Combining the hierarchical credibility model with a GLM (Ohlsson, 2008)
hierCredGLM.RdFit a random effects model using Ohlsson's methodology. In this function you explicitly specify the power parameter p.
See hierCredTweedie when you also want to estimate the p.
Usage
hierCredGLM(
formula,
data,
weights,
p = 1.5,
link.power = 0,
muHatGLM = TRUE,
epsilon = 1e-04,
maxiter = 500,
maxiterGLM = 500,
verbose = FALSE,
returnData = TRUE,
balanceProperty = TRUE,
y = TRUE,
...
)Arguments
- formula
object of type
formulathat specifies which model should be fitted. Syntax is the same as forlmerandglmer. For example,Yijkt ~ x1 + x2 + (1 | Industry / Branch).- 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 is1.5.- link.power
index of power link function, which is passed to
tweedie.link.power = 0produces a log-link. Defaults to the canonical link, which is1 - p.- muHatGLM
indicates which estimate has to be used in the algorithm for the intercept term. Default is
TRUE, which used the intercept as estimated by the GLM. IfFALSE, the estimate of the hierarchical credibility model is used.- epsilon
positive convergence tolerance \(\epsilon\); the iterations converge when 7 \(||\theta[k] - \theta[k - 1]||^2[[2]]/||\theta[k - 1]||^2[[2]] < \epsilon\). 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
speedglm.- 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
speedglm
Value
An object of type hierCredGLM with the following slots:
- call
the matched call
- HierarchicalResults
results of the hierarchical 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("dataCar")
fit = hierCredGLM(Y ~ area + (1 | VehicleType / VehicleBody), dataCar, weights = w,
p = 1.7)
fit
#> Call:
#> hierCredGLM(formula = Y ~ area + (1 | VehicleType/VehicleBody),
#> data = dataCar, weights = w, p = 1.7)
#>
#>
#> Combination of the hierarchical credibility model with a GLM
#>
#> Estimated variance parameters:
#> Var(V[jk]): 349.7505
#> Var(V[j]): 629.4394
#> Unique number of categories of VehicleType: 2
#> Unique number of categories of VehicleBody: 9
#>
#> Results contract-specific risk factors:
#>
#> Generalized Linear Model of class 'speedglm':
#>
#> Call: speedglm(formula = FormulaGLM, data = data, family = tweedie(var.power = p, link.power = 0), weights = data$wijkt, model = T, y = T, fitted = T)
#>
#> Coefficients:
#> (Intercept) areaB areaC areaD areaE areaF
#> 5.63463 0.04712 0.07701 -0.18517 0.13033 0.46341
#>
summary(fit)
#> Call:
#> hierCredGLM(formula = Y ~ area + (1 | VehicleType/VehicleBody),
#> data = dataCar, weights = w, p = 1.7)
#>
#>
#> Combination of the hierarchical credibility model with a GLM
#>
#> Estimated variance parameters:
#> Individual contracts: 4290291
#> Var(V[jk]): 349.7505
#> Var(V[j]): 629.4394
#> Unique number of categories of VehicleType: 2
#> Unique number of categories of VehicleBody: 9
#>
#> Results contract-specific risk factors:
#>
#> Generalized Linear Model of class 'speedglm':
#>
#> Call: speedglm(formula = FormulaGLM, data = data, family = tweedie(var.power = p, link.power = 0), weights = data$wijkt, model = T, y = T, fitted = T)
#>
#> Coefficients:
#> ------------------------------------------------------------------
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.63463 0.08794 64.0746 0.0000 ***
#> areaB 0.04712 0.13009 0.3622 0.7172
#> areaC 0.07701 0.11713 0.6575 0.5108
#> areaD -0.18517 0.15477 -1.1964 0.2315
#> areaE 0.13033 0.16747 0.7782 0.4364
#> areaF 0.46341 0.19218 2.4113 0.0159 *
#>
#> -------------------------------------------------------------------
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ---
#> null df: 67565; null deviance: 1405114;
#> residuals df: 67560; residuals deviance: 1400154;
#> # obs.: 67566; # non-zero weighted obs.: 67566;
#> AIC: NA; log Likelihood: ;
#> RSS: 21226516; dispersion: 314.1876; iterations: 11;
#> rank: 6; max tolerance: 5.91e-11; convergence: TRUE.
ranef(fit)
#> $sector
#> VehicleType Uj
#> 1: Common vehicle 0.9573893
#> 2: Uncommon vehicle 1.0525247
#>
#> $group
#> VehicleType VehicleBody Ujk
#> 1: Common vehicle HBACK 1.0252939
#> 2: Common vehicle SEDAN 0.9528740
#> 3: Common vehicle UTE 0.9971015
#> 4: Uncommon vehicle COUPE 1.0233226
#> 5: Uncommon vehicle HDTOP 1.0065053
#> 6: Uncommon vehicle MIBUS 1.0042123
#> 7: Uncommon vehicle PANVN 1.0023449
#> 8: Uncommon vehicle STNWG 0.9836377
#> 9: Uncommon vehicle TRUCK 1.0077063
#>
fixef(fit)
#> (Intercept) areaB areaC areaD areaE areaF
#> 5.63463077 0.04712428 0.07701333 -0.18517287 0.13033063 0.46341174
# }