  We discuss the possibility of applying some standard statistical methods (the
least square method, the maximum likelihood method, the method of statistical
moments for estimation of parameters) to deterministically chaotic
low-dimensional dynamic system (the logistic map) containing an observational
noise. A ``pure'' Maximum Likelihood (ML) method is suggested to estimate the
structural parameter of the logistic map along with the initial value $x_1$
considered as an additional unknown parameter. Comparisons with previously
proposed techniques on simulated numerical examples give favorable results (at
least, for the investigated combinations of sample size $N$ and noise level).
Besides, unlike some suggested techniques, our method does not require the a
priori knowledge of the noise variance. We also clarify the nature of the
inherent difficulties in the statistical analysis of deterministically chaotic
time series and the status of previously proposed Bayesian approaches. We note
the trade-off between the need of using a large number of data points in the ML
analysis to decrease the bias (to guarantee consistency of the estimation) and
the unstable nature of dynamical trajectories with exponentially fast loss of
memory of the initial condition. The method of statistical moments for the
estimation of the parameter of the logistic map is discussed. This method seems
to be the unique method whose consistency for deterministically chaotic time
series is proved so far theoretically (not only numerically).
