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Taylor: Stochastic loss reserving using GLM (Key definitions (x covariates…
Taylor:
Stochastic loss reserving using GLM
Selection of a GLM
Selection of a cumulant function, controlling the model’s assumed error distribution
As part of this, selection of index p (the parameter of the Tweedie distribution), which controls the relation between the model mean and variance
Selection of covariates , which are the explanatory variables considered to influence the cell mean
Selection of a link function, h(.), which specifies the functional relation between the cell mean and the associated covariates
Key definitions
x
covariates
u
mean
h
link function
Yi
is the i eme observation
B
in the vecto of estimated parameter
n
number of observation
p
number of parameters
teta
canonical parameter
phi
scale parameter
X
is the cumulative loss
A good fit model
Standardized pearson residuals
A histogram of Pearson residuals with a very right-skewed distribution with a light right tail (or whatever shape of the underlying error term is, typically Gamma)
A plot of Pearson residuals randomly scattered arround 0. If there are outliers, a constant (with weights equal to 1) is not appropriate. To correct this, we assign a weight to observations exhibiting greater variance equal to the ratio of the “normal variance” to the variance of the group of observations (effectively assigning them less weight in the GLM)
Standardized deviance residual
One would expect a model with a good fit to produce a histogram of residuals evenly around zero (normal-like). This is because deviance residuals remove the non-normality exhibited in the standardized Pearson residuals
Stocastic models suporting CL
ODP Mack model
Cross-classified Model
ODP Cross-classified
When modelling incremental losses under the (ODP) Cross-Classified model, the fitted betas may violate the condition that their sum is 1. In order to rectify this, we normalize the alphas and betas. Sum the original betas, divide each beta by this sum and multiply each alpha by this sum to get normalized parameters.
if ODP is used , they will be no heterostedasticity so no wi needed and the results will be the CL
members of exponential family
Normal, poisson, binomial et gamma
tweedy family
p=0 donne normal
p=1 donne ODP
p=3 donne gamma