On the cumulative quantile regression process

Let (X, Y) be a bivariate random vector and F(x) the marginal distribution function of X. The quantile regression (QR) function of Y on X is defined as r(u) = E[Y | F(X) = u] and the cumulative QR function (CQR) M(u) as its integral over [0, u]. The empirical counterpart based on a sample of size n is M _n (u). In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on M _in (u). A law of the iterated logarithm for the CQR process follows from our result.     

Composing the cumulative quantile regression function and the Goldie concentration curve

The model we discuss in this paper deals with inequality in distribution in the presence of a covariate. To elucidate that dependence, we propose to consider the composition of the cumulative quantile regression (CQR) function and the Goldie concentration curve, the standardized counterpart of which gives a fraction to fraction plot of the response and the covariate. It has the merit of enhancing the visibility of inequality in distribution when the latter is present. We shall examine the asymptotic properties of the corresponding empirical estimator. The associated empirical process involves a randomly stopped partial sum process of induced order statistics. Strong Gaussian approximations of the processes are constructed. The result forms the basis for the asymptotic theory of functional statistics based on these processes.     

Composite Quantile Regression Estimation for Left Censored Response Longitudinal Data

For left censored response longitudinal data, we propose a composite quantile regression estimator (CQR) of regression parameter. Statistical properties such as consistency and asymptotic normality of CQR are studied under relaxable assumptions of correlation structure of error terms. The performance of CQR is investigated via simulation studies and a real dataset analysis.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.     

部分线性单指标模型的复合分位数回归及变量选择

本文提出复合最小化平均分位数损失估计方法 (composite minimizing average check loss estimation,CMACLE)用于实现部分线性单指标模型(partial linear single-index models,PLSIM)的复合分位数回归(composite quantile regression,CQR).首先基于高维核函数构造参数部分的复合分位数回归意义下的相合估计,在此相合估计的基础上,通过采用指标核函数进一步得到参数和非参数函数的可达最优收敛速度的估计,并建立所得估计的渐近正态性,比较PLSIM的CQR估计和最小平均方差估计(MAVE)的相对渐近效率.进一步地,本文提出CQR框架下PLSIM的变量选择方法,证明所提变量选择方法的oracle性质.随机模拟和实例分析验证了所提方法在有限样本时的表现,证实了所提方法的优良性.     

一个有效估计:半参数非时齐扩散模型的局部线性复合分位回归估计

主要研究半参数非时齐扩散模型的参数估计问题.基于非时齐扩散模型的离散观测样本,首先得到漂移参数的局部线性复合分位回归估计,并证明估计量的渐近偏差、渐近方差和渐近正态性.其次,讨论了带宽的选择和局部线性复合分位回归估计关于局部线性最小二乘估计的渐近相对效,所得到的局部估计较局部线性最小二乘估计更为有效.最后,通过模拟说明了局部线性复合分位回归估计比局部线性最小二乘估计的模拟效果更好.     

Testing in linear composite quantile regression models

Composite quantile regression (CQR) can be more efficient and sometimes arbitrarily more efficient than least squares for non-normal random errors, and almost as efficient for normal random errors. Based on CQR, we propose a test method to deal with the testing problem of the parameter in the linear regression models. The critical values of the test statistic can be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and power evaluation is possible under the local alternatives. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from a walking behavior survey.     

Copula and composite quantile regression-based estimating equations for longitudinal data

Annals of the Institute of Statistical Mathematics - Composite quantile regression (CQR) is a powerful complement to the usual mean regression and becomes increasingly popular due to its robustness...     

An effective method to reduce the computational complexity of composite quantile regression

In this article, we aim to reduce the computational complexity of the recently proposed composite quantile regression (CQR). We propose a new regression method called infinitely composite quantile regression (ICQR) to avoid the determination of the number of uniform quantile positions. Unlike the composite quantile regression, our proposed ICQR method allows combining continuous and infinite quantile positions. We show that the proposed ICQR criterion can be readily transformed into a linear programming problem. Furthermore, the computing time of the ICQR estimate is far less than that of the CQR, though it is slightly larger than that of the quantile regression. The oracle properties of the penalized ICQR are also provided. The simulations are conducted to compare different estimators. A real data analysis is used to illustrate the performance.     

Systems of sets with multiplicative asymptotic density

The authors define a notion of system of sets with multiplicative asymptotic density in this paper. A criterion and one necessary condition for a given system {A _i } _i=1^∞ to be a system with multiplicative asymptotic density is given. Properties of certain special types of systems of sets with multiplicative asymptotic density are treated. This work is supported by The Ministry of Education, Youth and Sports of the Czech Republic. Project CQR 1M06047.     

A numerical method of nondifferentiable convex optimization with constraints

An iterative process is examined for minimizing a convex nondifferentiable functional on a convex closed set in a real Hilbert space. Convergence of the proposed process is proved. A two-sided bound on the optimal functional value is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 124–131, 1985.     

带有反馈机制的单服务台排队系统的泛函重对数律

本文考虑了一个带有贝努里反馈机制的单服务台排队系统.我们将该系统的一些数量指标如队长过程,忙期过程,负荷过程的泛函重对数律的问题转化为一个反射布朗运动相关的问题,利用已有的布朗运动的重对数率的结果,刻画了队长过程,忙期过程,负荷过程的重对数律.     

Functional estimation incorporating prior correlation information

This paper deals with the problem of estimating functional data from a functional noise model, i.e., on the basis of the observations of a discrete-time stochastic process in additive white noise which can be correlated with the process. Assuming prior information on the correlation functions involved and using principal component analysis for stochastic processes, a general suboptimum estimation procedure is derived. The proposed solution is valid for smoothing, filtering and prediction problems, can be applied to estimate any operation of the process, such as derivatives, and constitutes a computationally efficient algorithm.     

An approximate method for solving the convex programming problem

We consider an iterative process for maximization of a convex nondifferentiable functional in a real Hilbert space. Two-sided bounds on the optimal functional value are derived. Stability of the approximate solutions is considered. Convergence of the proposed iterative process is proved.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 122–129, 1986     

An integrated toolchain for model based functional safety analysis

The significant increase in the complexity and autonomy of the hardware systems renders the verification of the functional safety of each individual component as well as of the entire system a complex task and underlines the need for integrated, model based tools that would assist this process. In this paper the authors present such a tool, coupled with an approach to functional safety analysis, based on the integration of functional tests into the model itself. The analysis of the resulting model is done through a stochastic Bayesian model. This approach strives to both bypass the necessity for costly hardware testing and integrate the functional safety analysis into an intuitive component development process.     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

Sequential Estimation for a Functional of the Spectral Density of a Gaussian Stationary Process

Integral functional of the spectral density of stationary process is an important index in time series analysis. In this paper we consider the problem of sequential point and fixed-width confidence interval estimation of an integral functional of the spectral density for Gaussian stationary process. The proposed sequential point estimator is based on the integral functional replaced by the periodogram in place of the spectral density. Then it is shown to be asymptotically risk efficient as the cost per observation tends to zero. Next we provide a sequential interval estimator, which is asymptotically efficient as the width of the interval tends to zero. Finally some numerical studies will be given.     

On the structure of regular infinitely divisible point processes

A representation for the probability generating functional (p.g.fl.) of a regular infinitely divisible (i.d.) stochastic point process, motivated as a generalization of the Gauss-Poisson process, is presented. The functional is characterized by a sequence of Borel product measures. Necessary and sufficient conditions, in terms of these Borel measures, are given for this representation to be a p.g.fl., thus characterizing all regular i.d. point processes.     

The boundary value problem and the Jensen inequality for an entropy functional of a Markov diffusion process

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, expressed through an additive functional of the diffusion process, with a Lagrangian, determined by the parameters of a controlled stochastic equation. These results include a minimum condition for the entropy functional and the functional's Jensen inequality, which both are useful for the solution of important mathematical and applied problems.     

马尔可夫过程H-值可加泛函的向前向后鞅分解

本文研究了马尔可夫H-值可加泛函的向前向后鞅分解.利用Lyons-Meyer-Zheng鞅分解得到了泛函数极限定理所必需的极大不等式和紧性结果,在最小条件限度内得到了马尔可夫过程经验测度的泛函中心极限定理,将该定理从实值情形推广到了希尔伯特值情形.