Metric Sample Spaces of Continuous Geometric Curves and Estimation of Their Centroids

The metric sample space of Fréechet curves (Fréechet , 1934,1951,1961) is based on a generalization of regular curves that covers continuous curves in full generality. This makes it possible to deal with both smooth and non-smooth, even non-rectifiable geometric curves in statistical analysis. In the present paper this sample space is further extended in two directions that are relevant in practice: to incorporate information on landmark points in the curves and to impose invariance with respect to an arbitrary group of isometric spatial transformations. Properties of the introduced sample spaces of curves are studied, specially those concerning to the generation and representation of random curves by random functions.In order to provide measures of central tendency and dispersion of random curves, centroids and restricted centroids ofrandom curves are defined in a general metric framework, and methods for their consistent estimation are derived.