Dynamics of Statistical Experiments

Dmitri Koroliouk is a Doctor of Sciences (Hab.), leading researcher at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, and Professor at the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute". He is the author of 80 scientific publications and also holds 10 European patents for industrial invention.

• Preface ixList of Abbreviations xiIntroduction xiiiChapter 1 Statistical Experiments 11.1 Statistical experiments with linear regression 11.1.1 Basic definitions 11.1.2 Difference evolution equations 31.1.3 The equilibrium state 41.1.4 Stochastic difference equations 71.1.5 Convergence to the equilibrium state 91.1.6 Normal approximation of the stochastic component 111.2 Binary SEs with nonlinear regression 131.2.1 Basic assumptions 131.2.2 Equilibrium 151.2.3 Stochastic difference equations 171.2.4 Convergence to the equilibrium state 181.2.5 Normal approximation of the stochastic component 201.3 Multivariate statistical experiments 221.3.1 Regression function of increments 221.3.2 The equilibrium state of multivariate EPs 251.3.3 Stochastic difference equations 261.3.4 Convergence to the equilibrium state 281.3.5 Normal approximation of the stochastic component 291.4 SEs with Wright–Fisher normalization 311.4.1 Binary RFs 311.4.2 Multivariate RFIs 331.5 Exponential statistical experiments 351.5.1 Binary ESEs 361.5.2 Steady regime of ESEs 371.5.3 Approximation of ESEs by geometric Brownian motion 38Chapter 2 Diffusion Approximation of Statistical Experiments in Discrete–Continuous Time 432.1 Binary DMPs 442.1.1 DMPs in discrete–continuous time 452.1.2 Justification of diffusion approximation 472.2 Multivariate DMPs in discrete–continuous time 512.2.1 Evolutionary DMPs in discrete–continuous time 522.2.2 SDEs for the DMP in discrete–continuous time 532.2.3 Diffusion approximation of DMPs in discrete–continuous time 552.3 A DMP in an MRE 582.3.1 Discrete and continuous MRE 582.3.2 Proof of limit theorems 2.3.1 and 2.3.2 622.4 The DMPs in a balanced MRE 652.4.1 Basic assumptions 662.4.2 Proof of limit theorem 2.4.1 702.5 Adapted SEs 742.5.1 Bernoulli approximation of the SE stochastic component 752.5.2 Adapted SEs 772.5.3 Adapted SEs in a series scheme 792.6 DMPs in an asymptotical diffusion environment 842.6.1 Asymptotic diffusion perturbation 852.7 A DMP with ASD 912.7.1 Asymptotically small diffusion 912.7.2 EGs of DMP 942.7.3 AF of DMPs 97Chapter 3 Statistics of Statistical Experiments 1033.1 Parameter estimation of one-dimensional stationary SEs 1033.1.1 Stationarity 1033.1.2 Covariance statistics 1083.1.3 A priori statistics 1103.1.4 Optimal estimating function 1113.1.5 Stationary Gaussian SEs 1143.2 Parameter estimators for multivariate stationary SEs 1153.2.1 Vector difference SDEs and stationarity conditions 1163.2.2 Optimal estimating function 1183.2.3 Stationary Gaussian Markov SEs 1193.3 Estimates of continuous process parameters 1223.3.1 Diffusion-type processes 1223.3.2 Estimation of a continuous parameter 1233.4 Classification of EPs 1243.4.1 Basic assumption 1253.4.2 Classification of EPs 1263.4.3 Justification of EP models classification 1273.4.4 Proof of Theorem 3.4.1 1293.4.5 Interpretation of EPs 1333.4.6 Interpretation of EPs in models of collective behavior 1383.5 Classification of SEs 1393.5.1 The SA of SEs 1393.5.2 Classifiers 1403.5.3 Classification of SEs 1423.6 Evolutionary model of ternary SEs 1443.6.1 Basic assumptions 1443.6.2 The model interpretation and analysis 1463.7 Equilibrium states in the dynamics of ternary SEs 1493.7.1 Building a model 1493.7.2 The equilibrium state and fluctuations 1503.7.3 Classification of TSEs 151Chapter 4 Modeling and Numerical Analysis of Statistical Experiments 1534.1 Numerical verification of generic model 1534.1.1 Evolutionary processes with linear and nonlinear RFIs 1534.1.2 Generic model of trajectory generation 1564.2 Numerical verification of DMD 1584.2.1 Simulation of DMD trajectories 1584.2.2 Estimation of DMD parameters 1634.3 DMD and modeling of the dynamics of macromolecules in biophysics 1674.3.1 The model motivation 1684.3.2 Statistical model of a stationary DMD 1694.3.3 Stokes–Einstein kinetic diffusion model 1714.3.4 Verification of the model of stationary DMD by direct numerical simulation 1724.3.5 Numerical verification of DMD characteristics using the model of Stokes–Einstein 1734.3.6 The ability of the DMD model to detect the proportion of fast and slow particles 1764.3.7 Interpretation of the mixes of Brownian motions 182References 189Index 193