Loss-Alae fitting example.
Loss-Alae is a dataset that is provided in the R copula package, which documentation can be found there. This dataset corresponds to claims received by an insurer, where the two variables loss and alae respectively correspond to the amount of the loss and to associated expenses. There is a certain dependence structure between the two, and the actual distribution generating this data is of course unknown. Our task here is to provide a parametric model that approximates this distribution.
Let us first import the data :
using Copulas, Distributions, Plots
data = [
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]2×1500 Matrix{Float64}:
10.0 24.0 45.0 51.0 60.0 74.0 … 1.0e6 2.1736e6
3806.0 5658.0 321.0 305.0 758.0 8768.0 135653.0 134743.0Since the dataset is bivariate, we can have a glimpse of the bivariate distribution through a simple scatter plot:
loss = data[1,:]
alae = data[2,:]
scatter(loss,alae)
The observations seem to have extreme values. Let's move to log scales:
scatter(log.(loss),log.(alae))
This is a much better looking scatter plot. To fit a full distribution on this dataset, we have to understand what families of marginals and of dependence structure to use. Let us first look at the marginal behaviors:
plot(histogram(log.(loss)),histogram(log.(alae)))
This histogram look fairly Gaussian, and thus a good first guess for the distributions of the marginals would be LogNormal distributions. Now the dependence structure:
ranks = Copulas.pseudos(data)
scatter(ranks[1,:],ranks[2,:])
The vertical strides are there because of rounding in the input data. But still, we observe that there is dependence structure in both tails, and that the dependence structure seems fairly symmetric. Let us try to fit a few different copulas:
fit_gaussian = fit(SklarDist{GaussianCopula,Tuple{LogNormal,LogNormal}}, data)
fit_clayton = fit(SklarDist{ClaytonCopula,Tuple{LogNormal,LogNormal}}, data)
fit_gumbel = fit(SklarDist{GumbelCopula,Tuple{LogNormal,LogNormal}}, data)
fit_frank = fit(SklarDist{FrankCopula,Tuple{LogNormal,LogNormal}}, data)Let's check the negative loglikelihood on each of those models (note that they all have the same number of parameters):
nllhs = [-loglikelihood(fit,data) for fit in (fit_gaussian,fit_clayton,fit_gumbel,fit_frank)]4-element Vector{Float64}:
32216.375451240296
32430.051935405932
32174.890388465803
32206.44183201268So the Clayton looks a bit better. Let's look at the parametrization:
fit_claytonSklarDist{ClaytonCopula{2, Float64}, Tuple{Distributions.LogNormal{Float64}, Distributions.LogNormal{Float64}}}(
C: ClaytonCopula{2, Float64}(
G: Copulas.ClaytonGenerator{Float64}(0.9683325060462363)
)
m: (Distributions.LogNormal{Float64}(μ=9.37345394250046, σ=1.638106235266425), Distributions.LogNormal{Float64}(μ=8.521976324212277, σ=1.4298990317418996))
)
For the marginals, we can for example check quantile quantile plots (again, on log-scale)
n = size(data,2)
plot(
scatter(sort(log.(loss)), log.(quantile.(Ref(fit_clayton.m[1]),(1:n)./(n+1))), label="Loss"),
scatter(sort(log.(alae)), log.(quantile.(Ref(fit_clayton.m[2]),(1:n)./(n+1))), label="Alae")
)
These quantile-quantile plots are not perfect, we see that both tails are a bit wiggly. For the dependence structure, we can sample a new dataset from the fitted copula to check if the ranks behaviors looks like what we had before:
u = rand(fit_clayton.C, 1500)
scatter(u[1,:],u[2,:])
There are potential improvements that can be made to this fit:
- The tail dependency does not look like it is on the right side. To solve that, we could use
SurvivalCopulato fit a flipped version of the Clayton throughfit(SklarDist{SurvivalCopula{2,ClaytonCopula,(1,2)},Tuple{...}}, ...). - We could use other marginal proposals than
LogNormals and validate (e.g., through likelihood ratio tests) that the fits are OK. - We could keep marginals and/or the dependence structure empirical, through e.g.,
EmpiricalCopula.