Two Data Depths And Depth-weighted M-estimation

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 .