Mse is a generic function to calculate mean square error estimations in the chain ladder framework.Mse(ModelFit, FullTriangles, ...)## S3 method for class 'GMCLFit,triangles':
Mse(ModelFit, FullTriangles, ...)
## S3 method for class 'MCLFit,triangles':
Mse(ModelFit, FullTriangles, mse.method="Mack", ...)
predict."Mack" for the generazliation of the Mack (1993) approach, and "Independence" for the conditional resampling approach in Merz and Wuthrich (20Mse returns an object of class "MultiChainLadderMse" that has the following elements:$$\hat{mse}(\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\hat{Y}_{i,k}|D) \hat{B}_k + (\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\hat{Y}_{i,k} \otimes I) + \hat{\Sigma}_{\epsilon_{i_k}}.$$
$$\hat{mse}(\sum^I_{i=a_k}\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\sum^I_{i=a_k+1}\hat{Y}_{i,k}|D) \hat{B}_k + (\sum^I_{i=a_k}\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\sum^I_{i=a_k}\hat{Y}_{i,k} \otimes I) + \sum^I_{i=a_k}\hat{\Sigma}_{\epsilon_{i_k}} .$$
In the MCL model, the conditional mean square error from Merz and Wuthrich (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.
Zhang Y (2010). Prediction error of the general multivariate chain ladder model.
MultiChainLadder.