A partial overview of the theory of statistics with functional data

•The paper provides a survey on the current FDA (Functional Data Analysis) theory.•The probabilistic foundations of FDA are briefly presented.•The notions of mean, median and mode are discussed in the functional setting.•Functional regression and classification methods are reviewed.•Dimension reduction and bootstrap techniques are discussed in the FDA setup.

The theory and practice of statistical methods in situations where the available data are functions (instead of real numbers or vectors) is often referred to as Functional Data Analysis (FDA). This subject has become increasingly popular from the end of the 1990s and is now a major research field in statistics.The aim of this expository paper is to offer a short tutorial as well as a partial survey of the state of the art in FDA theory. Both the selection of topics and the references list are far from exhaustive. Many interesting ideas and references have been left out for the sake of brevity and readability.In summary, this paper provides:(a)A discussion on the nature and treatment of the functional data.(b)A review of some probabilistic tools especially suited for FDA.(c)A discussion about how the usual centrality parameters, mean, median and mode, can be defined and estimated in the functional setting.(d)Short accounts of the main ideas and current literature on regression, classification, dimension reduction and bootstrap methods in FDA.(e)Some final comments regarding software for FDA.