In this article the relations between the mathematical theory of function approximation, the maximum-likelihood method, the measures of optimality and the identification of parameters are presented. The estimator of the maximum likelihood for uniform noise is introduced on the basis of the generalized Cauchy probability density function (p.d.f.). The measures of optimality based on the maximum-likelihood method for the random variables with Gauss, Cauchy, Laplace and uniform p.d.f.s are presented. The theoretical statements are illustrated with a numerical experiment concerning the optimal parameter identification on the free-damped-noisy response of a single-degree-of-freedom system. The different types of noise and the different levels of the responses’ noisiness were used.