  In this article, we establish the mathematical foundations for modeling the
randomness of shapes and conducting statistical inference on shapes using the
smooth Euler characteristic transform. Based on these foundations, we propose
two chi-squared statistic-based algorithms for testing hypotheses on random
shapes. Simulation studies are presented to validate our mathematical
derivations and to compare our algorithms with state-of-the-art methods to
demonstrate the utility of our proposed framework. As real applications, we
analyze a data set of mandibular molars from four genera of primates and show
that our algorithms have the power to detect significant shape differences that
recapitulate known morphological variation across suborders. Altogether, our
discussions bridge the following fields: algebraic and computational topology,
probability theory and stochastic processes, Sobolev spaces and functional
analysis, analysis of variance for functional data, and geometric
morphometrics.
