bstraub {actuar} | R Documentation |
bstraub
calculates credibility premiums in the Bühlmann-Straub
credibility model.
bstraub(ratios, weights, heterogeneity = c("iterative", "unbiased"), TOL = 1e-06, echo = FALSE)
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 |
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.
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 |
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.
Vincent Goulet vincent.goulet@act.ulaval.ca and Sébastien Auclair
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.
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")