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摘要翻译：
随着现代科技的发展，功能数据在许多科学领域中被频繁地观测到。分析这类函数数据的一种流行方法是“先平滑，后估计”。也就是说，对函数数据的估计和假设检验等统计推断是基于用一种或另一种平滑技术得到的重构来替代潜在的单个函数来进行的。然而，对函数数据分析的这种替代效应知之甚少。本文研究了局部多项式核(LPK)平滑技术用于单个函数重构时的问题。我们发现在一些温和的条件下，替代效应可以渐近忽略。在此基础上，我们构造了泛函数据集的均值、协方差和噪声方差函数的基于LPK重构的估计，并推导了它们的渐近性。我们还提出了一个GCV规则来选择LPK重构的良好带宽。当均值函数也依赖于一些与时间无关的协变量时，我们考虑了一个泛函线性模型，其中均值函数与协变量线性相关，而协变量效应是时间的函数。构造了协变量效应和协方差函数的基于LPK重构的估计，并给出了它们的渐近性。此外，我们对一个关于协变效应的一般假设检验问题提出了一个基于$l^2$-范数的全局检验统计量，并给出了它的渐近随机表达式。通过仿真研究了GCV准则所选择的带宽对LPK重构和均值函数估计精度的影响。本文提出的方法是通过一个实际的气候学功能数据集的应用来说明的。
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英文标题：
《Statistical inferences for functional data》
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作者：
Jin-Ting Zhang, Jianwei Chen
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is ``smoothing first, then estimation.'' That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a $L^2$-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology.
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PDF链接：
https://arxiv.org/pdf/708.2207
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