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_i) = x^T * B
3. observations are stochastically independent

Question 7

Underlying assumptions of a standard linear regression (3)

Answer 7

1. errors are normally distributed
2. errors have constant variance
3. linear relationship between X and Y

Question 8

Difference b/w weighted linear regression and standard linear regression

Answer 8

weighted linear regression recognizes errors might have unequal variances

Question 9

Main differences between a GLM and a linear regression (2)

Answer 9

1. non-linear relationship between X and Y
2. non-normal error term

Question 10

Common estimation method for GLM parameters

Answer 10

MLE

Question 11

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

Answer 11

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 12

Measure of model goodness of fit

Taylor and McGuire

Answer 12

deviance

> > smaller = better

Question 13

Deviance formula (unscaled)

Answer 13

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

Question 14

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

Answer 14

scale parameter = deviance / ( n - p )

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

Question 15

Standardized Pearson Residuals (Taylor and McGuire)

Answer 15

= raw residual / std. dev. ( observation )

• unbiased and homoscedastic

Question 16

Problem with standardized Pearson residuals

Taylor and McGuire

Answer 16

does not remove skewness from the data

we prefer residuals that are approximately normally distributed

Question 17

Best residual to use for model assessment and why

Taylor and McGuire

Answer 17

deviance residuals

why: corrects any non-normality in the data

Question 18

Deviance residual

Answer 18

Rd_i = sqrt(d_i / phi^hat) * sign(Y_i - Y_i^hat)

Question 19

Types of stochastic models (4)

Taylor and McGuire

Answer 19

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

Question 20

Results of non-parametric Mack model (2)

Answer 20

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

Question 21

Special cases of parametric Mack models (2)

Answer 21

1. ODP Mack model
2. Tweedie Mack model

these remove Variance assumption from Mack, variance confined to the variance of an EDF distribution

Question 22

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

Answer 22

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

Question 23

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

Answer 23

under EDF and general Mack assumptions:

1. MLEs are the unbiased CL estimators
2. for ODP Mack and column dispersion parameters, CL estimators are MVUEs
3. cumulative loss and reserve estimates are also MVUEs

Question 24

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

Answer 24

minimum variance of all unbiased estimators, not just linear combinations

Question 25

EDF cross-classified model assumptions (2)

Answer 25

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

Question 26

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

Answer 26

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 27

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

Answer 27

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

Question 28

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

Answer 28

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 29

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

Answer 29

ODP Mack models link ratios

ODP cross-classified models incremental losses

Question 30

Matrix notation for GLM representation of ODP Mack model

Answer 30

Y = X * beta

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

Question 31

Matrix notation for GLM representation of ODP cross-classified model

Answer 31

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 32

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

Answer 32

can lead to parameter redundancy, need to alias a parameter (GLM software will do this automatically)

> > if parameters are aliased, estimates will not match the non-GLM version (rescale alpha and beta parameters)

Question 33

Forecast design matrix (GLM ODP cross-classified model)

Answer 33

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