# Methods for Generic Function Mse

### Description

`Mse`

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

### Usage

1 2 3 4 5 6 |

### 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{Σ}_{B_k} (\hat{Y}_{i,k} \otimes I) + \hat{Σ}_{ε_{i_k}}.*

*\hat{mse}(∑^I_{i=a_k}\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(∑^I_{i=a_k+1}\hat{Y}_{i,k}|D) \hat{B}_k + (∑^I_{i=a_k}\hat{Y}_{i,k}' \otimes I) \hat{Σ}_{B_k} (∑^I_{i=a_k}\hat{Y}_{i,k} \otimes I) + ∑^I_{i=a_k}\hat{Σ}_{ε_{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{β}_k \hat{β}_k') \odot \hat{mse}(\hat{Y}_{i,k}|D) + \hat{Σ}_{β_k} \odot (\hat{Y}_{i,k} \hat{Y}_{i,k}') + \hat{Σ}_{ε_{i_k}} +\hat{Σ}_{β_k} \odot \hat{mse}^E(\hat{Y}_{i,k}|D) .*

*\hat{mse}(∑^I_{i=a_k}\hat{Y}_{i,k+1}|D)=(\hat{β}_k \hat{β}_k') \odot ∑^I_{i=a_k+1}\hat{mse}(\hat{Y}_{i,k}|D) + \hat{Σ}_{β_k} \odot (∑^I_{i=a_k}\hat{Y}_{i,k} ∑^I_{i=a_k}\hat{Y}_{i,k}') + ∑^I_{i=a_k}\hat{Σ}_{ε_{i_k}}
+\hat{Σ}_{β_k} \odot ∑^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{Σ}_{β_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.`

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