In this paper we deepen the study of the nonlinear principal components introduced by Salinelli in 1998, referring to a real random variable. New insights on their probabilistic and statistical meaning are given with some properties. An estimation procedure based on spline functions, adapting
to a statistical framework the classical Rayleigh-Ritz method, is introduced.
Asymptotic properties of the estimator are proved, providing the rate of convergence
under suitable mild conditions. Some applications to the goodnessof- fit test and the construction of bivariate distributions are proposed.