This paper deals with an asymptotic problem of estimating an unknown parameter \(\theta\) of a function \(\varphi(\theta)\) in a statistical experiment \(({\mathcal X}^{(\varepsilon)}, A^{(\varepsilon)}, P_ \theta^{(\varepsilon)}\), \(\theta\in \Theta)\) generated by an observation \(X^{(\varepsilon)}\). We are interested in the following question: to what extent is the possible asymptotic accuracy of estimation or ``the asymptotic difficulty of a problem of statistical estimation'' related to the complexity of the structure of the parameter set \(\Theta\)? For the sake of definiteness, we suppose in what follows that the observations form a sample \(X_ 1,X_ 2, \dots,X_ n\) of size \(n\), and the \(X_ j\) have probability densities \(p(x,\theta)\) with respect to a \(\sigma\)-finite measure \(\mu\) in \(({\mathcal X}, A)\). \textit{N. N. Chentsov} [Sov. Math., Dokl. 3, 1559-1562 (1963); translation from Dokl. Akad. Nauk SSSR 147, 45-48 (1962; Zbl 0133.118)] seems to be one of the first who began considering the connection between the asymptotic difficulty of an estimation problem and the complexity of \(\Theta\). Thus, he has shown that for \(\Theta \subseteq \mathbb{R}^ d\), a reasonably defined asymptotic difficulty of an estimation problem is proportional to the dimension \(d\) of \(\Theta\). For infinite-dimensional \(\Theta\), it is reasonable to take as the characteristic of complexity different diameters or the \(\varepsilon\)-entropy. We note that from this point of view the work by Chentsov seems to be the first; he established relations between the accuracy of estimation of a distribution density \(p\) in the \(L_ 2(r)\) metric and the rate of decay of \(N\)-dimensional diameters (\(N\)-diameters) \(d_ N(\Theta)\) of the parameter set \(\Theta\) (the fact that \(p\in\Theta\) is supposed to be known). At present, a large number of papers have been devoted to the application of \(\varepsilon\)- entropy in estimation problems. Generally, both concepts, \(N\)-diameters and \(\varepsilon\)-entropy, play a fundamental role in approximation theory.