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Mahalanobis distance is a classical tool in multivariate analysis. We suggesthere an extension of this concept to the case of functional data. Moreprecisely, the proposed definition concerns those statistical problems wherethe sample data are real functions defined on a compact interval of the realline. The obvious difficulty for such a functional extension is thenon-invertibility of the covariance operator in infinite-dimensional cases.Unlike other recent proposals, our definition is suggested and motivated interms of the Reproducing Kernel Hilbert Space (RKHS) associated with thestochastic process that generates the data. The proposed distance is a truemetric; it depends on a unique real smoothing parameter which is fullymotivated in RKHS terms. Moreover, it shares some properties of its finitedimensional counterpart: it is invariant under isometries, it can beconsistently estimated from the data and its sampling distribution is knownunder Gaussian models. An empirical study for two statistical applications,outliers detection and binary classification, is included. The obtained resultsare quite competitive when compared to other recent proposals of theliterature.

On Mahalanobis distance in functional settings