| Mse-methods {ChainLadder} | R Documentation |
Methods for Generic Function Mse
Description
Mse is a generic function to calculate mean square error estimations in the chain-ladder framework.
Usage
Mse(ModelFit, FullTriangles, ...)
## S4 method for signature 'GMCLFit,triangles'
Mse(ModelFit, FullTriangles, ...)
## S4 method for signature '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 |
mse.method |
Character strings that specify the MSE estimation method. Only works for "MCLFit". Use |
... |
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 |
Author(s)
Wayne Zhang actuary_zhang@hotmail.com
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.
See Also
See also MultiChainLadder.