Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations
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Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations Alba M. Franco Pereira1,2
· Rosa E. Lillo2,3
Received: 9 March 2019 / Revised: 16 November 2019 / Accepted: 19 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Visualization techniques are very useful in data analysis. Their aim is to summarize information into a graph or a plot. In particular, visualization is especially interesting when one has functional data, where there is no total order between the data of a sample. Taking into account the information provided by the down–upward partial orderings based on the hypograph and the epigragh indexes, we propose new strategies to analyze graphically functional data. In particular, combining the two indexes we get an alternative way to measure centrality in a bunch of curves, so we get an alternative measure to the statistical depth. Besides, motivated by the visualization in the plane of the two measures for two functional data samples, we propose new methods for testing homogeneity between two groups of functions. The performance of the tests is evaluated through a simulation study and we have applied them to several real data sets. Keywords Data depth · Rank test · Epigraph · Hypograph · Functional data · Order statistics Mathematics Subject Classification 62G10 · 62H30 · 65S05
1 Introduction The data output sophistication in different research fields requires to advance in the statistical analysis of complex data. In functional data analysis (FDA), each observation is a real function xi (t), i = 1, . . . , n, t ∈ I , where I is an interval in R.
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Alba M. Franco Pereira [email protected]
1
Department of Statistics and OR, Complutense University of Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
2
UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Madrid, Spain
3
Department of Statistics, Carlos III University of Madrid, C/Madrid, 126, 28903 Getafe, Madrid, Spain
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A. M. F. Pereira, R. E. Lillo
Multivariate techniques such as principal components, analysis of variance and regression methods have already been extended to a functional context; the books by Ramsay and Silverman (2005a, b) and Ferraty and Vieu (2006) offer comprehensive introductions to FDA and its applications, while Horváth and Kokoszka (2012), Hsing and Eubank (2015) and Wang et al. (2016) review some recent developments for functional data. A fundamental task in functional data analysis is to provide an ordering within a sample of curves that allows the definition of order statistics such as ranks and L-statistics. An important tool to analyze these functional data aspects is the idea of statistical depth. Several definitions of depth for functional data have been introduced. See for example, Vardi and Zhang (2000), Fraiman and Muniz (2001), Cuevas et al. (2007), Cuesta-Albertos and Nieto-Reyes (2008), López-Pintado and Romo (2009) and López-Pintado and Romo (2011) and Sguera et al. (2014). The definition of depth
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