Dynamics of Statistical Experiments 1st Edition by Dmitri Koroliouk ISBN 978-1786305985 1786305984

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ISBN 10:  1786305984

ISBN 13: 978-1786305985

Author: Dmitri Koroliouk

This book is devoted to the system analysis of statistical experiments, determined by the averaged sums of sampling random variables. The dynamics of statistical experiments are given by difference stochastic equations with a speci?ed regression function of increments – linear or nonlinear. The statistical experiments are studied by the sample volume increasing (N ??), as well as in discrete-continuous time by the number of stages increasing (k ??) for different conditions imposed on the regression function of increments. The proofs of limit theorems employ modern methods for the operator and martingale characterization of Markov processes, including singular perturbation methods. Furthermore, they justify the representation of a stationary Gaussian statistical experiment with the Markov property, as a stochastic difference equation solution, applying the theorem of normal correlation. The statistical hypotheses verification problem is formulated in the classification of evolutionary processes, which determine the dynamics of the predictable component. The method of stochastic approximation is used for classifying statistical experiments.

Table of contents: 

Chapter 1. Statistical Experiments
1.1. Statistical Experiments with Linear Regression
1.1.1. Basic Definitions
1.1.2. Difference Evolution Equations
1.1.3. The Equilibrium State
1.1.4. Stochastic Difference Equations
1.1.5. Convergence to the Equilibrium State
1.1.6. Normal Approximation of the Stochastic Component
1.2. Binary SEs with Nonlinear Regression
1.2.1. Basic Assumptions
1.2.2. Equilibrium
1.2.3. Stochastic Difference Equations
1.2.4. Convergence to the Equilibrium State
1.2.5. Normal Approximation of the Stochastic Component
1.3. Multivariate Statistical Experiments
1.3.1. Regression Function of Increments
1.3.2. The Equilibrium State of Multivariate EPS
1.3.3. Stochastic Difference Equations
1.3.4. Convergence to the Equilibrium State
1.3.5. Normal Approximation of the Stochastic Component
1.4. SEs with Wright-Fisher Normalization
1.4.1. Binary RFs
1.4.2. Multivariate RFIs
1.5. Exponential Statistical Experiments
1.5.1. Binary ESES
1.5.2. Steady Regime of ESEs
1.5.3. Approximation of ESEs by Geometric Brownian Motion

Chapter 2. Diffusion Approximation of Statistical Experiments in Discrete-Continuous Time
2.1. Binary DMPS
2.1.1. DMPs in Discrete-Continuous Time
2.1.2. Justification of Diffusion Approximation
2.2. Multivariate DMPs in Discrete-Continuous Time
2.2.1. Evolutionary DMPs in Discrete-Continuous Time
2.2.2. SDEs for the DMP in Discrete-Continuous Time
2.2.3. Diffusion Approximation of DMPs in Discrete-Continuous Time
2.3. A DMP in an MRE
2.3.1. Discrete and Continuous MRE
2.3.2. Proof of Limit Theorems 2.3.1 and 2.3.2
2.4. The DMPs in a Balanced MRE
2.4.1. Basic Assumptions
2.4.2. Proof of Limit Theorem 2.4.1
2.5. Adapted SEs
2.5.1. Bernoulli Approximation of the SE Stochastic Component
2.5.2. Adapted SEs
2.5.3. Adapted SEs in a Series Scheme
2.6. DMPs in an Asymptotical Diffusion Environment
2.6.1. Asymptotic Diffusion Perturbation
2.7. A DMP with ASD
2.7.1. Asymptotically Small Diffusion
2.7.2. EGs of DMP
2.7.3. AF of DMPs

Chapter 3. Statistics of Statistical Experiments
3.1. Parameter Estimation of One-Dimensional Stationary SEs
3.1.1. Stationarity
3.1.2. Covariance Statistics
3.1.3. A Priori Statistics
3.1.4. Optimal Estimating Function
3.1.5. Stationary Gaussian SEs
3.2. Parameter Estimators for Multivariate Stationary SEs
3.2.1. Vector Difference SDEs and Stationarity Conditions
3.2.2. Optimal Estimating Function
3.2.3. Stationary Gaussian Markov SEs
3.3. Estimates of Continuous Process Parameters
3.3.1. Diffusion-Type Processes
3.3.2. Estimation of a Continuous Parameter
3.4. Classification of EPs
3.4.1. Basic Assumption
3.4.2. Classification of EPs
3.4.3. Justification of EP Models Classification
3.4.4. Proof of Theorem 3.4.1
3.4.5. Interpretation of EPs
3.4.6. Interpretation of EPs in Models of Collective Behavior
3.5. Classification of SEs
3.5.1. The SA of SEs
3.5.2. Classifiers
3.5.3. Classification of SES
3.6. Evolutionary Model of Ternary SEs
3.6.1. Basic Assumptions
3.6.2. The Model Interpretation and Analysis
3.7. Equilibrium States in the Dynamics of Ternary SES
3.7.1. Building a Model
3.7.2. The Equilibrium State and Fluctuations
3.7.3. Classification of TSES

Chapter 4. Modeling and Numerical Analysis of Statistical Experiments
4.1. Numerical Verification of Generic Model
4.1.1. Evolutionary Processes with Linear and Nonlinear RFIs
4.1.2. Generic Model of Trajectory Generation
4.2. Numerical Verification of DMD
4.2.1. Simulation of DMD Trajectories
4.2.2. Estimation of DMD Parameters
4.3. DMD and Modeling of the Dynamics of Macromolecules in Biophysics
4.3.1. The Model Motivation
4.3.2. Statistical Model of a Stationary DMD
4.3.3. Stokes-Einstein Kinetic Diffusion Model
4.3.4. Verification of the Model of Stationary DMD by Direct Numerical Simulation
4.3.5. Numerical Verification of DMD Characteristics Using the Model of Stokes-Einstein
4.3.6. The Ability of the DMD Model to Detect the Proportion of Fast and Slow Particles
4.3.7. Interpretation of the Mixes of Brownian Motions

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Tags: Dmitri Koroliouk, Statistical Experiments