Inferential Statistics

The importance of statistical science in the process of scientific discovery and industrial advancement lies in its ability to enable the formulation of inferences about phenomena of interest, to which risks of error or levels of confidence can be assigned. The calculation of these error risks is based on probability theory, but the principles and methods for associating these risks with inferences constitute a theoretical framework that serves as the foundation for all statistical methodologies.

This module is intended to provide a fairly comprehensive overview of these basic principles and the mathematical tools, results, and theorems used in inferential statistics. It covers the concepts of point and interval estimation, hypothesis testing, and fundamental concepts such as exponential families, the maximum likelihood principle, and the use of the p-value.
For implementing certain applications, appropriate tools from the R software package will be presented.

By the end of this module, students should be able to develop optimal statistical methodologies for estimation and hypothesis testing in certain families of parametric probability distributions. They should understand the limitations of the inferences made and be able to communicate these limitations to users. When working with small datasets, students should be able to make a reasoned choice regarding the best approach and perform the necessary calculations using the R software.

A college-level course in probability theory.
 

 
Recommended prerequisites: A bachelor's-level course in descriptive statistics would be an asset.

 CC + CT using the formula: final grade = max(CT, (CC + CT)/2)

1. Parametric statistical model

a) Parametric statistical model;

b) IID sampling model;

c) Review of asymptotic theorems (LGN, TCL, Delta method).

d) Concept of statistics—empirical characteristics of a sample and asymptotic laws.

2. Exponential family

a) Definition

b) Moments.

3. Fisher's score and information

a) Score;

b) Information from Fisher;

c) The case of the exponential family.

4. Comprehensive statistics

a) Completeness & Characterizations

b) Minimum exhaustive statistics; Complete statistics.

5. Point estimation

a) Risk. Quadratic risk = bias² + variance. Order of estimators. No optimal estimator.

b) Unbiased estimator: Fréchet's inequality. Efficient estimation & the exponential family. Rao's improvement. Optimal ESB & the Lehmann-Scheffe theorem.

c) Maximum likelihood estimation, asymptotic properties.

d) Estimation using the method of moments; asymptotic properties.

6. Ensemble estimation

a) Trusted region.

b) Pivot.

(c) Asymptotic confidence region.

7. Hypothesis Testing

a) Testing issues: assumptions, errors, losses, associated risks, level, and power. Test function. Pure testing vs. mixed testing.

b) Lack of an optimal test. Unbiased test. Convergent test.

c) Neyman's principle.

d) Simple hypothesis testing: Neyman's PP test.

e) One-tailed hypothesis tests: the monotone likelihood ratio family and UPP tests.

f) Two-tailed hypothesis tests: exponential family & UPPSB tests.

(g) Relationship between test acceptance regions and confidence regions.

(h) Asymptotic tests: Wald test, Rao's score test, maximum likelihood ratio test.

8. Two-sample problems

Comparison of Parameters: Estimation and Tests.

9. Suitability Tests

a) Chi-square test & application to the independence test.

b) Kolmogorov-Smirnov test.

c) Shapiro-Wilks normality tests.

Hours:
Lectures: 21 hours
Tutorials: 21 hours
Lab sessions:
Fieldwork: