Abstract
Sign, Wilcoxon and Mann-Whitney tests are nonparametric methods in one or two-sample problems. The nonparametric methods are alternatives used for testing hypothesis when the standard methods based on the Gaussianity assumption are not suitable to be applied. Recently, the functional data analysis (FDA) has gained relevance in statistical modeling. In FDA, each observation is a curve or function which usually is a realization of a stochastic process. In the literature of FDA, several methods have been proposed for testing hypothesis with samples coming from Gaussian processes. However, when this assumption is not realistic, it is necessary to utilize other approaches. Clustering and regression methods, among others, for non-Gaussian functional data have been proposed recently. In this paper, we propose extensions of the sign, Wilcoxon and Mann-Whitney tests to the functional data context as methods for testing hypothesis when we have one or two samples of non-Gaussian functional data. We use random projections to transform the functional problem into a scalar one, and then we proceed as in the standard case. Based on a simulation study, we show that the proposed tests have a good performance. We illustrate the methodology by applying it to a real data set.
Highlights
Different phenomena in diverse fields can be modeled by means of random observations that are represented as curves
In the case of two-sample problems, testing hypothesis that the generating distributions of two sets of curves are identical has been approached in several contexts, such as differences in mean curves, covariance functions or cumulative distribution functions (CDFs) [14]
Random projections have been recently applied in many contexts of the functional data analysis (FDA) [28], such as goodness-of-fit tests [29], clustering [30], and ANOVA [21], among others
Summary
Different phenomena in diverse fields can be modeled by means of random observations that are represented as curves. In the case of two-sample problems, testing hypothesis that the generating distributions of two sets of curves are identical has been approached in several contexts, such as differences in mean curves, covariance functions or cumulative distribution functions (CDFs) [14]. Some methods based on permutations and bootstrap have become very popular for testing hypothesis with functional data [15]. To the best of our knowledge, no studies on the adaptation of the sign, Wilcoxon and Mann-Whitney statistics [22,23] to the context of functional data have been conducted. The objective of this paper is to derive the sign, Wilcoxon, and Mann-Whitney statistics using data from functional variables.
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