Lorenz Curve for Truncated and Censored Data

In this paper, we consider a nonparametric estimator of the Lorenz curve and Gini index when the data are subjected to random left truncation and right censorship. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. A law of the iterated logarithm for the Lorenz curve is also derived. Lastly, we obtain a central limit theorem for the corresponding Gini index.     

关于洛伦兹曲线和基尼系数的统计推断

本文对于洛伦兹曲线的最一般定义,在无任何附加限制条件下,论证了基于样本数据所得到的洛仑兹曲线的通常估计量具有强相合性.在此基础上,证明了基尼系数和Schutz系数的通常估计量均具有强相合性.此外,导出了基尼系数估计量的渐进分布,并在此基础上给出了大样本情形下基尼系数的置信区间.     

Asymptotic theory of simultaneous estimation of Poisson means

The Poisson distribution is often a good approximation to the underlying sampling distribution and is central to the study of categorical data. In this paper, we propose a new unified approach to an investigation of point properties of simultaneous estimations of Poisson population parameters with general quadratic loss functions. The main accent is made on the shrinkage estimation. We build a series of estimators that could be represented as a convex combination of linear statistics such as maximum likelihood estimator (benchmark estimator), restricted estimator, composite estimator, preliminary test estimator, shrinkage estimator, positive rule shrinkage estimator (James-Stein type estimator). All these estimators are represented in a general integrated estimation approach, which allows us to unify our investigation and order them with respect to the risk. A simulation study with numerical and graphical results is conducted to illustrate the properties of the investigated estimators.     

Shrinkage estimator in normal mean vector estimation based on conditional maximum likelihood estimators

Estimation of normal mean vector has broad applications such as small area estimation, estimation of nonparametric functions and estimation of wavelet coefficients. In this paper, we propose a new shrinkage estimator based on conditional maximum likelihood estimator incorporating with Stein’s risk unbiased estimator (SURE) when data have the normality. We present some theoretical work and provide numerical studies to compare with some existing methods.     

Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models

Abstract

Maximum pseudo-likelihood estimation has hitherto been viewed as a practical but flawed alternative to maximum likelihood estimation, necessary because the maximum likelihood estimator is too hard to compute, but flawed because of its inefficiency when the spatial interactions are strong. We demonstrate that a single Newton-Raphson step starting from the maximum pseudo-likelihood estimator produces an estimator which is close to the maximum likelihood estimator in terms of its actual value, attained likelihood, and efficiency, even in the presence of strong interactions. This hybrid technique greatly increases the practical applicability of pseudo-likelihood-based estimation. Additionally, in the case of the spatial point processes, we propose a proper maximum pseudo-likelihood estimator which is different from the conventional one. The proper maximum pseudo-likelihood estimator clearly shows better performance than the conventional one does when the spatial interactions are strong.     

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

Optimal estimation of a signal perturbed by a sub-fractional Brownian motion

We consider the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion. We obtain the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian.     

具有重尾或异方差误差的双变量预测回归模型的中位无偏估计

In this paper, we consider median unbiased estimation of bivariate predictive regression models with non-normal, heavy-tailed or heterescedastic errors. We construct confidence intervals and median unbiased estimator for the parameter of interest. We show that the proposed estimator has better predictive potential than the usual least squares estimator via simulation. An empirical application to finance is given. And a possible extension of the estimation procedure to cointegration models is also described.     

基于轮换样本连续性调查的校准组合估计方法研究

校准估计是抽样调查中比较常用的一种利用辅助信息提高估计量精度的方法。回归组合估计量作为轮换样本连续性调查中使用的一种有效的估计量,是可以通过校准程序得到的。基于回归组合估计量和校准程序之间的关系,本文提出了轮换样本连续性抽样调查条件下的不同校准组合估计量及其方差估计。校准组合估计量的主要思想是在校准估计程序中将拼配样本和非拼配样本的辅助信息进行不同的组合利用。本文利用美国现时人口调查的微观数据进行数值模拟,来比较不同校准组合估计量的估计效率,模拟结果表明两步校准组合估计量和两步校准双组合估计量的表现相似,且估计精度都高于H-T估计量及回归组合估计量;而两步校准组合估计量由于其简便性更适合应用于实践中。最后以我国农村住户连续性抽样调查为例,设计一套符合我国实际的轮换样本连续性调查方案,并将提出的校准组合估计量运用于估计阶段,为中国政府统计调查提供一定的借鉴和参考.     

Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.     

Median Unbiased Estimation of Bivariate Predictive Regression Models with Heavy-tailed or Heteroscedastic Errors

In this paper, we consider median unbiased estimation of bivariate predictive regression models with non-normal, heavy-tailed or heteroscedastic errors. We construct confidence intervals and median unbiased estimator for the parameter of interest. We show that the proposed estimator has better predictive potential than the usual least squares estimator via simulation. An empirical application to finance is given. And a possible extension of the estimation procedure to cointegration models is also described.     

Concentration curve for truncated and censored data

Concentration curve is the inverse Lorenz curve. Together, they form the basis for most measures of distributional inequality. In this paper, we consider the empirical estimator of the concentration curve when the data are subjected to random left truncation and/or right censorship. Simultaneous strong Gaussian approximations for the associated Lorenz and normed concentration processes are established under appropriate assumptions. Functional laws of the iterated logarithm for the two processes are established as easy consequences. The construction provides a solid foundation for the study of functional statistics based on the two processes.     

Nonparametric estimation of multivariate density with direct and auxiliary data and application

We consider the problem of multivariate density estimation, using samples from the distribution of interest as well as auxiliary samples from a related distribution. We assume that the data from the target distribution and the related distribution may occur individually as well as in pairs. Using nonparametric maximum likelihood estimator of the joint distribution, we derive a kernel density estimator of the marginal density. We show theoretically, in a simple special case, that the implied estimator of the marginal density has smaller integrated mean squared error than that of a similar estimator obtained by ignoring dependence of the paired observations. We establish consistency of the marginal density estimator under suitable conditions. We demonstrate small sample superiority of the proposed estimator over the estimator that ignores dependence of the samples, through a simulation study with dependent and non-normal populations. The application of the density estimator in nonparametric classification is also discussed. It is shown that the misclassification probability of the resulting classifier is asymptotically equivalent to that of the Bayes classifier. We also include a data analytic illustration.     

Semiparametric estimation in regression with missing covariates using single-index models

We investigate semiparametric estimation of regression coefficients through generalized estimating equations with single-index models when some covariates are missing at random. Existing popular semiparametric estimators may run into difficulties when some selection probabilities are small or the dimension of the covariates is not low. We propose a new simple parameter estimator using a kernel-assisted estimator for the augmentation by a single-index model without using the inverse of selection probabilities. We show that under certain conditions the proposed estimator is as efficient as the existing methods based on standard kernel smoothing, which are often practically infeasible in the case of multiple covariates. A simulation study and a real data example are presented to illustrate the proposed method. The numerical results show that the proposed estimator avoids some numerical issues caused by estimated small selection probabilities that are needed in other estimators.

     

The Accuracy and the Computational Complexity of a Multivariate Binned Kernel Density Estimator

The computational cost of multivariate kernel density estimation can be reduced by prebinning the data. The data are discretized to a grid and a weighted kernel estimator is computed. We report results on the accuracy of such a binned kernel estimator and discuss the computational complexity of the estimator as measured by its average number of nonzero terms.     

Two linear-time algorithms for computing the minimum length polygon of a digital contour

The Minimum Length Polygon (MLP) is an interesting first order approximation of a digital contour. For instance, the convexity of the MLP is characteristic of the digital convexity of the shape, its perimeter is a good estimate of the perimeter of the digitized shape. We present here two novel equivalent definitions of MLP, one arithmetic, one combinatorial, and both definitions lead to two different linear time algorithms to compute them. This paper extends the work presented in Provençal and Lachaud (2009) [26], by detailing the algorithms and providing full proofs. It includes also a comparative experimental evaluation of both algorithms showing that the combinatorial algorithm is about 5 times faster than the other. We also checked the multigrid convergence of the length estimator based on the MLP.     

Asymptotically Local Minimax Estimation of Infinitely Smooth Density with Censored Data

The problem of the nonparametric minimax estimation of an infinitely smooth density at a given point, under random censorship, is considered. We establish the exact asymptotics of the local minimax risk and propose the efficient kernel-type estimator based on the well known Kaplan-Meier estimator.     

Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems

This work presents two simple and robust techniques based on time delay estimation for the respective control and synchronization of chaos systems. First, one of these techniques is applied to the control of a chaotic Lorenz system with both matched and mismatched uncertainties. The nonlinearities in the Lorenz system is cancelled by time delay estimation and desired error dynamics is inserted. Second, the other technique is applied to the synchronization of the Lü system and the Lorenz system with uncertainties. The synchronization input consists of three elements that have transparent and clear meanings.Since time delay estimation enables a very effective and efficient cancellation of disturbances and nonlinearities, the techniques turn out to be simple and robust. Numerical simulation results show fast, accurate and robust performance of the proposed techniques, thereby demonstrating their effectiveness for the control and synchronization of Lorenz systems.     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

Estimation in additive cox models by marginal integration

Assuming an additive model on the covariate effect in proportional hazards regression, we consider the estimation of the component functions. The estimator is based on the marginal integration method. Then we use a new kind of nonparametric estimator as the pilot estimator of the marginal integration. The pilot estimator is constructed by an analogy to the two-sample problems and by appealing to the principles of local partial likelihood and local linear fitting. We derive the asymptotic distribution of the marginal integration estimator of the component functions. The result of a simulation study is also given.