Fit the following credibility models: Bühlmann, Bühlmann-Straub, hierarchical, regression (Hachemeister) or linear Bayes.
cm(formula, data, ratios, weights, subset,
regformula = NULL, regdata, adj.intercept = FALSE,
method = c("Buhlmann-Gisler", "Ohlsson", "iterative"),
likelihood, ...,
tol = sqrt(.Machine$double.eps), maxit = 100, echo = FALSE)# S3 method for cm
print(x, ...)
# S3 method for cm
predict(object, levels = NULL, newdata, ...)
# S3 method for cm
summary(object, levels = NULL, newdata, ...)
# S3 method for summary.cm
print(x, ...)
Function cm computes the structure parameters estimators of the
model specified in formula. The value returned is an object of
class cm.
An object of class "cm" is a list with at least the following
components:
a list containing, for each level, the vector of linearly sufficient statistics.
a list containing, for each level, the vector of total weights.
a vector containing the unbiased variance components
estimators, or NULL.
a vector containing the iterative variance components
estimators, or NULL.
for multi-level hierarchical models: a list containing, the vector of credibility factors for each level. For one-level models: an array or vector of credibility factors.
a list containing, for each level, the vector of the number of nodes in the level.
the columns of data containing the
portfolio classification structure.
a list containing, for each level, the affiliation of a node to the node of the level above.
Regression fits have in addition the following components:
a list containing, for each node, the credibility
adjusted regression model as obtained with
lm.fit or lm.wfit.
if adj.intercept is TRUE, a transition
matrix from the basis of the orthogonal design matrix to the basis
of the original design matrix.
the terms object used.
The method of predict for objects of class "cm" computes
the credibility premiums for the nodes of every level included in
argument levels (all by default). Result is a list the same
length as levels or the number of levels in formula, or
an atomic vector for one-level models.
character string "bayes" or an object of
class "formula": a symbolic description of the
model to be fit. The details of model specification are given
below.
a matrix or a data frame containing the portfolio structure, the ratios or claim amounts and their associated weights, if any.
expression indicating the columns of data
containing the ratios or claim amounts.
expression indicating the columns of data
containing the weights associated with ratios.
an optional logical expression indicating a subset of observations to be used in the modeling process. All observations are included by default.
an object of class "formula":
symbolic description of the regression component (see
lm for details). No left hand side is needed
in the formula; if present it is ignored. If NULL, no
regression is done on the data.
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame)
containing the variables in the regression model.
if TRUE, the intercept of the regression
model is located at the barycenter of the regressor instead of the
origin.
estimation method for the variance components of the model; see Details.
a character string giving the name of the likelihood function in one of the supported linear Bayes cases; see Details.
tolerance level for the stopping criteria for iterative estimation method.
maximum number of iterations in iterative estimation method.
logical; whether to echo the iterative procedure or not.
an object of class "cm".
character vector indicating the levels to predict or to
include in the summary; if NULL all levels are included.
data frame containing the variables used to predict credibility regression models.
parameters of the prior distribution for cm;
additional attributes to attach to the result for the
predict and summary methods; further arguments to
format for the print.summary method;
unused for the print method.
The credibility premium at one level is a convex combination between the linearly sufficient statistic of a node and the credibility premium of the level above. (For the first level, the complement of credibility is given to the collective premium.) The linearly sufficient statistic of a node is the credibility weighted average of the data of the node, except at the last level, where natural weights are used. The credibility factor of node \(i\) is equal to $$\frac{w_i}{w_i + a/b},$$ where \(w_i\) is the weight of the node used in the linearly sufficient statistic, \(a\) is the average within node variance and \(b\) is the average between node variance.
The credibility premium of node \(i\) is equal to
$$y^\prime b_i^a,$$
where \(y\) is a matrix created from newdata and
\(b_i^a\) is the vector of credibility adjusted regression
coefficients of node \(i\). The latter is given by
$$b_i^a = Z_i b_i + (I - Z_I) m,$$
where \(b_i\) is the vector of regression coefficients based
on data of node \(i\) only, \(m\) is the vector of collective
regression coefficients, \(Z_i\) is the credibility matrix and
\(I\) is the identity matrix. The credibility matrix of node \(i\)
is equal to
$$A^{-1} (A + s^2 S_i),$$
where \(S_i\) is the unscaled regression covariance matrix of
the node, \(s^2\) is the average within node variance and
\(A\) is the within node covariance matrix.
If the intercept is positioned at the barycenter of time, matrices \(S_i\) and \(A\) (and hence \(Z_i\)) are diagonal. This amounts to use Bühlmann-Straub models for each regression coefficient.
Argument newdata provides the “future” value of the
regressors for prediction purposes. It should be given as specified in
predict.lm.
For hierarchical models, two sets of estimators of the variance components (other than the within node variance) are available: unbiased estimators and iterative estimators.
Unbiased estimators are based on sums of squares of the form $$B_i = \sum_j w_{ij} (X_{ij} - \bar{X}_i)^2 - (J - 1) a$$ and constants of the form $$c_i = w_i - \sum_j \frac{w_{ij}^2}{w_i},$$ where \(X_{ij}\) is the linearly sufficient statistic of level \((ij)\); \(\bar{X_{i}}\) is the weighted average of the latter using weights \(w_{ij}\); \(w_i = \sum_j w_{ij}\); \(J\) is the effective number of nodes at level \((ij)\); \(a\) is the within variance of this level. Weights \(w_{ij}\) are the natural weights at the lowest level, the sum of the natural weights the next level and the sum of the credibility factors for all upper levels.
The Bühlmann-Gisler estimators (method =
"Buhlmann-Gisler") are given by
$$b = \frac{1}{I} \sum_i \max \left( \frac{B_i}{c_i}, 0
\right),$$
that is the average of the per node variance estimators truncated at
0.
The Ohlsson estimators (method = "Ohlsson") are given by
$$b = \frac{\sum_i B_i}{\sum_i c_i},$$
that is the weighted average of the per node variance estimators
without any truncation. Note that negative estimates will be truncated
to zero for credibility factor calculations.
In the Bühlmann-Straub model, these estimators are equivalent.
Iterative estimators method = "iterative" are pseudo-estimators
of the form
$$b = \frac{1}{d} \sum_i w_i (X_i - \bar{X})^2,$$
where \(X_i\) is the linearly sufficient statistic of one
level, \(\bar{X}\) is the linearly sufficient statistic of
the level above and \(d\) is the effective number of nodes at one
level minus the effective number of nodes of the level above. The
Ohlsson estimators are used as starting values.
For regression models, with the intercept at time origin, only
iterative estimators are available. If method is different from
"iterative", a warning is issued. With the intercept at the
barycenter of time, the choice of estimators is the same as in the
Bühlmann-Straub model.
When formula is "bayes", the function computes pure
Bayesian premiums for the following combinations of distributions
where they are linear credibility premiums:
\(X|\Theta = \theta \sim \mathrm{Poisson}(\theta)\) and \(\Theta \sim \mathrm{Gamma}(\alpha, \lambda)\);
\(X|\Theta = \theta \sim \mathrm{Exponential}(\theta)\) and \(\Theta \sim \mathrm{Gamma}(\alpha, \lambda)\);
\(X|\Theta = \theta \sim \mathrm{Gamma}(\tau, \theta)\) and \(\Theta \sim \mathrm{Gamma}(\alpha, \lambda)\);
\(X|\Theta = \theta \sim \mathrm{Normal}(\theta, \sigma_2^2)\) and \(\Theta \sim \mathrm{Normal}(\mu, \sigma_1^2)\);
\(X|\Theta = \theta \sim \mathrm{Bernoulli}(\theta)\) and \(\Theta \sim \mathrm{Beta}(a, b)\);
\(X|\Theta = \theta \sim \mathrm{Binomial}(\nu, \theta)\) and \(\Theta \sim \mathrm{Beta}(a, b)\);
\(X|\Theta = \theta \sim \mathrm{Geometric}(\theta)\) and \(\Theta \sim \mathrm{Beta}(a, b)\).
\(X|\Theta = \theta \sim \mathrm{Negative~Binomial}(r, \theta)\) and \(\Theta \sim \mathrm{Beta}(a, b)\).
The following combination is also supported: \(X|\Theta = \theta \sim \mathrm{Single~Parameter~Pareto}(\theta)\) and \(\Theta \sim \mathrm{Gamma}(\alpha, \lambda)\). In this case, the Bayesian estimator not of the risk premium, but rather of parameter \(\theta\) is linear with a “credibility” factor that is not restricted to \((0, 1)\).
Argument likelihood identifies the distribution of \(X|\Theta
= \theta\) as one of
"poisson",
"exponential",
"gamma",
"normal",
"bernoulli",
"binomial",
"geometric",
"negative binomial" or
"pareto".
The parameters of the distributions of \(X|\Theta = \theta\) (when
needed) and \(\Theta\) are set in ... using the argument
names (and default values) of dgamma,
dnorm, dbeta,
dbinom, dnbinom or
dpareto1, as appropriate. For the Gamma/Gamma case, use
shape.lik for the shape parameter \(\tau\) of the Gamma
likelihood. For the Normal/Normal case, use sd.lik for the
standard error \(\sigma_2\) of the Normal likelihood.
Data for the linear Bayes case may be a matrix or data frame as usual;
an atomic vector to fit the model to a single contract; missing or
NULL to fit the prior model. Arguments ratios,
weights and subset are ignored.
Vincent Goulet vincent.goulet@act.ulaval.ca, Xavier Milhaud, Tommy Ouellet, Louis-Philippe Pouliot
cm is the unified front end for credibility models fitting. The
function supports hierarchical models with any number of levels (with
Bühlmann and Bühlmann-Straub models as
special cases) and the regression model of Hachemeister. Usage of
cm is similar to lm for these cases.
cm can also fit linear Bayes models, in which case usage is
much simplified; see the section on linear Bayes below.
When not "bayes", the formula argument symbolically
describes the structure of the portfolio in the form \(~ terms\).
Each term is an interaction between risk factors contributing to the
total variance of the portfolio data. Terms are separated by +
operators and interactions within each term by :. For a
portfolio divided first into sectors, then units and finally
contracts, formula would be ~ sector + sector:unit +
sector:unit:contract, where sector, unit and
contract are column names in data. In general, the
formula should be of the form ~ a + a:b + a:b:c + a:b:c:d +
....
If argument regformula is not NULL, the regression model
of Hachemeister is fit to the data. The response is usually time. By
default, the intercept of the model is located at time origin. If
argument adj.intercept is TRUE, the intercept is moved
to the (collective) barycenter of time, by orthogonalization of the
design matrix. Note that the regression coefficients may be difficult
to interpret in this case.
Arguments ratios, weights and subset are used
like arguments select, select and subset,
respectively, of function subset.
Data does not have to be sorted by level. Nodes with no data (complete
lines of NA except for the portfolio structure) are allowed,
with the restriction mentioned above.
Bühlmann, H. and Gisler, A. (2005), A Course in Credibility Theory and its Applications, Springer.
Belhadj, H., Goulet, V. and Ouellet, T. (2009), On parameter estimation in hierarchical credibility, Astin Bulletin 39.
Goulet, V. (1998), Principles and application of credibility theory, Journal of Actuarial Practice 6, ISSN 1064-6647.
Goovaerts, M. J. and Hoogstad, W. J. (1987), Credibility Theory, Surveys of Actuarial Studies, No. 4, Nationale-Nederlanden N.V.
subset, formula,
lm, predict.lm.
data(hachemeister)
## Buhlmann-Straub model
fit <- cm(~state, hachemeister,
ratios = ratio.1:ratio.12, weights = weight.1:weight.12)
fit # print method
predict(fit) # credibility premiums
summary(fit) # more details
## Two-level hierarchical model. Notice that data does not have
## to be sorted by level
X <- data.frame(unit = c("A", "B", "A", "B", "B"), hachemeister)
fit <- cm(~unit + unit:state, X, ratio.1:ratio.12, weight.1:weight.12)
predict(fit)
predict(fit, levels = "unit") # unit credibility premiums only
summary(fit)
summary(fit, levels = "unit") # unit summaries only
## Regression model with intercept at time origin
fit <- cm(~state, hachemeister,
regformula = ~time, regdata = data.frame(time = 12:1),
ratios = ratio.1:ratio.12, weights = weight.1:weight.12)
fit
predict(fit, newdata = data.frame(time = 0))
summary(fit, newdata = data.frame(time = 0))
## Same regression model, with intercept at barycenter of time
fit <- cm(~state, hachemeister, adj.intercept = TRUE,
regformula = ~time, regdata = data.frame(time = 12:1),
ratios = ratio.1:ratio.12, weights = weight.1:weight.12)
fit
predict(fit, newdata = data.frame(time = 0))
summary(fit, newdata = data.frame(time = 0))
## Poisson/Gamma pure Bayesian model
fit <- cm("bayes", data = c(5, 3, 0, 1, 1),
likelihood = "poisson", shape = 3, rate = 3)
fit
predict(fit)
summary(fit)
## Normal/Normal pure Bayesian model
cm("bayes", data = c(5, 3, 0, 1, 1),
likelihood = "normal", sd.lik = 2,
mean = 2, sd = 1)
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