actuar (version 0.9-3)

bstraub: Buhlmann-Straub Credibility Model

Description

bstraub computes structure parameters estimators in the B�hlmann-Straub credibility model and predict.bstraub computes the credibility premiums.

Usage

bstraub(ratios, weights,
        heterogeneity = c("iterative", "unbiased"),
        TOL = 1e-06, echo = FALSE)
## S3 method for class 'bstraub':
print(x, \dots)
## S3 method for class 'bstraub':
predict(object, \dots)
## S3 method for class 'bstraub':
summary(object, \dots)
## S3 method for class 'summary.bstraub':
print(x, \dots)

Arguments

ratios

a matrix of ratios (contracts in lines, years in columns).

weights

a matrix of weights corresponding to ratios.

heterogeneity

estimator of the between contract heterogeneity parameter used in premium calculation; "iterative" for the Bischel-Straub estimator; "unbiased" for the usual B�hlmann-Straub estimator (see below).

TOL

maximum relative error in the iterative procedure.

echo

logical; whether to echo iterative procedure or not.

x, object

an object of class "bstraub".

...

additional attributes to attach to the result for the summary method; further arguments to print for the print.summary method; unused for the print and <

Value

• For bstraub, an object of class "bstraub".

  An object of class "bstraub" is a list with the following components:

• model
  the name of the model used ("Buhlmann" or "Buhlmann-Straub");
• individual
  vector of contract weighted averages;
• collective
  collective premium estimator;
• weights
  vector of contracts total weights, as used in credibility factors;
• s2
  estimator of the within contract heterogeneity parameter;
• unbiased
  unbiased estimator of the between contract heterogeneity parameter;
• iterative
  iterative estimator of the between contract heterogeneity parameter.
• For predict.bstraub, a vector of credibility premiums.

Estimation of a

The B�hlmann-Straub unbiaised estimator (heterogeneity = "unbiased") of the between contracts heterogeneity parameter is $$\hat{a} = c \left( \sum_{i = 1}^I w_{i\cdot} (X_{iw} - X_{ww})^2 - (I - 1)\hat{s}^2 \right),$$ where $c = w_{\cdot\cdot}/(w_{\cdot\cdot}^2 - \sum_{i = 1}^I w_{i\cdot}^2)$ and $I$ is the number of contracts.

The Bishel-Straub pseudo-estimator (heterogeneity = "iterative") is obtained recursively as the solution of $$\hat{a} = \frac{1}{I - 1} \sum_{i=1}^I z_i (X_{iw} - X_{zw})^2.$$ The fixed point algorithm is used with a relative error of TOL as stopping criteria.

Details

The credibility premium of contract $i$ is given by $$z_i X_{iw} + (1 - z_i) X_{zw},$$ where $$z_{i} = \frac{w_{i\cdot} \hat{a}}{w_{i\cdot} \hat{a} + \hat{s}^2},$$ $X_{iw}$ is the weighted average of the ratios of contract $i$, $X_{zw}$ is the weighted average of the matrix of ratios using credibility factors and $w_{i\cdot}$ is the total weight of a contract. $\hat{s}^2$ is the estimator of the within contract heterogeneity and $\hat{a}$ is the estimator of the between contract heterogeneity.

Missing data are represented by NA in both the matrix of ratios and the matrix of weights. The function can cope with complete lines of NA in case a contract has no experience.

bstraub computes the structure parameters estimators and returns an object of class "bstraub". The method of summary for such objects displays further details and the method of predict computes the credibility premiums.

References

Goulet, V. (1998), Principles and Application of Credibility Theory, Journal of Actuarial Practice 6, 5--62.

Goovaerts, M. J. and Kaas, R. and van Heerwaarden, A. E. and Bauwelinckx, T. (1990), Effective actuarial methods, North-Holland.

See Also

cm for more general credibility models.

Examples

data(hachemeister)

## Credibility premiums calculated with the iterative estimator
fit <- bstraub(hachemeister[, 2:13], hachemeister[, 14:25])
fit 				# print method
summary(fit)			# more details
predict(fit)			# credibility premiums