【作者】 刘鹏；

【导师】 张忠占；

【作者基本信息】 北京工业大学 ， 统计学， 2021， 硕士

【摘要】 函数型数据广泛存在于医疗、经济、气象等各研究领域,因此函数型数据的统计分析方法有着巨大的应用价值,其基本原理是把观测数据的函数当做单个实体,而不仅仅是当做单个观测序列。检验函数型数据协方差是否相等是函数型数据方差分析的重要问题。之前的文章有提出基于L~2范数的检验方法,该方法在函数型数据相关性较小,且不存在局部峰值时表象良好,但上述情况在实际应用中不一定都能满足,为解决这个问题,本文采用基于上确界范数的检验。部分观测函数型数据是基于实际情况提出的问题,有别于传统方法假设函数型数据样本在相同区间上观测得到,这里假设样本只在总区间的子集上有观测结果,其余部分缺失。该问题有很大应用价值而相关领域的研究有待发展。本文第二章介绍了关于协方差函数检验的基础知识,第三章给出部分观测函数型数据的模型,并给出了协方差算子和均值函数的相合估计,接着给出了基于上确界范数的检验统计量。第四章研究了该检验统计量的分布和渐近性质。理论分布在实际应用中非常不便,因此提出用自助法研究实际问题,并给出了该方法的理论基础。最后在第五章利用数值模拟的方法验证了所提出方法的有效性。更多还原

【Abstract】 Functional data widely exist in medical,economic,meteorological and other research fields,so the statistical analysis methods of functional data have great application value.The basic idea of the functiongal data analysis is to treat the observed function as a single entity,not just as a single observation sequence.It is an important problem to test whether the covariances of functional variables are equal.Previous articles have proposed a test method based on the L2-norm,which is good when the correlation of functional data is small and there is no local spikes,but the above situation may not be satisfied in practical application.In order to solve this problem,this thesis uses the test based on the supremum-norm.Partially observed functional variables are often faced in the practical situations,which are different from the traditional method of assuming that the functional data samples are observed in the common interval.Here,it is assumed that the samples are only observed in a subset of the total interval,and the rest are missing.This problem has great application value,but the research in related fields needs to be developed.In the second chapter,we introduce the basic knowledge of test,we give the model of partial observation function data,and give the consistency estimates of covariance operator and mean function in chapter 3.Then we give the test statistic for null hypothesis of homogeneity based on supremum-norm,and study its asymptotic distribution.The theoretical distribution is very inconvenient in practical application,so we proposes to use the bootstrap method to study practical problems in chapter 4,and gives the theoretical basis of the method.Finally,the effectiveness of the proposed method is verified by numerical simulation in chapter 5.更多还原