Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

Altendji Belkais; Demongeot Jacques; Laksaci Ali; Rachdi Mustapha

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Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

机译：功能数据分析：根据随机左截断模型估算功能回归中的相对误差

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摘要

In this paper, we investigate the relationship between a functional random covariable and a scalar response which is subject to left-truncation by another random variable. Precisely, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional truncated data. Under some standard assumptions in functional data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure and to highlight its superiority over the classical kernel estimation, for different levels of simulated truncated data.

机译：在本文中，我们研究了功能随机协变量与标量响应之间的关系，标量响应受另一个随机变量的截断。精确地，我们使用均方根相对误差作为损失函数来构造这些功能截断数据的回归算子的非参数估计量。在功能数据分析中的一些标准假设下，我们建立了所构造估计量及其渐近正态性的几乎确定的一致性（随速率变化）。然后，在有限大小的样本上进行了仿真研究，以展示我们的估计程序的效率，并突出其针对不同水平的模拟截断数据的优于经典核估计的优势。

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来源
《Journal of nonparametric statistics》 |2018年第2期|472-490|共19页
作者
Altendji Belkais; Demongeot Jacques; Laksaci Ali; Rachdi Mustapha;
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作者单位

Univ Djillali Liabes Sidi Bel Abbes, Lab Math, Sidi Bel Abbes, Algeria;

Univ Grenoble Alpes, Fac Med Grenoble, Lab AGEIS, Equipe AGIM,EA 7407, La Tronche, France;

King Khalid Univ, Dept Math, Coll Sci, Abha, Saudi Arabia;

Univ Grenoble Alpes, Equipe AGIM, Lab AGEIS, UFR,SHS,EA 7407, BP 47, F-38040 Grenoble 09, France;

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正文语种 eng
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关键词
Functional data analysis (FDA); censored data; truncated data; small ball probability; functional regression; relative error;

机译：功能数据分析（FDA）;删失数据;截断数据;小球概率;功能回归;相对误差;

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1. On the Nonparametric Estimation of the Functional ψ-Regression for a Random Left-Truncation Model [J] . SALIHA DERRAR, ALI LAKSACI, ELIAS OULD SA?D Journal of statistical theory and practice . 2015,第3a4期

机译：随机左截断模型的函数ψ回归的非参数估计
2. Spline estimation of functional coefficient regression models for time series with correlated errors [J] . Michel H. Montoril, Pedro A. Morettin, Chang Chiann Statistics & Probability Letters . 2014,第Null期

机译：具有相关误差的时间序列的功能系数回归模型的样条估计
3. Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors [J] . Datta G.S., Rao J.N.K., Torabi M. Journal of Statistical Planning and Inference . 2010,第11期

机译：具有功能测量误差的嵌套误差线性回归模型下小面积均值的伪经验贝叶斯估计
4. A Probability Distribution Of Functional Random Variable With A Functional Data Analysis Application [C] . Etienne Cuvelier, Monique Noirhomme-Fraiture IEEE International Conference on Data Mining . 2006

机译：功能数据分析应用程序功能随机变量的概率分布
5. Essays on Nonparametric Identification: Identification of Dependent Multidimensional Unobserved Variables in a System of Linear Equations Identification and Estimation for Regressions with Errors in All Variables Identification of Nonparametrically Distributed Random Coefficients in Linear Panel Data Models. [D] . Ben-Moshe, Dan. 2012

机译：关于非参数识别的论文：线性方程组中相关多维多维观测变量的识别以及线性面板数据模型中非参数分布随机系数的所有变量中所有变量均具有误差的回归估计。
6. Semiparametric Estimation for Joint Modeling of Colorectal Cancer Risk and Functional Biomarkers Measured with Errors [O] . Qi Long, Xiaoxi Zhang, Roberd M. Bostick -1

机译：结直肠癌风险和功能生物标志物的联合建模的半甲酰估计
7. : "This paper deals with the functionality of the flaked stone assemblages from several Neolithic sites of the NE of the Iberian Peninsula. The chronological framework ranges from the mid-6th to the mid-4th millennium cal. BCE, which corresponds to the Early and Middle Neolithic. This long period ranges from the arrival of the first farming communities in the Iberian Peninsula, around 5600-5500 cal. BCE, to 3600-3500 cal. BCE, two millennia later, when Neolithic communities had stably occupied most of the available territory, from the mountainous areas of the Pyrenees to the coastal areas of the Mediterranean.In this context, the NE of the Iberian Peninsula is one of the areas where a major number of use-wear analyses have been carried out, especially for what concerns the Neolithic period. This is mainly due to the interest shown by the directors of the excavations and by the Spanish use-wear specialists as well; both participated and cooperated in order to enlarge the number of contexts studied, making the NE of the Iberian Peninsula one of the areas of Europe with the largest number of ‘functional studies’. As a result, today we have a reliable picture of the type of lithic tools used by the first farming groups, how they were used, and which needs they were satisfying.The methodology employed for this type of analysis is today broadly shared by most use-wear specialists. A stereoscopic microscope is used together with a reflected-light microscope for the analysis of the archaeological specimens. Afterwards, the observed use-wear traces are compared with the traces from experimental tools. This study primarily made use of the tools preserved at the ‘Traceoteque’ of the Institución Milà y Fontanals of the CSIC of Barcelona.This paper presents the results obtained for the different types of analysed sites and the relative lithic assemblages. In this way, tools from different contexts are compared: open-air sites, caves and rock-shelters, pit-sites, burials, and mining sites. Such a diversity of contexts allowed for the exploration of the existence of recurrences and differences in the functionality of the lithic tools from one site to another. The results obtained have also provided information on some aspects related to the economic processes carried out at the different sites, caves, rock-shelters and open-air contexts. It has been possible to gain fresh data on the types of tasks carried out and their relative importance for each site, their relation with the geographical and environmental context and the natural resources available there. In addition, analysing tools recovered from burials – mainly individual inhumations – provided information on the relationships between the buried individuals and the working tools, in particular exploring the relationships between sex and age patterns and the types of tools deposited as goods.In conclusion, use-wear analysis allowed for the exploration of a variety of issues; from technical aspects related to the production and management of the lithic resources to social aspects related to the subsistence activities and the individuals that carried them out." [O] . Gibaja Bao, Juan Francisco, Mazzucco, Niccolo 2015

机译：：“本文研究了伊比利亚半岛东北部几个新石器时代遗址的片状石材组合的功能。时间框架从公元前6世纪中叶到4世纪中叶，分别对应于早期和早期。新石器时代中期，这段时间的长短是从伊比利亚半岛的第一个农业社区到来，大约在公元前5600-5500 cal，到了两千年后的3600-3500 BCE，当时新石器时代的社区稳定地占据了大部分可用土地。从比利牛斯山脉山区到地中海沿岸地区。在这种情况下，伊比利亚半岛东北部是进行过大量使用磨损分析的地区之一，尤其是在关注方面新石器时代，这主要是由于发掘负责人和西班牙使用服装专家表现出的浓厚兴趣；他们参与和合作以扩大孔蒂的数量xts进行了研究，使伊比利亚半岛的NE成为“功能研究”数量最多的欧洲地区之一。结果，今天我们可以可靠地了解第一批耕作小组使用的石器工具的类型，使用方法以及它们满足的需求。今天，大多数使用方法广泛共享用于这种类型的分析的方法穿专家。立体显微镜与反射光显微镜一起用于分析考古样本。然后，将观察到的使用磨损痕迹与实验工具的痕迹进行比较。这项研究主要利用了巴塞罗那CSIC的米拉·丰塔纳尔研究所的“ Traceoteque”中保存的工具。本文介绍了针对不同类型的分析地点和相对石器组合所获得的结果。通过这种方式，可以比较来自不同环境的工具：露天场所，洞穴和岩石避难所，露天场所，埋葬场所和采矿场所。如此多样的上下文允许探索重复性的存在以及从一个站点到另一个站点的石器工具的功能差异。所获得的结果还提供了与在不同地点，洞穴，避难所和露天环境中进行的经济过程有关的某些方面的信息。可以获得有关所执行任务的类型及其对每个站点的相对重要性，它们与地理和环境环境以及那里可用自然资源的关系的新数据。此外，对从墓葬中回收的工具（主要是个人遗体）进行分析，可提供有关被掩埋的个体与工作工具之间关系的信息，尤其是探索性别与年龄模式之间的关系以及作为商品存放的工具的类型。磨损分析允许探索各种问题；从与石器资源的生产和管理有关的技术方面到与自给活动及其实施者有关的社会方面。”
8. Comparisons between Some Estimators in Functional Errors-in-Variables Regression Models. [R] . Carroll, R. J., Gallo, P. P. 1982

机译：功能性变量误差回归模型中某些估计量的比较。

Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

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