Structural properties and convergence results for contours of sample statistical depth functions

Zuo, Yijun; Serfling, Robert

doi:10.1214/aos/1016218227

Cited by 116 publications

(64 citation statements)

References 16 publications

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“…In the next section, trimmed regions and contours are defined through the idea of depth. Moreover, it is proved that the trimmed regions constructed with the band depth verify the properties in Zuo and Serfling (2000) for finite-dimensional observations. The notion of central region is introduced in the third section, where some of their estimators are analyzed.…”

Section: Introductionmentioning

confidence: 77%

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Depth-based inference for functional data

López‐Pintado

Romo

2007

Computational Statistics & Data Analysis

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We propose robust inference tools for functional data based on the notion of depth for curves. We extend the ideas of trimmed regions, contours and central regions to functions and study their structural properties and asymptotic behavior. Next, we introduce a scale curve to describe dispersion in a sample of functions. The computational burden of these techniques is not heavy and so they are also adequate to analyze high-dimensional data. All these inferential methods are applied to different real data sets. AbstractWe propose robust inference tools for functional data based on the notion of depth for curves. We extend the ideas of trimmed regions, contours and central regions to functions and study their structural properties and asymptotic behavior. Next, we introduce a scale curve to describe dispersion in a sample of functions. The computational burden of these techniques is not heavy and so they are also adequate to analyze high-dimensional data. All these inferential methods are applied to different real data sets.

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Section: Introductionmentioning

confidence: 77%

“…Zuo and Serfling (2000) studied the structural properties of regions and contours for statistical depth functions. Next we extend them to functional observations.…”

Section: Introductionmentioning

confidence: 99%

Depth-based inference for functional data

López‐Pintado

Romo

2007

Computational Statistics & Data Analysis

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“…Classical depth functions associate with any center of symmetry a maximal depth value. Together with the fact that depth decreases along any halfline originating from any deepest point, this leads to nested star-shaped (in most cases convex) depth regions, whatever the underlying distribution may be nonconvex (Zuo and Serfling (2000)). Distributions that are multimodal or have nonconvex support however are present in many economic applications (mixture models, multi-regime time series or issues solved by means of clustering procedures).…”

Section: Ranks Induced By Depth Functionsmentioning

confidence: 99%

“…In this paper we use the concept of local depth proposed by Paindaveine and Van Bever (2013) and implemented among others in Kosiorowski and Zawadzki (2014). Thorough presentation of the depth concept may be found in (Zuo and Serfling, 2000;Mosler, 2013;Nieto-Reyes and Battey, 2016 …”

Section: Ranks Induced By Depth Functionsmentioning

confidence: 99%

“…, 2x − x n . Sample properties of the local versions of depths result from general findings presented in (Zuo and Serfling, 2000). Implementations of local versions of several depths including projection depth, Student, simplicial, L p depth, regression depth and modified band depth can be found in free R package DepthProc (see Kosiorowski and Zawadzki (2014)).…”

Section: Ranks Induced By Depth Functionsmentioning

confidence: 99%

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Detecting a structural change in functional time series using local Wilcoxon statistic

2017

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Functional data analysis (FDA) (Ramsay et al. (2009);Ramsay and Silverman (2005)) is a part of modern multivariate statistics that analyses data providing information about curves, surfaces or anything else varying over a certain continuum. In economics and empirical finance we often have to deal with time series of functional data, where we cannot easily decide, whether they are to be considered as homogeneous or heterogeneous. At present a discussion on adequate tests of homogenity for functional data is carried (see e.g. Flores et al. (2015)). We propose a novel statistic for detetecting a structural change in functional time series based on a local Wilcoxon statistic induced by a local depth function proposed in Paindaveine and Van Bever (2013).

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Process Capability Analysis for NonLinear Profiles Using Depth Functions

Guevara

Vargas

2013

Qual. Reliab. Engng. Int.

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There are practical situations in which the quality of a process or product can be better characterized by a functional relationship between a response variable and one or more explanatory variables, which is called profile. Such profiles frequently can be represented adequately using linear or nonlinear models. While there are several studies in monitoring profiles, there are few studies to evaluate the capability of a process with profile quality characteristic; specifically, there is no method in the literature to analyze process capability characterized by nonlinear profiles. In this paper, we propose two methods to measure the capability of these processes, based on the concept of functional depth. These methods do not have distributional assumptions and extend to functional data the Process Capability Indexes proposed by Clements 1 to measure the capability of a process characterized by a random variable. Performance of the proposed methods is evaluated through simulation studies. An example illustrates the applicability of these methods.

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Structural properties and convergence results for contours of sample statistical depth functions

Cited by 116 publications

References 16 publications

Depth-based inference for functional data

Depth-based inference for functional data

Detecting a structural change in functional time series using local Wilcoxon statistic

Process Capability Analysis for NonLinear Profiles Using Depth Functions

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