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Introduction to

Generalized Linear

Models

Statistica per le assicurazioni

Classical Multiple Linear Regression

• Yi = ß

• ß

X

i 1

• ß

X

i 2

…+ ß

p

X

ip

+ e

i

• Yi are the response variables
• Xij are predictors
• i subscript denotes ith^ observation
• j subscript identifies jth^ predictor

Problems with Traditional Model

• Number of claims is discrete
• Claim sizes are skewed to the right
• Probability of an event is in [0,1]
• Variance is not constant across data points i
• Nonlinear relationship between X ’s aŶd Y ’s

Generalised Linear Models (1)

• Fewer restrictions
• Y can model number of claims, probability of renewing, loss severity, loss ratio, etc.
• Y ĐaŶ ďe ŶoŶliŶear fuŶĐtioŶ of X’s
• Classical linear regression model is a special case

Generalized Linear Models - GLMs

• Same goal

Predict : μ

i

= E[ Y

i

]

Generalized Linear Models - GLMs

• g ( μ )=x

ß

• g ( ) is named link function (it is a monotonic and differentiable function - such as log or square root)
• E[ Yi ] = μi = g-1 ( ß 0 + ß 1 Xi 1 + ß 2 Xi 2 …+ ßpXip )

 Yi can be Normal, Poisson, Gamma, BiŶoŵial, PoissoŶ, …

 Variance can be modeled

Why Exponential Family?

• Distributions in Exponential Family can model a variety of problems
• Standard algorithm for finding coefficients ß 0 , ß 1 , …, ßm

Estimating Coefficients ß

, ß

, …, ß

m

• Classical linear regression uses least squares
• GLMs use Maximum Likelihood Method
• Solution will exist for distributions in exponential family

The generalized linear model

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• The choice of a ( θ ) determines the response

distribution.

• The choice of g ( μ ), link function, determines

how the mean is related to the explanatory

variables x.

• The setup in (1 and (2 states that, given x , μ is

determined through g ( μ ). Given μ , θ is

determined through ˙ a ( θ ) = μ. Finally given θ , y

is determined as a draw from the exponential

density specified in a ( θ ).

• Observations on y are assumed to be

independent.

Steps in generalized linear modeling

Given a response variable y , constructing a GLM consists of the following

steps:

(i) Choose a response distribution f ( y ) and hence choose a ( θ )

(ii) Choose a link g ( μ ). This choice is sometimes simplified by choosing the

so called Dzcanonicaldz link corresponding to each response distribution.

(iii) Choose explanatory variables x in terms of which g ( μ ) is to be modeled.

iv) Collect observations y 1 ,... , yn on the response y and corresponding

values x 1 ,... , xn on the explanatory variables x.

Successive observations are assumed to be independent

(v) Fit the model by estimating β and, if unknown, φ by ML

(vi) Given the estimate of β , generate predictions (or fitted values) of y for

different settings of x and examine how well the model fits by examining the

departure of the fitted values from actual values, as well as other model

diagnostics.

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