bstraub {actuar}R Documentation

Buhlmann-Straub Credibility Model

Description

bstraub calculates credibility premiums in the Bühlmann-Straub credibility model.

Usage

bstraub(ratios, weights,
        heterogeneity = c("iterative", "unbiased"),
        TOL = 1e-06, echo = FALSE)

Arguments

ratios matrix of ratios (contracts in lines, years in columns)
weights 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 Bühlmann-Straub estimator (see below)
TOL maximum relative error in the iterative procedure
echo boolean, whether to echo iterative procedure or not

Details

The credibility premium of contract i is given by

z[i] X[iw] + (1 - z[i]) X[zw],

where

z[i] = (w[i.] a)/(w[i] a + 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.] is the total weight of a contract. s^2 is the estimator of the within contract heterogeneity and a is the estimator of the between contract heterogeneity.

Missing data are represent 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.

Value

A list with the following components:

premiums vector of credibility premiums
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

Estimation of a

The Bühlmann-Straub unbiaised estimator (heterogeneity = "unbiased") of the between contracts heterogeneity parameter is

a = c sum(w[i.] * (X[iw] - X[ww])^2 - (I - 1) * s^2),

where c = w[..]/(w[..]^2 - sum(w[i.]^2)) and I is the number of contracts.

The Bishel-Straub pseudo-estimator (heterogeneity = "iterative") is obtained recursively as the solution of

a = 1/(I - 1) sum(z[i] * (X[iw] - X[zw])^2).

The fixed point algorithm is used up, with a relative error of TOL stopping criteria.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Sébastien Auclair

References

Goulet, V. (1998), Principles and Application of Credibility Theory, Journal of Actuarial Practice, Volume 6, ISSN 1064-6647.

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

Examples

data(hachemeister)

## Credibility premiums calculated with the iterative estimator
bstraub(hachemeister$claims, hachemeister$weights)

## Credibility premiums calculated with the unbiased estimator
bstraub(hachemeister$claims, hachemeister$weights, heterogeneity = "unbiased")

[Package actuar version 0.1-3 Index]