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## Syllabus

EN IT

### Learning Objectives

LEARNING OUTCOMES:

The course is designed to provide an in-depth knowledge of the main aspects of statistical inference (point estimation and hypothesis testing), both from a conceptual and a technical point of view. Techniques for small and large samples will be provided.

KNOWLEDGE AND UNDERSTANDING: The student is expected to learn the main inferential techniques and to acquire the tools to evaluate the goodness of the different methods.

APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will be able to formalize practical problems and solve specific analytical problems such as finding and comparing estimators, comparing different inferential methods and implementing hypothesis testing techniques.

MAKING JUDGEMENTS:
At the end of the course, the students will be able to apply the knowledge learned and to critically interpret quantitative data related to economic and financial phenomena.

COMMUNICATION SKILLS:
Students will acquire the technical language typical of statistics and be able to comunicate in a clear and unambiguous way the concepts learned during the course.

LEARNING SKILLS:
At the end of the course the students will be able to formalize and to solve pratical problems, showing to be able to implement independently the methods learned.

### Prerequisites

Students should be familiar with mathematical concepts (including the connection between
exponential and logarithmic functions), basic calculus (derivatives, integrals, and function
analysis), and elementary concepts of probability and statistics (e.g., descriptive statistics,
univariate and multivariate random variables, independence of random variables, Gaussian
distribution, and basic concepts of inference).

Suggested Refences;
Simon, C. P., & Blume, L. (1994). Mathematics for economists. New York: Norton.
Mood, A. M., Graybill, F. A., & Boes, D. C. (2007). Introduction to the Theory of Statistics,
3rd Edn. McGraw-Hill

### Program

The program is structured into three thematic areas:
1. Sample Random Variables and Statistics (10 hours)
2. Point and Interval Estimation (16 hours)
3. Hypothesis Testing: Criteria and Construction of Optimal Tests (10 hours)

Specifically, the following topics will be covered:
Brief Recall of Probability

Multiple Random Variables. Recall of Asymptotic Theory (2 hours)

Sampling and Sampling Distributions (6 hours)

Sufficiency and Likelihood Principle (2 hours)
Inference and Point Estimation

Comparison of Estimators (2 hours)

Bayesian Estimators (2 hours)

Confidence Intervals (2 hours)
Hypothesis Testing: Optimal Tests

Neyman-Pearson Lemma (2 hours)

Likelihood Ratio Test (2 hours)

Asymptotic Tests: Likelihood Ratio Tests, Score Test, Wald Test (2 hours)

p-value Approach to Hypothesis Testing (2 hours)

Nonparametric Inference (2 hours)

### Books

Required: Casella, George, and Roger L. Berger. Statistical inference. Cengage Learning,
2021.

### Bibliography

Testi consigliati::

K. Knight. Mathematical statistics. Chapman Hall/CRC (2000).
N. Mukhopadhyay. Probability and Statistical Inference, Dekker-CRC Press (2000).
T. H. Wonnacott and R. J. Wonnacott. Statistics: Discovering Its Power. John Wiley
Sons; International Ed edition (1982).
A. Mood, F. Graybill and D. Boes. Introduction to the theory of statistics, McGraw-Hill
(1974).

### Teaching methods

Lessons and practices in class

### Exam Rules

The final exam will consist of a written test and a discussion based on the written test. The
written test will include exercises and both open-ended and closed-ended theoretical
questions covering the entire syllabus.
The final grade is on a scale of thirty points. The theoretical questions can earn up to 12
points. The inference exercise, which covers all aspects of likelihood theory and hypothesis
testing, can earn up to 12 points. The remaining 6 points are allocated to inference
exercises that require intuitive skills, demonstrating the ability to quickly analyze and
understand concepts to solve problems.
Throughout the course, students will complete one or two homework assignments, and
there will be one or two surprise multiple-choice quizzes in class (which can increase the
final grade by 2 points if the exam is taken in the winter session).
During the exam, the evaluation will assess whether the student has acquired the ability to
formalize practical problems and solve specific analytical questions (e.g., determining and
comparing estimators, evaluating different inference methods, implementing hypothesis
testing methods). Students will be evaluated on their ability to apply acquired knowledge
and critically interpret results.

o Unsuitable: major deficiencies and/or inaccuracies in knowledge and
understanding of topics; limited ability to analyze and synthesize; frequent
generalizations.

o 18-20: Barely sufficient knowledge and understanding of topics with
possible imperfections; Sufficient ability to analyze synthesis and
independent judgment.

o 21-23: Routine knowledge and understanding of topics; Correct analysis and
synthesis skills with coherent logical argumentation.

o 24-26: Fair knowledge and understanding of topics; Good analytical and
synthesis skills with rigorously expressed arguments.

o 27-29: Comprehensive knowledge and understanding of topics; Remarkable
skills of analysis, synthesis. Good independent judgment.

o 30-30L: Excellent level of knowledge and understanding of topics.
Remarkable analytical and synthesis skills and independent judgment.
Arguments expressed in an original way.