Methods for Generic Function Mse

Mse is a generic function to calculate mean square error estimations in the chain-ladder framework.

Usage

Mse(ModelFit, FullTriangles, ...)

# S4 method for GMCLFit,triangles
Mse(ModelFit, FullTriangles, ...)
# S4 method for MCLFit,triangles
Mse(ModelFit, FullTriangles, mse.method="Mack", ...)

Arguments

ModelFit: An object of class "GMCLFit" or "MCLFit".
FullTriangles: An object of class "triangles". Should be the output from a call of predict.
mse.method: Character strings that specify the MSE estimation method. Only works for "MCLFit". Use "Mack" for the generazliation of the Mack (1993) approach, and "Independence" for the conditional resampling approach in Merz and Wuthrich (2008).
...: Currently not used.

Details

These functions calculate the conditional mean square errors using the recursive formulas in Zhang (2010), which is a generalization of the Mack (1993, 1999) formulas. In the GMCL model, the conditional mean square error for single accident years and aggregated accident years are calcualted as:

$$\hat{mse}(\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\hat{Y}_{i,k}|D) \hat{B}_k + (\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\hat{Y}_{i,k} \otimes I) + \hat{\Sigma}_{\epsilon_{i_k}}.$$

$$\hat{mse}(\sum^I_{i=a_k}\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\sum^I_{i=a_k+1}\hat{Y}_{i,k}|D) \hat{B}_k + (\sum^I_{i=a_k}\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\sum^I_{i=a_k}\hat{Y}_{i,k} \otimes I) + \sum^I_{i=a_k}\hat{\Sigma}_{\epsilon_{i_k}} .$$

In the MCL model, the conditional mean square error from Merz and Wüthrich (2008) is also available, which can be shown to be equivalent as the following:

$$\hat{mse}(\hat{Y}_{i,k+1}|D)=(\hat{\beta}_k \hat{\beta}_k') \odot \hat{mse}(\hat{Y}_{i,k}|D) + \hat{\Sigma}_{\beta_k} \odot (\hat{Y}_{i,k} \hat{Y}_{i,k}') + \hat{\Sigma}_{\epsilon_{i_k}} +\hat{\Sigma}_{\beta_k} \odot \hat{mse}^E(\hat{Y}_{i,k}|D) .$$

$$\hat{mse}(\sum^I_{i=a_k}\hat{Y}_{i,k+1}|D)=(\hat{\beta}_k \hat{\beta}_k') \odot \sum^I_{i=a_k+1}\hat{mse}(\hat{Y}_{i,k}|D) + \hat{\Sigma}_{\beta_k} \odot (\sum^I_{i=a_k}\hat{Y}_{i,k} \sum^I_{i=a_k}\hat{Y}_{i,k}') + \sum^I_{i=a_k}\hat{\Sigma}_{\epsilon_{i_k}} +\hat{\Sigma}_{\beta_k} \odot \sum^I_{i=a_k}\hat{mse}^E(\hat{Y}_{i,k}|D) .$$

For the Mack approach in the MCL model, the cross-product term $\hat{\Sigma}_{\beta_k} \odot \hat{mse}^E(\hat{Y}_{i,k}|D) $in the above two formulas will drop out.

Value

Mse returns an object of class "MultiChainLadderMse" that has the following elements:

mse.ay: condtional mse for each accdient year
mse.ay.est: conditional estimation mse for each accdient year
mse.ay.proc: conditional process mse for each accdient year
mse.total: condtional mse for aggregated accdient years
mse.total.est: conditional estimation mse for aggregated accdient years
mse.total.proc: conditional process mse for aggregated accdient years
FullTriangles: completed triangles

References

Zhang Y (2010). A general multivariate chain ladder model.Insurance: Mathematics and Economics, 46, pp. 588-599.

Zhang Y (2010). Prediction error of the general multivariate chain ladder model.

Author