Taylor and McGuire Flashcards
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
Reason Taylor and McGuire’s parametric MVUE results are stronger than regular Mack results
minimum variance of all unbiased estimators, not just linear combinations
EDF cross-classified model assumptions (2)
- stochastic independence of response variable
2. explicit row and column parameters, where column parameters sum to 1
Theorem 3.2 from Taylor and McGuire (EDF and ODP cross-classified results)
under the EDF cross-classified assumptions, restricted to an ODP distribution with constant dispersion parameter, the MLE fitted values and forecasts are the same as the usual CL method
Theorem 3.3 from Taylor and McGuire (MVUE for cross-classified models)
if theorem 3.2 applies and the fitted and forecasted values are corrected for bias, then they are MVUEs
Alpha and beta parameter calculations for non-GLM version of ODP cross-classified model, order of calculations, and forecasted incremental losses
order: alpha increasing, beta decreasing
alpha ( 1 ) = latest cumulative loss
all other alphas = latest cumulative loss / ( 1 - sum of already calculated betas )
beta = sum ( incremental losses in column ) / sum (already calculated alphas )
forecasted incremental losses = alpha * beta for given row and column
Difference b/w ODP Mack model and ODP cross-classified model under GLM representations of CL models
ODP Mack models link ratios
ODP cross-classified models incremental losses
Matrix notation for GLM representation of ODP Mack model
Y = X * beta
where Y = individual predicted age-to-age factors (all)
X = identity matrix
beta = volume weighted age-to-age factors
Matrix notation for GLM representation of ODP cross-classified model
Y = X * beta
where Y = estimated incremental losses (all)
X = identity matrix
beta = ln of all alpha and beta parameters (single column in order)
Problem with GLM representation of ODP cross-classified model and consequence
can lead to parameter redundancy (aka model is over-specified and parameters are aliased)
> > if parameters are aliased, estimates will not match the non-GLM version (rescale alpha and beta parameters)
Forecast design matrix (GLM ODP cross-classified model)
same as regular GLM representation but only shows estimates of future incremental losses
