Combining the hierarchical credibility model with a GLM (Ohlsson, 2008)
hierCredTweedie.RdFit a random effects model using Ohlsson's methodology. In this function you estimate the power parameter p. See hierCredGLM when
you want fix p.
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.
- 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 \(||\theta[k] - \theta[k - 1]||^2[[2]]/||\theta[k - 1]||^2[[2]] < \epsilon\). Here, \(\theta[k]\) is the parameter vector at the \(k^{th}\) iteration.
- maxiter
maximum number of iterations.
- verbose
logical indicating if output should be produced during the algorithm.
- returnData
logical indicating if input data has to be returned.
- cpglmControl
a list of parameters to control the fitting process in the GLM part. By default,
cpglmControl = list(bound.p = c(1.01, 1.99))which restricts the range of the power parameter p to [1.01, 1.99] in the fitting process. This list is passed tocpglm.- balanceProperty
logical indicating if the balance property should be satisfied.
- optimizer
a character string that determines which optimization routine is to be used in estimating the index and the dispersion parameters. Possible choices are
"nlminb"(the default, seenlminb),"bobyqa"(bobyqa) and"L-BFGS-B"(optim).- y
logical indicating whether the response vector should be returned as a component of the returned value.
- ...
arguments passed to
cpglm.
Value
An object of type hierCredTweedie 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.
Details
When estimating the GLM part, this function uses the cpglm function from the cplm package.
References
Ohlsson, E. (2008). Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal 2008(4), 301–314.
See also
hierCredTweedie-class, fitted.hierCredTweedie, predict.hierCredTweedie, ranef-actuaRE,
weights-actuaRE, hierCredibility, hierCredGLM, cpglm, plotRE,
adjustIntercept, BalanceProperty
@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
Examples
# \donttest{
data("dataCar")
fit = hierCredTweedie(Y ~ area + (1 | VehicleType / VehicleBody), dataCar,
weights = w, epsilon = 1e-6)
fit
#> Call:
#> hierCredTweedie(formula = Y ~ area + (1 | VehicleType/VehicleBody),
#> data = dataCar, weights = w, epsilon = 1e-06)
#>
#>
#> Combination of the hierarchical credibility model with a GLM
#>
#> Estimated variance parameters:
#> Var(V[jk]): 349.3256
#> Var(V[j]): 628.8985
#> Unique number of categories of VehicleType: 2
#> Unique number of categories of VehicleBody: 9
#>
#> Results contract-specific risk factors:
#>
#>
#> Call:
#> cpglm(formula = FormulaGLM, link = "log", data = data, weights = wijkt,
#> control = cpglmControl, optimizer = optimizer)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -8.095 -5.970 -4.573 -2.928 86.137
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.63662 0.08834 63.803 <2e-16 ***
#> areaB 0.04772 0.13048 0.366 0.7146
#> areaC 0.07692 0.11742 0.655 0.5124
#> areaD -0.18574 0.15671 -1.185 0.2359
#> areaE 0.13053 0.16712 0.781 0.4348
#> areaF 0.46389 0.18823 2.464 0.0137 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Estimated dispersion parameter: 175.29
#> Estimated index parameter: 1.5705
#>
#> Residual deviance: 2360054 on 67560 degrees of freedom
#> AIC: 116884
#>
#> Number of Fisher Scoring iterations: 6
#>
summary(fit)
#> Call:
#> hierCredTweedie(formula = Y ~ area + (1 | VehicleType/VehicleBody),
#> data = dataCar, weights = w, epsilon = 1e-06)
#>
#>
#> Combination of the hierarchical credibility model with a GLM
#>
#> Estimated variance parameters:
#> Individual contracts: 4289401
#> Var(V[jk]): 349.3256
#> Var(V[j]): 628.8985
#> Unique number of categories of VehicleType: 2
#> Unique number of categories of VehicleBody: 9
#>
#> Results contract-specific risk factors:
#>
#>
#> Call:
#> cpglm(formula = FormulaGLM, link = "log", data = data, weights = wijkt,
#> control = cpglmControl, optimizer = optimizer)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -8.095 -5.970 -4.573 -2.928 86.137
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.63662 0.08834 63.803 <2e-16 ***
#> areaB 0.04772 0.13048 0.366 0.7146
#> areaC 0.07692 0.11742 0.655 0.5124
#> areaD -0.18574 0.15671 -1.185 0.2359
#> areaE 0.13053 0.16712 0.781 0.4348
#> areaF 0.46389 0.18823 2.464 0.0137 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Estimated dispersion parameter: 175.29
#> Estimated index parameter: 1.5705
#>
#> Residual deviance: 2360054 on 67560 degrees of freedom
#> AIC: 116884
#>
#> Number of Fisher Scoring iterations: 6
#>
#> $call
#> cpglm(formula = FormulaGLM, link = "log", data = data, weights = wijkt,
#> control = cpglmControl, optimizer = optimizer)
#>
#> $deviance
#> [1] 2360054
#>
#> $aic
#> [1] 116884
#>
#> $contrasts
#> NULL
#>
#> $df.residual
#> [1] 67560
#>
#> $iter
#> [1] 6
#>
#> $na.action
#> NULL
#>
#> $deviance.resid
#> Min 1Q Median 3Q Max
#> -8.095052 -5.969509 -4.572944 -2.927895 86.137065
#>
#> $coefficients
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.63661846 0.08834343 63.8034805 0.00000000
#> areaB 0.04771903 0.13047847 0.3657234 0.71457260
#> areaC 0.07691849 0.11741546 0.6550968 0.51240767
#> areaD -0.18573986 0.15670655 -1.1852719 0.23591415
#> areaE 0.13052750 0.16711877 0.7810464 0.43477797
#> areaF 0.46389185 0.18823319 2.4644530 0.01372473
#>
#> $dispersion
#> [1] 175.2852
#>
#> $vcov
#> (Intercept) areaB areaC areaD areaE
#> (Intercept) 0.007804562 -0.007804562 -0.007804562 -0.007804562 -0.007804562
#> areaB -0.007804562 0.017024631 0.007804562 0.007804562 0.007804562
#> areaC -0.007804562 0.007804562 0.013786391 0.007804562 0.007804562
#> areaD -0.007804562 0.007804562 0.007804562 0.024556942 0.007804562
#> areaE -0.007804562 0.007804562 0.007804562 0.007804562 0.027928682
#> areaF -0.007804562 0.007804562 0.007804562 0.007804562 0.007804562
#> areaF
#> (Intercept) -0.007804562
#> areaB 0.007804562
#> areaC 0.007804562
#> areaD 0.007804562
#> areaE 0.007804562
#> areaF 0.035431734
#>
#> $p
#> [1] 1.570459
#>
ranef(fit)
#> $sector
#> VehicleType Uj
#> 1: Common vehicle 0.9557004
#> 2: Uncommon vehicle 1.0509018
#>
#> $group
#> VehicleType VehicleBody Ujk
#> 1: Common vehicle HBACK 1.0248171
#> 2: Common vehicle SEDAN 0.9524791
#> 3: Common vehicle UTE 0.9969568
#> 4: Uncommon vehicle COUPE 1.0232340
#> 5: Uncommon vehicle HDTOP 1.0064096
#> 6: Uncommon vehicle MIBUS 1.0041678
#> 7: Uncommon vehicle PANVN 1.0022996
#> 8: Uncommon vehicle STNWG 0.9831763
#> 9: Uncommon vehicle TRUCK 1.0076169
#>
fixef(fit)
#> (Intercept) areaB areaC areaD areaE areaF
#> 5.63661846 0.04771903 0.07691849 -0.18573986 0.13052750 0.46389185
# }