A Copula-Based family of Bivariate Composite Models for Claim Severity Modelling Girish Aradhyey George Tzougasand Deepesh Bhatiy1

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A Copula-Based family of Bivariate Composite Models for Claim Severity

Modelling

Girish Aradhye†, George Tzougas∗and Deepesh Bhati†1

†Department of Statistics, Central University of Rajasthan, Ajmer, India.

∗Department of Actuarial Mathematics and Statistics, Heriot-Watt University,

Edinburgh, EH14 4AS, United Kingdom.

Abstract

In this paper we consider bivariate composite models for modeling jointly diﬀer-

ent types of claims and their associated costs in a ﬂexible manner. For expository

purposes, the Gumbel copula is paired with the composite Weibull-Inverse Weibull,

Paralogistic-Inverse Weibull and Inverse Burr-Inverse Weibull marginal models. The

resulting bivariate copula-based composite models are ﬁtted on motor insurance bod-

ily injury and property damage data from a European motor insurance company and

their parameters are estimated via the inference functions for margins method.

Keywords: Classical Composite Technique, Copula, Dependence Parameter,

Gumbel Copula, Inverse Weibull Distribution, Inverse Burr Distribution, Paralogis-

tic Distribution, Weibull Distribution.

1 Introduction

Over the last few decades, there has been a vast increase of actuarial research works on

modelling the cost of a speciﬁc claim type in non-life insurance based a variety of claim

severity modelling approaches due to the peculiar characteristics of the claim severity dis-

tribution which poses several challenges as it often ranges over several magnitudes, from

small and moderate claim sizes with a high frequency, as well as a few major ones with a

low frequency. Additionally, claim size data are unimodal and heavily skewed to the right,

see, for instance, Bakar et al. 2015). As it can be clearly understood, when the data spans

over a wide range of magnitudes, selecting a probability distribution that can eﬃciently ﬁt

small and/or moderate and large claims becomes crucial for insurance pricing, reserving,

and risk management. Some popular methods that were developed in the literature for

addressing this issue are (i) the transformation of random variable (r.v.) (see, for example,

Vernic, 2006, Adcock et al., 2015, Kazemi and Noorizadeh, 2015 and Eling, 2012, Bhati

and Ravi, 2018), mixture of two or more distributions (Lin, 2010, Verbelen et al., 2015,

1Corresponding author: deepesh.bhati@curaj.ac.in

arXiv:2210.05091v1 [stat.AP] 11 Oct 2022

Miljkovic and Grün, 2016), (ii) the method of compounding (see, for instance, Punzo et

al., 2018) and (iii) the method of composition of distributions (see, for example, Cooray

and Ananda, 2005, Scollnik, 2007, Ciumara, 2006, Cooray, 2009, Scollnik and Sun, 2012,

Nadarajah and Bakar, 2014 and the references therein). Regarding the most recent stud-

ies on composite models, which are the main research focus of this work, it is worth noting

that Nadarajah and Bakar (2014) proposed diﬀerent composite models by considering the

Burr, Loglogistic, Paralogistic and Generalized Pareto distributions for the tail of the data

and truncated densities before and after the threshold point. Calderin-Ojeda and Kwok

(2016) established alternative composite models by matching the two families of distribu-

tions at the modal value and they proposed the Lognormal-Stoppa and Weibull-Stoppa

models for modelling the claim size data. Bhati et al. (2019) constructed composite

models based on the Mode-Matching method and they considered diﬀerent choices for

the tail and head distribution. Grun and Miljkovic (2019), instead of creating composite

models incrementally, consider 256 composite models that are derived from sixteen para-

metric distributions which are frequently used in actuarial science. Wang et al. (2020)

put special emphasis on modelling extreme claims using a variety of composite models

and threshold selection techniques, including heuristic methods, the Minimum AMSE of

the Hill estimator, the exponentiality test, and the Gertensgarbe plot. Finally, Fung et

al. (2022) introduce a mixture composite claim severity regression model extending the

setup of Reynkens et al. (2017), who used a ﬁnite mixture distribution for the body and

a Pareto-type distribution for the tail of the distribution by incorporating explanatory

variables on all three parts of the claim size distribution: clustering probabilities, body

part, and tail part. At this point it should be noted that, even if the literature conserving

composite models in the univariate setting is abundant, their bivariate extensions have

not been investigated so far. Nevertheless, in non-life insurance it is common for the ac-

tuary to observe the existence of dependence structures between diﬀerent types of claims

and their associated costs either from the same type of coverage or from multiple types of

coverage, such as, for example, motor and home insurance bundled into one single policy.

In this paper, motivated by a European Motor Third Party Liability (MTPL) insurance

dataset, which is described in Section 5, we introduce a family of bivariate composite

models for joint modelling of the costs of positively correlated bodily injury and prop-

erty damage claims. The proposed class of bivariate composite models is constructed by

pairing a continuous copula distribution with two marginal composite distributions. The

modelling framework we develop can account for the dependence structure between the

two claim types in a versatile manner since it allows for a variety of alternative copula

functions which can fully specify the dependence structure separately from the univariate

marginal composite models (see, Joe 1997). Furthermore, depending on the choice of

the composite marginal models, the proposed family of bivariate composite models can

be used to model the correlation between small and/or moderate and large claim sizes

which can be the result of the same accident. We exemplify our approach by employing

the Gumbel copula for modelling the dependence between bodily injury and property

damage claims based on three bivariate composite models namely the bivariate composite

Weibull - Inverse Weibull, bivariate composite Paralogistic - Inverse Weibull and bivari-

ate composite Inverse Burr - Inverse Weibull models. The parameters of the models are

estimated using the inference function for margins (IFM) method whcih consists of esti-

mating univariate parameters from separately maximizing the marginal composite models

and then estimating the dependence parameters from the bivariate likelihoods which are

derived based on the Gumbel copula.

The rest of the paper is structured as follows. In section 2, the alternative marginal

composite models we consider herein are derived based on the Classical Composition (CC)

technique. Section 3 presents the construction of the proposed bivariate composite mod-

els based on the Gumbel copula for modelling the dependence structure between diﬀerent

claim types. Model estimation via the IFM method along with the computational aspects

of ﬁtting the proposed bivariate composite models are discussed in section 4. In Section

5 we describe the MTPL dataset that we use for our empirical analysis and we provide

estimation via the IFM method and model comparison for the proposed models. Finally,

concluding remarks can be found in section 6.

2 Modelling framework : The Classical Composition

Technique

Bakar et al. (2015) proposed various composite models using unrestricted mixing weight

(r), the right truncated and left truncated density truncated at threshold (θ) for Head

and Tail distributions respectively. The resulting probability density function (pdf) of the

composite model can be written as

f(x) = 





rf∗

1(x|Ξ1, θ)for 0< x ≤θ,

(1 −r)f∗

2(x|Ξ2, θ)for θ < x < ∞,

(1)

where r∈[0,1] and θ> 0.

The function f∗

1(x|Ξ1, θ) = f1(x|Ξ1)

F1(θ|Ξ1)and f∗

2(x|Ξ2, θ) = f2(x|Ξ2)

1−F2(θ|Ξ2)are the adequate truncation

of the pdfs f1and f2upto and after an unknown threshold value θrespectively.

•The value of weight parameter ris obtained by continuity condition imposed at

threshold θi.e. rf∗

1(θ|Ξ1, θ) = (1 −r)f∗

2(θ|Ξ2, θ). Hence, we get

r=r(θ, Ξ1,Ξ2) = f2(θ|Ξ2)F1(θ|Ξ1)

f2(θ|Ξ2)F1(θ|Ξ1) + f1(θ|Ξ1)(1 −F2(θ|Ξ2)).(2)

•Further, imposing the diﬀerentiability condition at threshold value θ rf0

1(θ|Ξ1, θ) =

(1 −r)f0

2(θ|Ξ2, θ2), makes the density smooth.

These above conditions reduces the number of parameters and makes the resulting density

continuous and diﬀerentiable. We henceforth refer this technique as Classical Composition

(CC) technique.

3 The Bivariate Composite Model

Let Y= (Y(1),Y(2))and yi= (y(1)

i, y(2)

i)respectively be the claims vector and its corre-

sponding realizations of the two types of claims. Suppose that, Y(j)

i,j= 1,2follows the

composite H-Inverse Weibull (IW) model with pdf given by

fj(y(j)

i) =











r(j)

H,IW f∗

H(y(j)

i; Ξ(j)),for 0< y(j)

i≤θ(j)

(1 −r(j)

H,IW )

α(j)

y(j)

i γ(j)

y(j)

i!α(j)

exp









− γ(j)

y(j)

i!α(j)









1−exp





−γ(j)

θ(j)α(j)





for θ(j)< y(j)

i<∞.

(3)

The cumulative distribution function (cdf) of composite H-Inverse Weibull model may

be written as

Fj(y(j)

i) =











r(j)

H,IW

FH(y(j)

i;Ξ(j)

FH(θ(j);Ξ(j)

1),for 0< y(j)

i≤θ(j)

r(j)

H,IW + (1 −r(j)

H,IW )

exp









− γ(j)

y(j)

i!α(j)









−exp





−γ(j)

θ(j)α(j)





1−exp





−γ(j)

θ(j)α(j)





,for θ(j)< y(j)

i<∞.

(4)

For i= 1,2· · · , n and j= 1,2. Where r(j)

H,IW ∈[0,1],Ξ(j)>0are the parameters

associated with the head density (H) of the marginal composite model, α(j)>0, threshold

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ACopula-BasedfamilyofBivariateCompositeModelsforClaimSeverityModellingGirishAradhyey,GeorgeTzougasandDeepeshBhatiy1yDepartmentofStatistics,CentralUniversityofRajasthan,Ajmer,India.DepartmentofActuarialMathematicsandStatistics,Heriot-WattUniversity,Edinburgh,EH144AS,UnitedKingdom.AbstractInthispape...

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A Copula-Based family of Bivariate Composite Models for Claim Severity Modelling Girish Aradhyey George Tzougasand Deepesh Bhatiy1

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