Functional data analysis with rough sample paths?

Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is imposed to allow for the use of smoothing techniques that annihilate the noise. At the same time, imposing the smoothness assumption excludes a considerable range of stochastic processes, most notably diffusion processes. Under perfect observation of the sample paths, such processes would not need to be excluded from the realm of functional data analysis. In this paper, we introduce a careful modification of existing methods, dubbed the 'reflected triangle estimator', and show that this allows for the functional data analysis of processes with nowhere differentiable sample paths, even when these are discretely and noisily observed, including under irregular and sparse designs. Our estimator matches the established rates of convergence for processes with smooth paths, and furthermore attains the same optimal rates as one would get under perfect observation. Thus, with reflected triangle estimation, the scope of applicability of much of the methodology developed for discretely/irregularly/noisily/sparsely sampled functional data is considerably extended. By way of simulation it is shown that the advantages furnished are reflected in practice, hinting at potential closer links with the field of diffusion inference.