Combining the Buhlmann-Straub credibility model with a GLM (Ohlsson, 2008)

Fit a single-level random effects model using Ohlsson's methodology combined with Buhlmann-Straub credibility. This is the single-level analogue of hierCredGLM.

Usage

buhlmannStraubGLM(
  formula,
  data,
  weights,
  p = 1.5,
  link.power = 0,
  muHatGLM = FALSE,
  epsilon = 1e-04,
  maxiter = 500,
  maxiterGLM = 500,
  verbose = FALSE,
  returnData = TRUE,
  balanceProperty = TRUE,
  y = TRUE,
  ...
)

Arguments

formula: object of type formula that specifies which model should be fitted. Syntax follows lmer: e.g., Y ~ x1 + x2 + (1 | Cluster). Only one random effect is allowed.
data: an object that is coercible by as.data.table, containing the variables in the model.
weights: variable name of the exposure weight.
p: the value for the power parameter of the Tweedie distribution, which is passed to tweedie. Default is 1.5.
link.power: index of power link function, which is passed to tweedie. link.power = 0 produces a log-link. Defaults to the canonical link, which is 1 - p.
muHatGLM: indicates which estimate has to be used in the algorithm for the intercept term. Default is TRUE, which uses the intercept as estimated by the GLM. If FALSE, the estimate of the Buhlmann-Straub credibility model is used.
epsilon: positive convergence tolerance \(\epsilon\); the iterations converge when \(||\theta[k] - \theta[k - 1]||^2_2/||\theta[k - 1]||^2_2 < \epsilon\). Here, \(\theta[k]\) is the parameter vector at the \(k^{th}\) iteration.
maxiter: maximum number of iterations.
maxiterGLM: maximum number of iterations when fitting the GLM part. Passed to glm.
verbose: logical indicating if output should be produced during the algorithm.
returnData: logical indicating if input data has to be returned.
balanceProperty: logical indicating if the balance property should be satisfied.
y: logical indicating whether the response vector should be returned as a component of the returned value.
...: arguments passed to glm

Value

An object of type buhlmannStraubGLM with the following slots:

call: the matched call
CredibilityResults: results of the Buhlmann-Straub credibility model.
fitGLM: the results from fitting the GLM part.
iter: total number of iterations.
Converged: logical indicating whether the algorithm converged.
LevelsCov: object that summarizes the unique levels of each of the contract-specific covariates.
fitted.values: the fitted mean values, resulting from the model fit.
prior.weights: the weights (exposure) initially supplied.
y: if requested, the response vector. Default is TRUE.

References

Campo, B.D.C. and Antonio, Katrien (2023). Insurance pricing with hierarchically structured data an illustration with a workers' compensation insurance portfolio. Scandinavian Actuarial Journal, doi: 10.1080/03461238.2022.2161413

Ohlsson, E. (2008). Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal 2008(4), 301–314.

Examples

# \donttest{
data("hachemeister", package = "actuar")
# Prepare data
X = as.data.frame(hachemeister)
Df = reshape(X, idvar = "state",
             varying = list(paste0("ratio.", 1:12), paste0("weight.", 1:12)),
             direction = "long")
# Add a covariate
Df$time_factor = factor(Df$time)
# Fit model
fit = buhlmannStraubGLM(ratio.1 ~ time_factor + (1 | state), Df,
                        weights = weight.1, p = 1.5)
summary(fit)
#> Call:
#> buhlmannStraubGLM(formula = ratio.1 ~ time_factor + (1 | state), 
#>     data = Df, weights = weight.1, p = 1.5)
#> 
#> Buhlmann-Straub GLM credibility model
#> 
#> Convergence: YES 
#> Number of iterations: 1 
#> 
#> GLM Summary:
#> 
#> Call:
#> glm(formula = FormulaGLM, family = tweedie(var.power = p, link.power = 0), 
#>     data = data, weights = data$wijt, model = TRUE, y = TRUE)
#> 
#> Coefficients:
#>               Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)    7.28881    0.03296 221.171  < 2e-16 ***
#> time_factor2  -0.03024    0.04527  -0.668  0.50734    
#> time_factor3   0.03562    0.04558   0.782  0.43827    
#> time_factor4   0.14086    0.04520   3.117  0.00309 ** 
#> time_factor5   0.10941    0.04628   2.364  0.02216 *  
#> time_factor6   0.20572    0.04524   4.547 3.70e-05 ***
#> time_factor7   0.11839    0.04425   2.675  0.01018 *  
#> time_factor8   0.12531    0.04600   2.724  0.00896 ** 
#> time_factor9   0.14831    0.04678   3.171  0.00265 ** 
#> time_factor10  0.22036    0.04535   4.859 1.30e-05 ***
#> time_factor11  0.22157    0.04520   4.902 1.13e-05 ***
#> time_factor12  0.27532    0.04380   6.285 9.18e-08 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for Tweedie family taken to be 609.1103)
#> 
#>     Null deviance: 90265  on 59  degrees of freedom
#> Residual deviance: 29027  on 48  degrees of freedom
#> AIC: NA
#> 
#> Number of Fisher Scoring iterations: 3
#> 
#> 
#> Variance parameters from Buhlmann-Straub model:
#>   Sigma (within-group variance): 29131863 
#>   Tau (between-group variance): 69999.22 
#> 
#> Random effects at the state level:
#> 
#> Key: <state>
#>    state        zj        Uj
#>    <num>     <num>     <num>
#> 1:     1 0.9961348 1.2293438
#> 2:     2 0.9808662 0.9044844
#> 3:     3 0.9724230 1.0816545
#> 4:     4 0.9142906 0.8278582
#> 5:     5 0.9893672 0.9566591
ranef(fit)
#> $MLFj
#> Key: <state>
#>    state        Uj
#>    <num>     <num>
#> 1:     1 1.2293438
#> 2:     2 0.9044844
#> 3:     3 1.0816545
#> 4:     4 0.8278582
#> 5:     5 0.9566591
#> 
# }