Mathematical Statistics Learning Guide
Systematically consolidate the core knowledge framework of mathematical statistics, organized in a logically progressive sequence of foundational theory, inferential analysis, and applied analysis, covering probability theory, statistical methods, and practical applications as three major dimensions.
Basic Theory Construction - 3
Foundations of Probability Theory and Properties of Random Variables
Principles of Statistics and Data Description
Statistical Inference and Parameter Estimation
Parameter Estimation Theoretical Framework
- Methods of Constructing Point Estimates (e.g., Moment Estimation, Maximum Likelihood Estimation)
- Principle of Confidence Interval Calculation in Interval Estimation
- Criteria for Estimator Performance (Unbiasedness, Efficiency, Consistency)
- Asymptotic Normality Theory in Parameter Estimation
- Prior and Posterior Distributions in Bayesian Statistics
Theoretical Foundation of Hypothesis Testing
- Steps and Types of Hypothesis Testing (One-Sided/Two-Sided Tests)
- Calculation of p-value and Determination of Significance Level
- Distribution Characteristics of Test Statistics (Normal Distribution, Chi-Squared Distribution, etc.)
- Analysis of Type I and Type II Errors in Hypothesis Testing
- Power Calculation and Optimization Strategies
Regression Analysis and Correlation Study
Linear Regression Model and Parameter Estimation
- The mathematical form and hypothesis testing of a linear regression model
- Least Squares Method for Solving Regression Coefficients
- Significance Testing of Regression Equations and Residual Analysis
- Detection and Handling Methods for Multicollinearity
- Diagnosis and validation of regression models, such as R² and adjusted R²
Correlation Analysis and Variable Relationships
- Calculation and Interpretation of Correlation Coefficients (Pearson Correlation Coefficient, Spearman Rank Correlation)
- Distinguishing Correlation from Causal Relationship
- Variable transformation and nonlinear relationship handling
- Covariance Matrix and Analysis of Covariance (ANCOVA)
- Robustness Test in Correlation Analysis
Data Distribution and Sampling Theory
Properties of Common Probability Distributions
- Normal Distribution, Poisson Distribution, Binomial Distribution Parameter Estimation
- Chi-squared distribution, t-distribution, F-distribution properties and applications
- Method of Parameter Estimation (such as Maximum Likelihood Estimation)
- Mathematical Expectation and Variance Calculation of Distribution Functions
- Verification of properties of distribution functions such as symmetry and monotonicity