85159 - Fundamental Concepts of Statistics

Learning outcomes

By the end of the course the student acquires the fundamental notions of theory of probability and statistical inference. In particular, the student is able to investigate the properties of random variables, including transformations and convergence, and to solve estimation problems and hypothesis testing by classical parametric inference in an effective and coherent way.

Course contents

1. Fundamental Instruments
   
   Characteristic functions (CF); properties; MGFs; properties; guided exercises
   
   
2. Empirical Distribution & Order Statistics
   
   EDF; order-stats distributions; guided exercises
   
   
3. Fundamental Theorems
   
   Markov, Chebyshev, Taylor, Jensen; proofs; guided exercises
   
   
4. Convergence of RVs
   
   Definitions; convergence types; sufficient conditions; WLLN, SLLN; links; guided exercises
   
   
5. Continuous Mapping & CLT
   
   CMT, Slutsky, CLT, Delta method, mean value theorem; guided exercises
   
   
6. Statistical Inference
   
   Point estimation (bias, MSE, efficiency); guided exercises
   
   
7. Likelihood & Fisher Information
   
   C-S inequality; likelihood; Fisher information; Cramér–Rao; guided exercises
   
   
8. Asymptotic Properties of Estimators
   
   Consistency, Glivenko–Cantelli, ARE; asymptotic normality; guided exercises
   
   
9. Classes of Estimators
   
   MoM, MLE; guided exercises
   
   
10. Interval Estimation
    
    CIs, Pivotal statistics, asymptotic CIs, bootstrap methods; guided exercises
    
    
11. Validation Methods
    
    Monte-Carlo simulation; guided exercises
    
    
12. Hypothesis Testing
    
    Test Statistics; Significance Level; p-Value; LRT, Wilks theorem; guided exercises
    
    
13. Numerical Methods
    
    Newton–Raphson, Gradient Descent Algorithm, Fisher Scoring method; guided exercises

Readings/Bibliography

Recommended readings that also include additional exercises:

• George Casella, Roger L. Berger, Statistical Inference, 2nd Edition, 2002, Duxbury Pr (Cengage)

• All of Statistics, A Coincise Course in Statistical Inference, Larry Wasserman, Springer.

Teaching methods

• Interactive lectures using a tablet projected on a large screen.
  
  
• Systematic question–answer dialogue between lecturer and students.
  
  

Assessment methods

Students may choose between two assessment modalities:

• Full examination: an 80-minute written exam covering the entire syllabus, offered in the three regular examination sessions during the academic year.
  
  
• Split examination: students may take a midterm examination approximately halfway through the course and a second written examination covering the remaining part of the syllabus during the first regular examination session immediately following the end of the course. The final grade is computed as the arithmetic mean of the grades obtained in the two examinations.

• Aids permitted: official cheat sheet (provided). No other materials, calculators or electronic devices.

• The final examination is graded on a scale from 0 to 32, with grades of 31 and 32 recorded as 30 cum laude.
  
  
• Registration to a chosen exam session is mandatory through the AlamaEsami web site.
  
  
• The classes allow students to learn the fundamental concepts in an interactive format (Q&A) and closely follow the resolution of the exercises.

The assessments aim at verifting the following detailed learning outcomes:

Knowledge (KNOW):

1. State and prove core probabilistic inequalities (Markov, Chebyshev, Jensen) and limit theorems (WLLN, SLLN, CLT).
2. Study empirical distribution functions and statistical functionals to derive properties of sample functions.
3. Study and apply classical estimation principles (method of moments, maximum likelihood) and study their large-sample properties (consistency, asymptotic normality, ARE, CR-bound).
4. Define, interpret and apply key concepts in interval estimation (confidence intervals) and hypothesis testing (p-value, LRT, Wilks’ theorem).
5. Study resampling methods (bootstrap) for inference (confidence intervals)
6. Learn simulation-based validation methods (Monte Carlo) for finite sample properties of estimators and inferential methods.
7. Learn numerical methods for computing estimators (Newton Method, FSM)

Skills (DO):

1. Use inequalities, theorems and moment generating functions to derive properties and distributions of sequences of random variables.
2. Apply limit theorems to derive asymptotic distributions of sample functions (estimators, test statistics) and approximate sampling errors.
3. Use fundamental concepts of statistics to prove statements.
4. Implement estimation algorithms (Newton–Raphson, FSM), using R programming.
5. Evaluate the finite sample properties of estimators and inferential methods via Monte Carlo simulation, using R programming.
6. Use sampling based procedures (bootstrap) to construct confidence intervals, using R programming.
7. Communicate statistical reasoning clearly in a 2-hour written exam under time pressure.

Teaching tools

• The teaching material presented in class is conveniently made available to the student through Unibo Virtuale.
  
  
• This digital access, with a username and password reserved for students enrolled at the University of Bologna, ensures students can study at their own pace and convenience.
  
  
• Office hours can be delivered using Teams.
  
  
• The instructor responds to e-mail messages duly signed by the student with Name, Surname and enrollment number, which concern appointment requests for clarifications about the correction of the in-class exercises.

Office hours

See the website of Marco Novelli