# Actuarial Statistics

## Objectives

At the end of this course the student will have acquired the knowledge, skills and competences that will allow him:

- To be able to create distributions through core operations

- To know the main distributions and their relationships used for the insurer’s claims

- To analyze the tail of a distribution and know how to classify it

- To be able to understand the basic concepts of extreme theory applied to actuarial sciences

- To know how to use statistical techniques for typical data concerning the number and amount of claims

## General characterization

##### Code

12232

##### Credits

6.0

##### Responsible teacher

Rui Manuel Rodrigues Cardoso

##### Hours

Weekly - 4

Total - 70

##### Teaching language

Português

### Prerequisites

Mathematical Analysis I, II. Probability and Statistics at a medium level.

### Bibliography

- Bowers, Newton, Gerber, Hickman, Jones and Nesbitt. (1997) Actuarial Mathematics (second edition). Itasca,

Illinois: The Society of Actuaries.

- Dickson, D.C.M., Hardy, M.R. & Waters, H.R. (2020) Actuarial Mathematics for Life Contingent Risks, (third edition). Cambridge

University Press.

- Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. (2008) Modern Actuarial Risk Theory - using R (second

edition), Springer.

- Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2019) Loss Models: From Data To Decisions (fifth edition),

Wiley.

- Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2012) Loss Models: Further Topics,

Wiley.

### Teaching method

In the theoretical and practical lectures, it will be explained and discussed the successive topics of the course program. The topics are introduced by the teacher, consolidated using as much as possible with real examples drawn from the insurance industry, followed by a brief discussion and use of computational means to support problem solving.

### Evaluation method

Two tests, to be carried out during the academic period and exam (s) according to the academic calendar.

Each of the tests has a weight of 50% for the calculation of the final grade, being exempt from examination the student who has a weighted average greater than or equal to 9.5, with both tests evaluated at least 7.5 values.

## Subject matter

1 Sum of independent random variables

1.1 Some results

1.2 Convolutions

2 Creating new distributions

2.1 Multiplication by a constant

2.2 Raising to a power and exponentiation

2.3 Mixing

2.4 Splicing

3 Distribution families

3.1 Parametric families

3.2 Limiting distributions

3.3 Relationships between distributions

3.4 Exponential family

4 Tails of distributions

4.1 Classification

4.2 Equilibrium distribution

4.3 Tail behavior

5 Extreme value distributions

5.1 Distribution of the maximum

5.2 Maximum domain of attraction

5.3 Generalized Pareto distribution

5.4 Limiting distributions of excesses

6 Estimation

6.1 Kaplan-Meier estimator

6.2 Nelson-Aalen estimator

6.3 Kernel density models

6.4 Estimation to complete data

6.5 Estimation to modified data

6.6 Estimation to truncated data

## Programs

Programs where the course is taught: