bstraub {actuar} | R Documentation |
bstraub
computes structure parameters estimators in the
Bühlmann-Straub credibility model and predict.bstraub
computes
the credibility premiums.
bstraub(ratios, weights, heterogeneity = c("iterative", "unbiased"), TOL = 1e-06, echo = FALSE) ## S3 method for class 'bstraub': print(x, ...) ## S3 method for class 'bstraub': predict(object, ...) ## S3 method for class 'bstraub': summary(object, ...) ## S3 method for class 'summary.bstraub': print(x, ...)
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
Bühlmann-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 predict methods. |
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 method of
summary
for such objects displays further
details and the method of predict
computes the
credibility premiums.
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.
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 with a relative error of TOL
as stopping criteria.
Vincent Goulet vincent.goulet@act.ulaval.ca, Sébastien Auclair and Louis-Philippe Pouliot
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.
cm
for more general credibility models.
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