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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

The student should have covered the material of a good undergraduate program both in mathematics and in statistics. Students should be comfortable with algebra (including the connection between logarithms and exponents), basic calculus (derivatives and integration), basic concepts of probability and statistics (e.g. descriptive statistics, probability and sample spaces, independence, random variables: univariate and multivariate, Gaussian distribution, basic concept of estimation).

### Program

Topics include: random sampling; principles of data reduction; point estimation; hypothesis
testing; confidence intervals.
In Particular the following topics will be covered::
Brief review of probability:
-Random samples and asymptotic methods
-Sampling and sums of random variables
-Laws of large numbers and central limit theorem
Principles of Data Reduction: Sufficiency
The Likelihood Principle: the Likelihood Function
Point Estimation
-Methods of Finding Estimators: Methods of Moments,
-Maximum Likelihood Estimators.
-Finite Sample Properties: Unbiasedness and Efficiency.
-Asymptotic Properties: Asymptotic Unbiasedness, Consistency
and Efficiency.
Comparison betweene stimators
Bayesian estimators
-Fisher Information and the Cramer-Rao theorem.
Confidence Intervals
Hypothesis Testing
-Methods of Finding Tests: Neyman Pearson lemma
-Large sample tests: Likelihood Ratio Tests, Score Test, Wald Test
-Methods of Evaluating Tests: the Power Function, Most Powerful Tests.
-The p-value.
Nonparametric Inference

### Books

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

### Bibliography

Suggested texts:

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

Final Exam will have a written and an oral part. The written test will consist on exercises and open-ended and multiple choice questions on theory, covering all the program.
During the course students will be ask to solve one or two homework assignments and to
take one o two surprise multiple choice test during the class.

During the exam students are expected to be able to formalize pratical probelms and solve specific analytical problems such as finding and comparing estimators, comparing different inferential methods and implementing hypothesis testing techniques. The students will be evaluated for the ablitity to apply the knowledge learned and to critically interpret the findings.