Taylor and McGuire

Question 1

Mean and variance of exponential dispersion family (EDF) of distributions

Answer 1

mean = mu

variance = dispersion parameter * variance function

Question 2

Variance function for the Tweedie sub-family of EDFs

Answer 2

variance = mu ^ p
And p between 0 and 1 (inclusive)

in words: variance is proportional to a power of the mean

Question 3

Relationship between p for the Tweedie distribution and tail heaviness

Answer 3

tail heaviness increases as p increases

Question 4

Mean and variance of a Tweedie distribution

Answer 4

mean = mu = [ ( 1 - p ) * theta ] ^ ( 1 / ( 1 - p ) )

where theta = location parameter

variance = dispersion parameter * mu ^ p

Question 5

General GLM format (in matrix notation)

Taylor and McGuire

Answer 5

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)

Question 6

Conditions for the structure of a GLM (3)

Taylor and McGuire

Answer 6

1. each observation is a member of the EDF
2. h(mu-sub i) = x-sub i-transpose * beta
3. observations are stochastically independent

Question 7

GLM version of a standard linear regression

Answer 7

mean, mu = sumproduct ( x, beta)

Question 8

Underlying assumptions of a standard linear regression (3)

Answer 8

1. errors are normally distributed
2. errors have constant variance
3. linear relationship

Question 9

Difference b/w weighted linear regression and standard linear regression

Answer 9

weighted linear regression recognizes errors might have unequal variances

Question 10

Model generalizations to get from a linear regression to a GLM (2)

Answer 10

1. non-linear relationship

2. non-normal errors

Question 11

Common estimation method for GLM parameters

Answer 11

MLE

Question 12

Requirements for selection of a GLM and purpose of each (4)

Answer 12

selection of:

1. cumulant function (controls the shape of the distribution)
2. index, p (controls relationship b/w mean and variance in an EDF)
3. covariates (x’s = explanatory variables)
4. link function (controls relationship b/w mean and covariates)

Question 13

Measure of model goodness of fit

Taylor and McGuire

Answer 13

deviance

> > smaller = better

Question 14

Deviance formula (unscaled)

Answer 14

deviance = 2 * sum ( log-likelihood (perfect model ) - log-likelihood ( actual model ) )

Question 15

Scale parameter calculated from deviance and corresponding distribution
(Taylor and McGuire)

Answer 15

scale parameter = deviance / ( n - p )

> > Chi-square distribution w/ ( n - p ) df

Question 16

Standardized Pearson Residuals (Taylor and McGuire)

Answer 16

= raw residual / std. dev. ( observation )

• unbiased and homoscedastic

Question 17

Problem with standardized Pearson residuals

Taylor and McGuire

Answer 17

reproduces any non-normality from the observations

Question 18

Best residual to use for model assessment and why

Taylor and McGuire

Answer 18

deviance residuals

why: corrects any non-normality in the data

Question 19

Deviance residual

Answer 19

= 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

Question 20

Types of stochastic models (4)

Taylor and McGuire

Answer 20

1. non-parametric Mack model
2. parametric Mack models
3. cross-classified models
4. GLM representations of CL models

Question 21

Results of non-parametric Mack model (2)

Answer 21

1. estimators of CL age-to-age factors are MVUE’s among estimators that are unbiased linear combinations
2. unbiased CL reserve estimates

Question 22

Special cases of parametric Mack models (2)

Answer 22

1. ODP Mack model

2. Tweedie Mack model

Question 23

Assumption required to turn a non-parametric Mack model into a parametric one

Answer 23

require that incremental observations (given claims to date) come from the EDF distributions

Question 24

Theorem 3.1 from Taylor and McGuire (3 MVUE results for parametric models)

Answer 24

under EDF and general Mack assumptions:

1. MLEs are the usual CL estimators (and are unbiased)
2. for special case of ODP Mack model and if dispersion parameters are just column dependent, then CL estimators are MVUEs
3. cumulative loss and reserve estimates are also MVUEs

> > CL estimators = age-to-age factors

Question 25

Reason Taylor and McGuire's parametric MVUE results are stronger than regular Mack results

Answer 25

minimum variance of all unbiased estimators, not just linear combinations

Question 26

EDF cross-classified model assumptions (2)

Answer 26

1. stochastic independence of response variable | 2. explicit row and column parameters, where column parameters sum to 1

Question 27

Theorem 3.2 from Taylor and McGuire (EDF and ODP cross-classified results)

Answer 27

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

Question 28

Theorem 3.3 from Taylor and McGuire (MVUE for cross-classified models)

Answer 28

if theorem 3.2 applies and the fitted and forecasted values are corrected for bias, then they are MVUEs

Question 29

Alpha and beta parameter calculations for non-GLM version of ODP cross-classified model, order of calculations, and forecasted incremental losses

Answer 29

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

Question 30

Difference b/w ODP Mack model and ODP cross-classified model under GLM representations of CL models

Answer 30

ODP Mack models link ratios ODP cross-classified models incremental losses

Question 31

Matrix notation for GLM representation of ODP Mack model

Answer 31

Y = X * beta where Y = individual predicted age-to-age factors (all) X = identity matrix beta = volume weighted age-to-age factors

Question 32

Matrix notation for GLM representation of ODP cross-classified model

Answer 32

Y = X * beta where Y = estimated incremental losses (all) X = identity matrix beta = ln of all alpha and beta parameters (single column in order)

Question 33

Problem with GLM representation of ODP cross-classified model and consequence

Answer 33

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)

Question 34

Forecast design matrix (GLM ODP cross-classified model)

Answer 34

same as regular GLM representation but only shows estimates of future incremental losses