Inferential statistics

The importance of statistical science in the process of scientific discovery and industrial advancement is that it allows the formulation of inferences about phenomena of interest to which one can associate risks of error or degrees of confidence. The calculation of these risks of error is based on probability theory, but the principles and methods for associating these risks with inferences constitute a theoretical corpus that serves as a basis for all statistical methodologies.

This module is intended to be a fairly complete presentation of these basic principles and of the tools, results and mathematical theorems used in inferential statistics. It develops the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families and the principle of maximum likelihood and the use of p-value.
For the implementation of certain applications, the adapted tools of the R software will be presented.

At the end of this module, the student will be able to develop the optimal statistical methodologies for estimation and hypothesis testing in certain families of parametric probability laws. He/she should understand the limitations of the inferences produced and be able to communicate them to users. Faced with small data sets, he or she will have to know how to choose the best approach in a reasoned manner and perform the necessary calculations with the R software.

A course in probability calculus at the Bachelor's level.
 

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

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

1. Parametric statistical model

a) Parametric statistical model;

b) Sampling model iid ;

c) Recall on asymptotic theorems (LGN, TCL, Delta-method).

d) Notion of statistics - empirical characteristics of a sample & asymptotic laws.

2. Exponential family

a) Definition

b) Moments.

3. Fisher score and information

a) Score;

b) Fisher information;

c) Case of the exponential family.

4. Comprehensive statistics

a) Completeness & characterizations

b) Minimum exhaustive statistics; Complete statistics.

5. Point estimate

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

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

c) Maximum likelihood estimation, asymptotic properties.

d) Estimation by the method of moments, asymptotic properties.

6. Ensemblistic estimation

a) Region of trust.

b) Pivot.

c) Asymptotic confidence region.

7. Hypothesis testing

a) Testing problem: hypotheses, errors, losses, associated risks, level and power. Test function. Pure test vs. mixed test.

b) No optimal test. Unbiased test. Convergent test.

c) Neyman's principle.

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

e) One-sided hypothesis tests: monotonic likelihood ratio family & UPP tests.

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

g) Relationship between test acceptance regions and confidence regions.

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

8. Two sample problems

Comparison of parameters: estimation and tests.

9. Suitability tests

a) Chi2 test & application to the test of independence.

b) Kolmogorov-Smirnov test.

c) Shapiro-Wilks normality tests.

Hourly volumes:
CM: 21h
TD: 21h
TP:
Field: