3 Hypothesis Testing
In this section, we study another central topic in statistics: Hypothesis Testing. The section is organized as follows:
- 1.
- Fundamentals
- 2.
- Multiple Testing and Error Rate Control
- 3.
- Causal Inference
- 4.
- Conformal Prediction
- 5.
- Watermark Detection
This section mainly follows STAT 300A, STAT 300C, and STATS 361 at Stanford University [13, 2, 23], and Mathematical Statistics at Peking University, taught by Fang Yao [27].
3.1 Fundamentals
3.1.1 Neyman-Pearson Lemma
3.1.2 Monotone Likelihood Ratio
3.1.3 Composite Null
3.1.4 Method of Undetermined Multipliers
3.1.5 UMP Invariant Tests
3.1.6 Confidence Regions
3.2 Multiple Testing and Error Rate Control
3.2.1 Bonferroni’s Test and Fisher’s Test
3.2.2 Higher Criticism
3.2.3 False Discovery Rate
3.2.4 E-Values
3.3 Causal Inference
3.3.1 Randomized Controlled Trials
3.4 Conformal prediction
3.4.1 Fundamentals
3.4.2 Approaches
3.5 Application: Watermark Detection
3.5.1 The testing problem
3.5.2 Pivots and robust power
3.1.1 Neyman-Pearson Lemma
3.1.2 Monotone Likelihood Ratio
3.1.3 Composite Null
3.1.4 Method of Undetermined Multipliers
3.1.5 UMP Invariant Tests
3.1.6 Confidence Regions
3.2 Multiple Testing and Error Rate Control
3.2.1 Bonferroni’s Test and Fisher’s Test
3.2.2 Higher Criticism
3.2.3 False Discovery Rate
3.2.4 E-Values
3.3 Causal Inference
3.3.1 Randomized Controlled Trials
3.4 Conformal prediction
3.4.1 Fundamentals
3.4.2 Approaches
3.5 Application: Watermark Detection
3.5.1 The testing problem
3.5.2 Pivots and robust power