Taylor

Question 1

Describe the effect of p in the Tweedie distribution

Answer 1

The tail heaviness of Tweedie distributions increases as p increases. (generates more widely dispersed residuals)

Question 2

Describe what we want to see from standardized Pearson residuals

Answer 2

residuals are revolving around 0 and homoscedasticity (constant variance) when plotting against covariates

Question 3

4 conditions for non-parametric Mack model (same model detailed in the Mack 1994 paper)

Answer 3

M1: Accident years are stochastically independent (similar to Mack assumption 2)
M2: For each accident year, the Xkj from a Markov chain (means that Xkj is only dependent on X-1)
M3a: For each AY k and Development period j, The expected incremental loss for the next development period is the incremental loss from current development period times LDF (similar to Mack Assumption 1)
M3b: For each AY and Development period, the variance for j+1 is some variance times Xkj (similar to Mack assumption 3)

Question 4

2 results from Non-Parametric Mack model

Answer 4

1. The conventional chain ladder estimators (ldfs) are unbiased AND minimum variance estimators among estimators that are unbiased linear combinations of LDFs
2. The conventional chain ladder estimator for reserves are unbiased

Question 5

Why is the Mack model stochastic and non-parametric

Answer 5

It’s stochastic because it considers the means AND variances of observations.
It’s non-parametric because it does not consider the distribution of the observations

Question 6

Describe the Theorem 3.1 of parametric Mack model (EDF Mack model results)

Answer 6

Assuming the data array is triangle (J=K)
If Mack’s original variance assumption M3b holds, then the maximum likelihood estimators (MLEs) of the LDFs are the conventional chain-ladder estimators (which are unbiased).
If ODP Mack Model AND the dispersion parameters are just column dependent (AY doesn’t matter), then the conventional chain-ladder estimators are minimum variance unbiased estimators (MVUEs). and the cumulative loss estimates and reserves estimates are also MVUEs.

Question 7

Assumptions for the EDF cross-classified model

Answer 7

1. The random variables Ykj are stochastically independent
2. Y follows EDF distribution. E(Yjk) = ak*bj. sum of all bj is 1

Question 8

Assumption of the ODP cross-classified model

Answer 8

Given the assumptions from the EDF cross-classified model.
Assume:
Ykj is restricted to an ODP distribution.
The dispersion parameter are identical for all cells

Question 9

Describe the Theorem 3.2 for ODP cross-classified model results

Answer 9

MLE fitted values and forecasts Ykj are the same as those given by the conventional chain-ladder method

Question 10

Describe the Theorem 3.3 for ODP cross-classified model results

Answer 10

The MLEs Ykj will not be unbiased for EDF model. However, if we assume it’s ODP and that the fitted values and forecasts Ykj and reserves are corrected for bias, then they are MVUEs for Ykj and reserves

Question 11

Why are theorems 3.2 and 3.3 more remarkable than 3.1

Answer 11

They state that the forecasts from the ODP Mack and ODP cross-classified models are identical and the same as those from the conventional chain-ladder method despite the different formulations.
Forecasts can be obtained from the ODP cross-classified model without any explicit consideration of its parameters by working as if the model were the ODP Mack model

Question 12

What happen when model is over-specified

Answer 12

leads to parameter redundancy.
Parameters are said to be aliased.
GLM software will set one of the parameters to zero and remove the redundancy

Question 13

What do you need to select to specify a GLM

Answer 13

1. Error distribution (one of the EDF)
2. Explanatory variables
3. Link function h