两类数据深度及深度加权M估计
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摘要
数据深度(或称为统计深度函数)作为一种多维数据的排序方法,已经被广泛应用于质量控制、多元回归、置信区域、聚类判别、非参检验和风险度量等众多领域.本文利用多元分析方法并结合凸分析方法,研究了两类数据深度,包括凸深度和投影型数据深度(包括弱投影深度和投影不变数据深度),提出了线性模型下基于数据深度的加权M估计。本文主要包括三大部分:
1.提出了凸数据深度的定义,并指出其实际上为一种理想数据深度,证明了一种数据深度为凸深度的充要条件,并研究了凸深度的若干性质:拟凹性、截尾区域、中位数、加权均值和截尾均值。介绍了凸深度的三种构造方法。
2.提出了投影型数据深度:弱投影深度(包含强投影深度)和投影不变数据深度。证明了数据深度为弱投影深度(包含强投影深度)和投影不变数据深度的充要条件,减弱了文献[93]中的定理2中关于强投影深度的充要条件,并指出了定理3中的一个遗漏条件。在实例的讨论中首次得到了半空间深度为强投影深度的实例。研究了文献[93]提出的广义数据投影深度和广义投影深度加权均值的若干稳健性质,丰富了现有文献的结论。
3.首次提出了线性模型下基于数据深度的加权M估计,证明了该估计有不随空间维数变化的比较好的崩溃点,并讨论在误差序列为φ混合,φ混合,ρ混合,(?)混合序列和NA序列,在矩条件较弱的条件下得到了该估计的强相合性,在实质上改进了现有结论。
Data depths (statistical depth functions) have been widely applied to quality control, mulitivariate regression, clustering and calssification,nonparametric test and risk measure. In this dissertation, we study some data depths, including convex data depth,weak projection depth,strong projection depth and projection invariance depth. We put forword a weighted M-estimation based on data depth.
The thesis consists mainly of three parts:
1.Convex data depth is introduced and it is pointed out that the convex depth is an ideal data depth. A sufficient and necessary condition for convex depth is proved. Some properties of convex depth are studied, including qusi-concavity,trimmed region,median,weighted mean and trimmed weighted mean. Three constructive methods of convex data depth are introduced.
2. Projection type data depths:weak projection depth ( including strong projection depth) and projection invariance depth are defined. sufficient and necessary conditions for the weak projection depth ( including strong projection depth) and projection invariance depth are proved. One sufficient and necessary condition in Theorem 2 in a paper by Rainer Dyckerhoff (Allg. Stat. Arch. 88:163-190, 2004) is weakened. One missing condition in Theorem 3 in the paper is stated . An example that Halfspace depth can be a strong projection depth is given firstly. Some robust properties of generalized projection depth and generalized projection depth mean are stduied . Existing results are improved.
3. Weighted M-estimation based on data depth is defined fistly in linear model.High breakdown point of the estimation independent of dimension is achieved. NA sequence and some mixing sequences:φ,φ,ρand (ρ|~) considered as error sequences in linear model are discussed, and the strong consistency of the estimation is obtained in lower moment condition. Result is greatly better than the corresponding result .
1.提出了凸数据深度的定义,并指出其实际上为一种理想数据深度,证明了一种数据深度为凸深度的充要条件,并研究了凸深度的若干性质:拟凹性、截尾区域、中位数、加权均值和截尾均值。介绍了凸深度的三种构造方法。
2.提出了投影型数据深度:弱投影深度(包含强投影深度)和投影不变数据深度。证明了数据深度为弱投影深度(包含强投影深度)和投影不变数据深度的充要条件,减弱了文献[93]中的定理2中关于强投影深度的充要条件,并指出了定理3中的一个遗漏条件。在实例的讨论中首次得到了半空间深度为强投影深度的实例。研究了文献[93]提出的广义数据投影深度和广义投影深度加权均值的若干稳健性质,丰富了现有文献的结论。
3.首次提出了线性模型下基于数据深度的加权M估计,证明了该估计有不随空间维数变化的比较好的崩溃点,并讨论在误差序列为φ混合,φ混合,ρ混合,(?)混合序列和NA序列,在矩条件较弱的条件下得到了该估计的强相合性,在实质上改进了现有结论。
Data depths (statistical depth functions) have been widely applied to quality control, mulitivariate regression, clustering and calssification,nonparametric test and risk measure. In this dissertation, we study some data depths, including convex data depth,weak projection depth,strong projection depth and projection invariance depth. We put forword a weighted M-estimation based on data depth.
The thesis consists mainly of three parts:
1.Convex data depth is introduced and it is pointed out that the convex depth is an ideal data depth. A sufficient and necessary condition for convex depth is proved. Some properties of convex depth are studied, including qusi-concavity,trimmed region,median,weighted mean and trimmed weighted mean. Three constructive methods of convex data depth are introduced.
2. Projection type data depths:weak projection depth ( including strong projection depth) and projection invariance depth are defined. sufficient and necessary conditions for the weak projection depth ( including strong projection depth) and projection invariance depth are proved. One sufficient and necessary condition in Theorem 2 in a paper by Rainer Dyckerhoff (Allg. Stat. Arch. 88:163-190, 2004) is weakened. One missing condition in Theorem 3 in the paper is stated . An example that Halfspace depth can be a strong projection depth is given firstly. Some robust properties of generalized projection depth and generalized projection depth mean are stduied . Existing results are improved.
3. Weighted M-estimation based on data depth is defined fistly in linear model.High breakdown point of the estimation independent of dimension is achieved. NA sequence and some mixing sequences:φ,φ,ρand (ρ|~) considered as error sequences in linear model are discussed, and the strong consistency of the estimation is obtained in lower moment condition. Result is greatly better than the corresponding result .
引文
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