Taylor and McGuire Flashcards
(34 cards)
Mean and variance of exponential dispersion family (EDF) of distributions
mean = mu
variance = dispersion parameter * variance function
Variance function for the Tweedie sub-family of EDFs
variance = mu ^ p
And p between 0 and 1 (inclusive)
in words: variance is proportional to a power of the mean
Relationship between p for the Tweedie distribution and tail heaviness
tail heaviness increases as p increases
Mean and variance of a Tweedie distribution
mean = mu = [ ( 1 - p ) * theta ] ^ ( 1 / ( 1 - p ) )
where theta = location parameter
variance = dispersion parameter * mu ^ p
General GLM format (in matrix notation)
Taylor and McGuire
link function = transposed covariate matrix * beta matrix
where betas are the linear response variables, and the link function transforms the mean of each observation into a linear function of the parameters (betas)
Conditions for the structure of a GLM (3)
Taylor and McGuire
- each observation is a member of the EDF
- h(mu-sub i) = x-sub i-transpose * beta
- observations are stochastically independent
GLM version of a standard linear regression
mean, mu = sumproduct ( x, beta)
Underlying assumptions of a standard linear regression (3)
- errors are normally distributed
- errors have constant variance
- linear relationship
Difference b/w weighted linear regression and standard linear regression
weighted linear regression recognizes errors might have unequal variances
Model generalizations to get from a linear regression to a GLM (2)
- non-linear relationship
2. non-normal errors
Common estimation method for GLM parameters
MLE
Requirements for selection of a GLM and purpose of each (4)
selection of:
- cumulant function (controls the shape of the distribution)
- index, p (controls relationship b/w mean and variance in an EDF)
- covariates (x’s = explanatory variables)
- link function (controls relationship b/w mean and covariates)
Measure of model goodness of fit
Taylor and McGuire
deviance
> > smaller = better
Deviance formula (unscaled)
deviance = 2 * sum ( log-likelihood (perfect model ) - log-likelihood ( actual model ) )
Scale parameter calculated from deviance and corresponding distribution
(Taylor and McGuire)
scale parameter = deviance / ( n - p )
> > Chi-square distribution w/ ( n - p ) df
Standardized Pearson Residuals (Taylor and McGuire)
= raw residual / std. dev. ( observation )
- unbiased and homoscedastic
Problem with standardized Pearson residuals
Taylor and McGuire
reproduces any non-normality from the observations
Best residual to use for model assessment and why
Taylor and McGuire
deviance residuals
why: corrects any non-normality in the data
Deviance residual
= sgn ( actual - fitted ) * ( d-sub i / scale parameter ) ^ .5
where sgn function = -1 if negative, 0 if 0, and 1 if positive
and d-sub i is the contribution of the i-th observation to the unscaled deviance
Types of stochastic models (4)
Taylor and McGuire
- non-parametric Mack model
- parametric Mack models
- cross-classified models
- GLM representations of CL models
Results of non-parametric Mack model (2)
- estimators of CL age-to-age factors are MVUE’s among estimators that are unbiased linear combinations
- unbiased CL reserve estimates
Special cases of parametric Mack models (2)
- ODP Mack model
2. Tweedie Mack model
Assumption required to turn a non-parametric Mack model into a parametric one
require that incremental observations (given claims to date) come from the EDF distributions
Theorem 3.1 from Taylor and McGuire (3 MVUE results for parametric models)
under EDF and general Mack assumptions:
- MLEs are the usual CL estimators (and are unbiased)
- for special case of ODP Mack model and if dispersion parameters are just column dependent, then CL estimators are MVUEs
- cumulative loss and reserve estimates are also MVUEs
> > CL estimators = age-to-age factors
