RISK INS / 2.1 & 2.2

In these lecture notes, we still assume that distributions are known, or we at least have an idea of what distributions could be. However, the goal is to fit data to distributions and find an optimal distribution parameter.

We do this, using maximum likelihood estimation (MLE). MLE is something you have seen multiple times before and therefore this topic should not be too difficult for you. However, we will introduce some new, slightly different, techniques that are often applied to specific insurance data.

Maximum Likelihood Estimation

Basics

Individual Data

Likelihood Function

Individual Data

Log-likelihood Function

Grouped Data

Grouped Data

(Log) Likelihood Function

Maximum Likelihood Estimation

SOC

2 forms of Incompletely Observed Data

Censored Data

2 forms of Incompletely Observed Data

Truncated Data

Why Standard Regression (OLS) Fails for Insurance

Replacing Standard Regression by

Data Transformations

Replacing Standard Regression by

Generalised Linear Models (GLMs)

Video

Dispersion

Replacing Standard Regression by

Generalised Linear Models (GLMs)

Why does it work? Fundamental Parts

Generalised Linear Models (GLMs)

Variance and Exposure Weights

Generalised Linear Models (GLMs)

Canonical Links

Generalised Linear Models (GLMs)

Null vs Full Models

Generalised Linear Models (GLMs)

Residuals & Goodness of Fit

Multiplicative Rating System

Multiplicative Rating System

Four Ways to Solve the Multipliers: First Two

Multiplicative Rating System

Four Ways to Solve the Multipliers: Last Two

How do we solve for these parameters when the equations contain unknowns on both sides (e.g., α_i. depends on β_j, and β_j depends on α_i)?

Example

Numerical illustration of the above methods

Scaled Deviance

Deviance

Deviance Formulas by Distribution

Estimating the Dispersion Parameter

Comparing Nested Models (Analysis of Deviance)

Information Criteria

9.5: Case study: Analyzing a simple automobile portfolio

View studies in book