Formula for Exponential Distribution Family (EDF)
- y is the value of an observation Y
- θ is a location parameter called the canonical parameter
- φ is a dispersion parameter called the scale parameter
- b(θ) is the cumulant function, which determines the shape of the distribution
- e c(y,φ)is a normalizing factor producing unit total mass for the distribution
Expected Value and Variance of EDF are as
Deriving Poisson from EDF
Tweedie sub-family distribution
Tweedie sub-family Expected Value and Variance
Selections to specify a GLM
- Error distribution (one of EDF distribution, index p)
- Explanatory variables xis
- Link function h(), i.e. identity (no tranformation), log, logit
GLM Deviance
Standardized Pearson Residuals and Model Validation
Standardized pearson residual should be random around zero (unbiased) and have constant variance (homoscedasticity)
Standardized deviance residuals
Standardized deviance should be normally distributed
Non-Parametric Mack Model Assumptions
- Accident years are stochastically independent (aka. just independent)
- for each AY k, the cumulative loss Xk,j form a Markov Chain
- for each accident year k and development period j, see below (fj is the ATA factor of age j)
Results of Mack Model
Result 1: the conventional chain ladder estimators fkj (ATA factor) are unbiased and minimum variance estimators (MVUE) that are unbiased linear combinations of the fkj
Result 2: the conventional chain ladder estimator Rk (reserve for AY k) is unbiased
Parametric Mack model assumptions
Same assumption as Non-Parametric Mack except:
- variance assumption is removed and driven by EDF selected
Theorem 1 regarding the EDF Mack Model
- if the original Mack assumption also holds, then the MLEs of the fj parameters
