of the regression parameters of the zeta distribution

The zeta distribution with regression parameters has been rarely used in statistics because of the difficulty of estimating the parameters by traditional maximum likelihood. We propose an alternative method for estimating the parameters based on an iteratively reweighted least-squares algorithm. The quadratic distance estimator (QDE) obtained is consistent, asymptotically unbiased and normally distributed; the estimate can also serve as the initial value required by an algorithm to maximize the likelihood function. We illustrate the method with a numerical example from the insurance literature; we compare the values of the estimates obtained by the quadratic distance and maximum likelihood methods and their approximate variance–covariance matrix. Finally, we calculate the bias, variance and the asymptotic efficiency of the QDE compared to the maximum likelihood estimator (MLE) for some values of the parameters.     

左截断相依数据下条件分位数的双核局部线性估计

本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.     

Nonparametric regression with current status data

We apply nonparametric regression to current status data, which often arises in survival analysis and reliability analysis. While no parametric assumption on the distributions has been imposed, most authors have employed parametric models like linear models to measure the covariate effects on failure times in regression analysis with current status data. We construct a nonparametric estimator of the regression function by modifying the maximum rank correlation (MRC) estimator. Our estimator can deal with the cases where the other estimators do not work. We present the asymptotic bias and the asymptotic distribution of the estimator by adapting a result on equicontinuity of degenerate U-processes to the setup of this paper.     

Information amount and higher-order efficiency in estimation

By means of second-order asymptotic approximation, the paper clarifies the relationship between the Fisher information of first-order asymptotically efficient estimators and their decision-theoretic performance. It shows that if the estimators are modified so that they have the same asymptotic bias, the information amount can be connected with the risk based on convex loss functions in such a way that the greater information loss of an estimator implies its greater risk. The information loss of the maximum likelihood estimator is shown to be minimal in a general set-up. A multinomial model is used for illustration.     

非参数回归函数的kn近邻估计的渐近性质及其Bootstrap逼近

本文研究回归函数的kn-近邻估计的渐近性质,得到了回归函数的kn－近邻估计的渐近正态性和它的Bootstrap统计量的相合性,在高阶矩存在的条件下,我们证明了回归函数的kn-近邻估计的Bootstrap逼近比正态逼近更精确。     

Multivariate Locally Weighted Polynomial Fitting and Partial Derivative Estimation

Nonparametric regression estimator based on locally weighted least squares fitting has been studied by Fan and Ruppert and Wand. The latter paper also studies, in the univariate case, nonparametric derivative estimators given by a locally weighted polynomial fitting. Compared with traditional kernel estimators, these estimators are often of simpler form and possess some better properties. In this paper, we develop current work on locally weighted regression and generalize locally weighted polynomial fitting to the estimation of partial derivatives in a multivariate regression context. Specifically, for both the regression and partial derivative estimators we prove joint asymptotic normality and derive explicit asymptotic expansions for their conditional bias and conditional convariance matrix (given observations of predictor variables) in each of the two important cases of local linear fit and local quadratic fit.     

带有辅助信息的ROC曲线两种估计的比较

在医学研究中,常常使用受试者操作特性曲线(ROC)曲线来研究两样本的比较问题。Lloyd构造了ROC曲线的核平滑估计,并给出了其渐近偏差以及渐近标准差。此外,当还可以获悉某一处理组上的辅助信息时,Zhou,Zhou & Ma利用经验似然的方法构造了ROC曲线的核平滑经验似然估计。本文利用"亏量"这个概念比较了带有辅助信息的情况下,对核平滑经验似然估计与完全经验似然估计进行了比较。并给出了核平滑经验似然估计优于完全经验似然估计的结论,并且随着样本容量的增大,该亏量也是无限增大的。     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

左截断右删失数据分位差估计及其渐近性质

我们研究了左截断右删失数据分位差,基于左截断右删失数据乘积限构造了分位差的经验估计,同时克服经验估计的非光滑性,提出了分位数差的核光滑估计.利用经验过程理论推导出这两个估计的渐近偏差和渐近方差,并且在左截断右删失数据下研究了这两个分位差的大样本性质,获得分位差估计的相合性和渐近正态性.同时给出计算模拟以验证光滑分位差估计的表现,在均方损失的意义下模拟结果表明光滑估计比经验估计具有更好的性质.     

Estimation and Bootstrap with Censored Data in Fixed Design Nonparametric Regression

We study Beran's extension of the Kaplan-Meier estimator for thesituation of right censored observations at fixed covariate values. Thisestimator for the conditional distribution function at a given value of thecovariate involves smoothing with Gasser-Müller weights. We establishan almost sure asymptotic representation which provides a key tool forobtaining central limit results. To avoid complicated estimation ofasymptotic bias and variance parameters, we propose a resampling methodwhich takes the covariate information into account. An asymptoticrepresentation for the bootstrapped estimator is proved and the strongconsistency of the bootstrap approximation to the conditional distributionfunction is obtained.     

对于金融资产漂移函数估计的双域组合方法(英文)

Time-domain state-domain methods are common approaches in modern financial analysis.Economic conditions vary time,drift function depends on time and price level for a given state variable.In this paper,to consistently estimate the bivariate drift function,our purpose a new dynamic integrated estimator by combing time-and state-domain methods for estimating drift function.And we establish its asymptotic properties and illustrates it outperforms some old ones by simulations.     

Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions

The problem of estimating the Markov renewal matrix and the semi-Markov transition matrix based on a history of a finite semi-Markov process censored at time T (fixed) is addressed for the first time. Their asymptotic properties are studied. We begin by the definition of the transition rate of this process and propose a maximum likelihood estimator for the hazard rate functions and then we show that this estimator is uniformly strongly consistent and converges weakly to a normal random variable. We construct a new estimator for an absolute continous semi-Markov kernel and give detailed derivation of uniform strong consistency and weak convergence of this estimator as the censored time tends to infinity. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the bias of the least squares estimator for the first order autoregressive process

The paper provides an exact formula for the bias of the parameter estimator of the first order autoregressive process and derives the asymptotic bias.     

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.     

Asymptotic normality of wavelet density estimator under censored dependent observations

In this paper, we discuss the asymptotic normality of the wavelet estimator of the density function based on censored data, when the survival and the censoring times form a stationary ??-mixing sequence. To simulate the distribution of estimator such that it is easy to perform statistical inference for the density function, a random weighted estimator of the density function is also constructed and investigated. Finite sample behavior of the estimator is investigated via simulations too.     

Local asymptotic behavior of regression splines for marginal semiparametric models with longitudinal data

In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f. Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly u...     

分数布朗运动模型中贝叶斯估计的渐近正态性

本文研究了由分数布朗运动驱动的线性随机微分方程中贝叶斯估计的渐近趋势．利用分数布朗运动的随机积分理论和Girsanov公式，得到了在平方损失函数下贝叶斯估计的渐近正态性．     

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.