bstraub {actuar}R Documentation

Buhlmann-Straub Credibility Model

Description

bstraub computes structure parameters estimators in the Bühlmann-Straub credibility model and predict.bstraub computes the credibility premiums.

Usage

bstraub(ratios, weights,
        method = c("unbiased", "iterative"),
        tol = sqrt(.Machine$double.eps), maxit = 100,
        echo = FALSE, old.format = TRUE)

## S3 method for class 'bstraub':
predict(object, levels = NULL, newdata, ...)
## S3 method for class 'bstraub.old':
predict(object, ...)

Arguments

ratios a matrix of ratios (contracts in lines, years in columns).
weights a matrix of weights corresponding to ratios.
method estimator of the between contract heterogeneity parameter used in premium calculation; "unbiased" for the usual Bühlmann-Straub estimator, "iterative" for the Bischel-Straub estimator (see below).
tol maximum relative error in the iterative procedure.
maxit maximum number of iterations.
echo logical; whether to echo iterative procedure or not.
old.format logical; if TRUE, return results in the deprecated pre-0.9-4 format.
object an object of class "bstraub".
levels, newdata, ... unused arguments.

Details

Direct usage of this function is deprecated. The function will not be exported in future versions of the package. Use cm instead.

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 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 methods of predict compute the credibility premiums.

Value

For bstraub with old.format = TRUE , an object of class "bstraub.old". This format is deprecated. An object of class "bstraub.old" is a list with the following components:

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;
cred vector of credibility factors;
means a list containing the collective premium estimator and vector of contract weighted averages.
weights a list containing the total portfolio weight and the vector of contracts total weights, as used in credibility factors;
unbiased a vector containing the unbiased variance components estimators.
iterative a vector containing the iterative variance components estimators, or NULL.
cred vector of credibility factors.
nodes a list containing the number of contracts in the portfolio.


For predict.bstraub, a vector of credibility premiums.

Estimation of a

The Bühlmann-Straub unbiased 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 with a relative error of tol as stopping criteria.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca, Sébastien Auclair and Louis-Philippe Pouliot

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

Examples

data(hachemeister)

## Credibility premiums calculated with the iterative estimator
fit <- bstraub(hachemeister[, 2:13], hachemeister[, 14:25],
               old.format = FALSE)
fit                             # a list
predict(fit)                    # credibility premiums

[Package actuar version 0.9-7 Index]