The problem of comparison of second-order properties of two samples of random curves is considered. We propose and investigate tests for comparing covariance operators using the common empirical Karhunen-Loève expansion and truncated approximation of the Hilbert-Schmidt distance of the empirical covariance operators. The work is motivated by the study of the mechanical properties of short strands of DNA. We present results for a dataset of DNA minicircles obtained through the electron microscope. We also deal with the problem of alignment of random three-dimensional closed curves. In the last part of the talk we present a method that is resitant to the presence of atypical observations in functional samples. We introduce a class of dispersion operators and develop a spectrally truncated score test. The talk is based on joint work with Victor Panaretos and John Maddocks.
