  We study the statistical distribution of components in the non-perturbative
parts of energy eigenfunctions (EFs), in which main bodies of the EFs lie. Our
numerical simulations in five models show that deviation of the distribution
from the prediction of random matrix theory (RMT) is useful in characterizing
the process from nearly-integrable to chaotic, in a way somewhat similar to the
nearest-level-spacing distribution. But, the statistics of EFs reveals some
more properties, as described below. (i) In the process of approaching quantum
chaos, the distribution of components shows a delay feature compared with the
nearest-level-spacing distribution in most of the models studied. (ii) In the
quantum chaotic regime, the distribution of components always shows small but
notable deviation from the prediction of RMT in models possessing classical
unterparts, while, the deviation can be almost negligible in models not
possessing classical counterparts. (iii) In models whose Hamiltonian matrices
possess a clear band structure, tails of EFs show statistical behaviors
obviously different from those in the main bodies, while, the difference is
smaller for Hamiltonian matrices without a clear band structure.
