Methods for Generic Function Mse

Description

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

Usage

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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 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{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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