Koroliouk D. Dynamic models of statistical experiments, their analysis and modeling

The dissertation is devoted to the system analysis of statistical experiments (SE), which are determined by the averaged sums of sampling random variables. The dynamics of SE is given by a difference stochastic equation with specified regression function of increments (RFI), linear or nonlinear, which depends on the value of the SE at the previous step, as well as stochastic component, which is characterized by martingale differences with predetermined first two moments. The proof of limit theorems uses modern methods of operator and martingale characterization of Markov processes, including singular perturbation methods. An important novelty in statistics of statistical experiments is the use of two new principles. One of them is the SE characterization by twocomponent vector of the trajectory and its increments. The linearity of the RFI provides the stationary SE characterization by the covariance matrix of twocomponent vector. The second one is that, the introduced covariance statistics allow to solve the key problems of SE statistics: to obtain estimators of the directing parameter and of the stochastic component dispersion. One established the consistency with probability 1 of the directing action parameter estimator using martingale characterization for the a priori estimator's deviation. For the first time introduced an optimal estimating function given by the quadratic variation of the martingale. Established the optimality of the consistent directing action parameter estimator and an estimator of the stochastic component dispersion is obtained. It was found that in the case of a Gaussian distribution of a stationary SE, the optimal maximum likelihood estimate coincides with the minimum of the optimal evaluation function. Moreover, it was justified, using the theorem on normal correlation, the representation of a stationary Gaussian SE having additional Markov property, as the solution of a stochastic difference equation. There proposed an effective solution of the problem of SE exit from a predefined interval using the analysis of large deviations for Markov processes. The action functional of a discrete Markov diffusion (DMD), defined by the solution of the variational problem for exponential generator of DMD, coincides with the corresponding action functional for the process of Ornstein-Uhlenbeck type. This reduces the problem of DMD exit from interval to a simplified task for dynamic systems using the potential. The problem of statistical hypotheses verification is formulated in the form of classification of evolutionary processes (EP), which determine the dynamics of the predictable component. The classification of EP is given in 3 versions: with the original directing parameters, with directing parameter and 2 points of equilibrium, with directing parameter and equilibrium point of binary SE. The method of stochastic approximation is used for SE classification. The interpretation of evolutionary processes, as well as of statistical experiments, provides the clearness and effectiveness of SE models applications in processing of experimental data in biophysics (fluorescence microscopy), as well as in sociology (interpretation of the collective behavioral models in the learning process).