Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3062 
https://internationalpubls.com 

Bayesian Bonus-Malus Premiums Under Different Loss Functions 

in Car Insurance 

 

Ouchen Imene1, Sadoun Ahmed1 And Remita Mohamed Riad1,2 
1LaPS laboratory, Badji Mokhtar-Annaba University, BP 12, 23000, 

Annaba, Algeria 
imene.ouchen@univ-annaba.dz, ahmed.sadoun@univ-annaba.dz 

2National School of Artificial Intelligence, Algiers, Algeria. 
riad.remita@ensia.edu.dz 

 
 

Article History: 

Received: 02-01-2025 

Revised: 25-02-2025 

Accepted: 20-03-2025 

Abstract—  

In the traditional bonus-malus system, automobile insurance premiums are typically 

calculated based solely on claim frequency. This study introduces an alternative 

approach, incorporating both claim frequency and claim severity into the 

determination of premiums. Furthermore, the research explores the premiums under 

various loss functions, including quadratic, linex, and entropy, applied to both 

frequency and severity. The Bayesian approach is employed to compute the bonus-

malus premiums. Additionally,this work relies on the R program to provide a 

numerical application based on a real automobile insurance dataset. By incorporating 

diverse loss functions, this approach aims to enhance flexibility, achieve balance, and 

provide greater control over premiums, all while ensuring the solvency of the 

insurance company. 

 

 

1. INTRODUCTION 

In actuarial science, a key responsibility involves designing a pricing framework that fairly 

distributes the burden of claims among insured individuals. This is achieved by using the most 

appropriate models to calculate insurance premiums. One such method is the bonus-malus 

system (BMS), which adjusts premiums based on an individual’s claim history. It is widely 

implemented, particularly in automobile insurance, to ensure that all policyholders pay 

premiums commensurate with their risk levels. Safe driving without claims results in a 

premium reduction as a reward, whereas a reported incident leads to a premium increase. 

The BMS serves two primary purposes for insurance companies. First, it encourages 

policyholders to operate their vehicles more carefully throughout the year, aiming to reduce 

the number of claims. Second, it ensures that policyholders pay premiums aligned with their 

individual risk levels as reflected in their claims history [1]. The fundamental concept of this 

system is that higher claim frequencies lead to higher premiums. Traditionally, BMSs have 

relied solely on the stochastic nature of claim frequency [4]. However, since claims may vary 

significantly in size, it is important to design a BMS that accounts for both the frequency and 

severity of claims. Although several authors have already proposed this idea [2, 7, 9], their 

models typically determine premiums using only the classical quadratic loss function. 

mailto:imene.ouchen@univ-annaba.dz
mailto:ahmed.sadoun@univ-annaba.dz
mailto:riad.remita@ensia.edu.dz


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3063 
https://internationalpubls.com 

In insurance, actuaries use loss functions as analytical tools to evaluate the costs associated 

with deviations from expected outcomes. These functions are critical in risk assessment, 

premium calculation, and decision-making processes. Measuring different forms of loss 

enhances an insurer’s ability to manage and mitigate risk. Loss functions help insurers select 

models and strategies that align with their risk tolerance. By accurately evaluating risks, 

insurers can set fair and financially sustainable premium rates. 

There are several types of loss functions used in statistics. The most common is the quadratic 

loss function, also known as the mean squared error. In automobile insurance, it calculates the 

squared difference between actual and estimated values, assigning greater weight to larger 

errors. Another important loss function is the linex loss function, which is asymmetric. This 

asymmetry allows for an unequal assessment of underestimation versus overestimation of 

risk—particularly useful when underestimating significant claims has more serious 

consequences than overestimating minor ones. Lastly, the entropy loss function measures the 

proportional gap between observed and expected values. It is particularly sensitive to rare but 

important deviations, making it suitable for capturing infrequent but high-impact events. 

In this study, we aim to calculate bonus-malus premiums using multiple loss functions while 

incorporating both claim frequency and severity. While most existing research focuses on a 

single loss function, we examine the use of different loss functions across various scenarios, 

and we compare the resulting premiums with those obtained using only one loss function. 

Additionally, we provide a numerical application using a real automobile insurance dataset and 

employ R software to compute the premiums, allowing for a detailed comparison and 

identification of the most beneficial approach for insurance providers. 

This article is structured as follows: 

• The first two sections address claim frequency and claim severity, each divided into 

five subsections: the selected distribution, the Bayesian approach, and the premiums 

computed using the quadratic, linex, and entropy loss functions. 

• The third section presents the final premiums calculated based on both claim 

frequency and severity. 

• The fourth section offers a numerical application, using a real dataset and R software, 

and analyzes the outcomes based on the final premiums discussed in the third section. 

• The final section provides a summary of the study’s key findings. 

 

2. CLAIMFREQUENCYDISTRIBUTIONUSINGPOISSON-AKASH 

2.1. Poisson-Akash Distribution.We generallyusethePoissondistribution in car 
insurance to explain the random frequency of the claim.The claims 
numberkissupposedtofollowthePoissondistributionwithparameter𝜃and probability mass 
function as 

𝑃(𝑙|θ) =
𝑒−θθ𝑙

𝑙!
, 𝑙 = 0,1,2, … , θ > 0, 

withthefollowingexpectedvalue 

𝐸[𝑙|θ] = 𝑉𝑎𝑟[𝑙|θ] = θ. 

𝜃 is assumed to follow the Akash distribution with parameter 𝛾and a probability density 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3064 
https://internationalpubls.com 

function expressed as follows: 

π(θ) =
γ3

γ2 + 2
(1 + θ2)𝑒−γθ. 

Then,themixeddistributionofthePoissonwithAkashdistribution(seeShanker in 2016 
[11]) is: 

𝑓(𝑙) = ∫ 𝑃(𝑙|θ)π(θ)𝑑θ
∞

0

=
γ3

γ2 + 2

γ2 + 2γ + 𝑙2 + 3𝑙 + 3

(1 + γ)𝑙+3
(1) 

2.2. Bayesian method. The Bayesian technique is one of the most commonly employed 
computational solutions for the bonus-malus premium calculation, 
whichhasbeenwellexploredin [5].Thismethod’sprimarygoal 
istodeterminetheposteriordistributionfunction. 

𝑙1, 𝑙2 … , 𝑙𝑡is asample ofsizet.Theclaimstotalnumbergeneratedbyaninsured over 

tyearsis𝑁 = ∑ 𝑙𝑖
𝑡
𝑖=1 ,where𝑙𝑖istheclaim’snumberestablishedbyaninsuredinyears𝑖, 𝑖 =

1, 2, … , 𝑡.Thelikelihoodfunctionis: 

𝐿(θ; 𝑙1,𝑙2, … , 𝑙𝑛) = ∏
𝑒−θθ𝑙𝑖

𝑙𝑖!

𝑡

𝑖=1

=
1

∏ 𝑙𝑖
𝑡
𝑖=1 !

𝑒−𝑡θθ∑ 𝑙𝑖 ∝ 𝑒−𝑡θθ𝑁. (2) 

ThePriordistributionis: 

π(θ) =
γ3

γ + 2
(1 + θ2)𝑒−γθ ∝ (1 + θ2)𝑒−γθ. (3) 

Inorder to determine the posterior distribution for an insured with𝑘1,𝑘2, … , 𝑘𝑡 

claimhistory,weemploytheBayes’theorem.Theproductofthepriordistribution in (3) and the 

likelihood function in (2) represents the posterior distribution function as follows: 

π∗(θ|𝑙1, … , 𝑙𝑛) ∝ 𝑃(𝑙1, … , 𝑙𝑛|θ)π(θ) = 𝑒−θ𝑡θ𝑁(1 + θ2)𝑒−γθ 

= θ𝑁(1 + θ2)𝑒−θ(𝑡+γ). 

Finaly,fortheclaimfrequency,theposteriordistributionfunctionis(for moredetailsreferto[9]) 

π∗(θ|𝑙1, … , 𝑙𝑛) =
(𝑡 + γ)𝑛+3

Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]
θ𝑁(1 + θ2)𝑒−θ(𝑡+γ). 

2.3. Premium under quadratic loss function. In this paper, we aim to 

determinethenetpremiumwhichcorrespondstothemeanoftheclaimnumber 

ofeachinsured.Assumethattheclaimhistoryis 𝑙1, 𝑙2 … , 𝑙𝑡.Themeanof 

theposteriordistributionfunctionforPoisson-Akashdistribution(whichisthe expected 

number of claims) is obtained by: 

θ𝑡+1̂ = 𝐸[θ|𝑙1, … , 𝑙𝑡] = 𝐸[𝑙1, … , 𝑙𝑛|θ] = ∫ θπ∗(θ𝑖|𝑙1, … , 𝑙𝑛)
∞

0

 

=
(𝑛 + 1)(𝑛 + 2)(𝑛 + 3) + (𝑛 + 1)(𝑡 + γ)2

(𝑡 + γ)3 + (𝑡 + γ)(𝑛 + 2)(𝑛 + 1)
. 

Consider that 100 represents the initial premium at time t=0.Consequently, 
thepremiumattimet+1isdefinedthereby: 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
θ𝑡+1̂

𝐸[𝑥]
 

= 100
θ(θ2 + 2)

𝜃2
+ 6

(𝑛 + 1)(𝑛 + 2)(𝑛 + 3) + (𝑛 + 1)(𝑡 + γ)2

(𝑡 + γ)3 + (𝑡 + γ)(𝑛 + 2)(𝑛 + 1)
. (4) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3065 
https://internationalpubls.com 

2.4. Premium under Linex Loss function:The following subsection 
examinestheasymmetriclinexlossfunction.Itexhibitsanexponentialincrease on 
one side of zero and a linear increase on the other one, as demonstrated in [8, 10, 
12].Itisarticulatedas: 

𝐿(θ̂, θ) = exp (𝑎(θ̂ − θ)) − 𝑎(θ̂ − θ) − 1,       𝑎 ≠ 0, 

where 𝜃 denotes the estimator of 𝜃under the linex loss function that minimizes the 

aforementioned equation (see [13]), 𝜃can be defined as: 

θ�̂� = −
1

α
ln(𝐸[𝑒−αθ|𝑋]). 

Then 

E[𝑒−αθ|𝑙1, … , 𝑙𝑛] = ∫ 𝑒−αθ
∞

0

π∗(θ𝑖|𝑙1, … , 𝑙𝑛)dθ 

∝ ∫ θ𝑁+2
∞

0

𝑒−(𝑡+γ+α)θdθ + ∫ θ𝑁
∞

0

𝑒−(𝑡+γ+α)θdθ 

=
(𝑡 + γ)𝑛+3(𝑛 + 1 + 𝑡 + γ + α)

[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2](𝑡 + γ + α)𝑛+2
. 

Finally  

   θ�̂� = −
1

𝑎
ln (

(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ + 𝑎)2

(𝑡 + γ + 𝑎)𝑛+3

(𝑡 + γ)𝑛+3

(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2
) . 

 
Attimet=0,thepremiumis100.Thepremiumattimet+1isexpressedas 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
 

                          ∗ [−
1

𝑎
𝑙𝑛 (

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾 + 𝑎)2

(𝑡 + 𝛾 + 𝑎)𝑛+3
∗

∗
(𝑡 + 𝛾)𝑛+3

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2
)] . (5) 

PremiumunderEntropyLossfunction:Thispartinterpretstheentropylossfunction.Inv
ariousactualscenarios,itappearsmoresensibleto express the loss as the ratio 
θ̂

θ
;forthiscontext,[6]presenteda loss function referred to as entropy.This loss function serves 

as a powerful asymmetric loss which assigns credit for both underestimation and 
overestimation, defined by the following structure. 

L(θ�̂� − θ) = (
θ�̂�

θ
)

𝑞

− q ln (
θ�̂�

θ
) − 1;     q ≠ 0 

where𝜃�̂�denotestheestimateof𝜃thatminimizestheaforementionedequationundertheentropylossf

unction. Theexpressionfor𝜃�̂�isprovidedas follows: 

θ�̂� = (𝐸[θ−𝑝|𝑙1, … , 𝑙𝑛])
−

1

𝑝 

𝐸[θ−𝑝|𝑙1, … , 𝑙𝑛] = ∫ θ−𝑝
∞

0

 Π∗(θ|𝑙1, … , 𝑙𝑛) 𝑑θ 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3066 
https://internationalpubls.com 

= ∫ θ−𝑝
∞

0

 θ𝑁(1 + θ2) 𝑒−θ(𝑡+γ) 𝑑θ 

= [
Γ(𝑛 − 𝑝 + 3)

(𝑡 + θ)𝑛−𝑝+3
+

+
Γ(𝑛 − 𝑝 + 1)

(𝑡 + θ)𝑛−𝑝+1
]

(𝑡 + 𝛾)𝑛+3

Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2]
 

𝐸[𝜃−𝑝|𝑙1, … , 𝑙𝑛] = [
Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + 𝛾)2

(𝑡 + 𝜃)𝑛−𝑝+3
] 

                               ∗
(𝑡 + 𝛾)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2]]
 

Finally 

θ�̂� =

= [(
Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + γ)2

(𝑡 + θ)𝑛−𝑝+3
) ∗

(𝑡 + γ)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]]
]

−
1

𝑝

(6) 

 

𝑃remiu𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
∗ [(

Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + γ)2

(𝑡 + θ)𝑛−𝑝+3
) 

∗
(𝑡 + γ)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]]
]

−
1

𝑝

. (7) 

 

3. CLAIMSEVERITYDISTRIBUTIONUSINGINVERS-

GAMMALINDLEY 

3.1. Mixing Distribution. X denote a random variable that signifies the 
claimsizeassociatedwitheachinsurer.LetusconsiderthatXadherestoan Inverse-
Gamma distribution, characterized by the following probability density function: 

𝑓(𝑥|λ) =
λα𝑥−α−1

Γ(α)
𝑒−

λ

𝑥. 

The Invers-Gamma expected value is 

𝐸[𝑋] =
λ

α − 1
. 

Let λ be a random variable that adheres to a Lindley distribution characterized by the parameter 

β. Then, the representation of the PDF of λ is provided as: 

π(λ) =
β2

β + 1
(λ + 1)𝑒−βλ. 

Thus, the combination of the Invers-Gamma and Lindley distributions is derived thereby ([9]): 

𝑓(𝑥) = ∫
λα

Γ(α)
𝑥−α−1𝑒−

λ

𝑥
β2

β + 1
(λ + 1)𝑒−βλ

∞

0

dλ 

=
𝑥−α−1β2(α + 1)

(β + 1) (β +
1

𝑥
)

α+2 [α + 1 + β +
1

𝑥
] . (8) 

3.2. BayesianMethod.𝑁 = ∑ 𝑙𝑖
𝑡
𝑖=1  representsthetotalnumberofclaims submitted by an 

insured over a duration of t years. Define x as the amount of claim𝑙where 𝑙ranges from 1 

to 𝑁.The subsequent expression identifies the likelihood function: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3067 
https://internationalpubls.com 

L(λ |x1,…,xn)  = f(x1,…,xn|λ ) = ∏
𝜆 𝛼 𝑥−𝛼 −1

Γ (𝛼 )
𝑒

−𝜆 
1

𝑥𝑖

n

i=1

 

  =
λ nα 

(Γ (α ))
n ∏ x−α −1

n

i=1

e
−λ  ∑

1

xi

n
i=1   

∝  λ nα e
−λ  ∑

1

xi

n
i=1 . 

The following π(λ) denotes the prior distribution 

π(λ) ∝ (λ + 1)e−βλ. 
To determine the posterior distribution function, we utilize Bayes' Theorem in the following 

manner: 

π∗(λ|𝑥1,…,𝑥𝑛) ∝ 𝑓(𝑥1,…,𝑥𝑛|λ)π(λ) ∝ λ𝑛α(1 + λ)𝑒
−λ(β+∑

1

𝑥𝑖

𝑛
𝑖=1 )

. 
 

Thus, the posterior distribution function for the claim severity is represented as (for more detail 

see[9]): 

π∗(λ|𝑥1,…,𝑥𝑛) =
(β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛+2

Γ(α𝑛 + 1) (α𝑛 + 2 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 )

λ𝑛α(1 + λ)𝑒
−λ(β+∑

1

𝑥𝑖

𝑛
𝑖=1 )

. (9) 

 
3.3. Premium under quadratic loss function.For an insured given a claim history 

𝑙1, 𝑙2, … , 𝑙𝑡, the mean of the posterior distribution function for Poisson- Akash distribution 

(or the predicted claim number) is: 

λ𝑡+1
̂ = 𝐸[λ|𝑥1, … , 𝑥𝑛] = ∫ λπ∗(λ|𝑥1, … , 𝑥𝑛)𝑑λ

∞

0

 

=
(α𝑛 + 2) (α𝑛 + 3 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

(β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) (α𝑛 + 2 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

. 

According to  

𝐸[λ|𝑥1, … , 𝑥𝑛] = λ̂. 
We have 

𝐸[𝑥1, … , 𝑥𝑛|λ] =
λ̂

α − 1
. 

Consequently 

𝐸[𝑥1, … , 𝑥𝑛|λ] =
(α𝑛 + 2) (α𝑛 + 3 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

(α − 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) (α𝑛 + 2 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

. (10) 

3.4. PremiumunderLinexLossfunction: 

Inthissection,also,weusedtheVarian’sasymmetriclinexlossfunction, 

definedbythefollowingequation: 

𝐿(λ̂, λ) = exp (𝑎(λ̂ − λ)) − 𝑎(λ̂ − λ) − 1,       𝑎 ≠ 0. 

Consider λ̂ is the estimator of λ under the Linex loss fonction which minimizes the precedent 

equation, λ̂ is given by: 

λ�̂� = −
1

𝑐
ln 𝐸 (𝑒−𝑐λ|𝑋) 

Then 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3068 
https://internationalpubls.com 

𝐸(𝑒−𝑐λ|𝑋) = ∫ 𝐵𝑒−𝑐λ
∞

0

π∗(λ𝑖|𝑥1, … , 𝑥𝑛)𝑑λ = 𝐵 ∫ 𝑒−𝑐λ
∞

0

λα𝑛(1 + λ)𝑒
−λ(β+∑

1

𝑥𝑖

𝑛
𝑖=1 )

𝑑λ 

= [
Γ(α𝑛 + 1)

(𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛+1 +
Γ(α𝑛 + 2)

(𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛+2] 

                                   ∗
(β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛+2

Γ(α𝑛 + 1) [α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

𝐸(𝑒−𝑐λ|𝑋) 

= (
β + ∑

1

𝑥𝑖

𝑛
𝑖=1

𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)

α𝑛+2

(
α𝑛 + 1 + 𝑐 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1

α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

) 

Finally 

λ�̂� =
1

α − 1
(−

1

𝑐
) ln [(

β + ∑
1

𝑥𝑖

𝑛
𝑖=1

𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)

α𝑛+2

(
α𝑛 + 1 + 𝑐 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1

α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)] 

3.5. Premium under Entropy Loss function:In this part, we also, utilized the 

Entropy loss function in the following manner: 

λ�̂� = [𝐸(λ−𝑝|𝑋)]
−

1

𝑝 

Then  

                        𝐸(λ−𝑝|𝑋) = ∫ 𝐵λ−𝑝∞

0
π∗ (λ𝑖|𝑥1, … , 𝑥𝑛)𝑑λ = 𝐵 ∫ λ−𝑝∞

0
λα𝑛(1 +

λ)𝑒
−λ(β+∑

1

𝑥𝑖

𝑛
𝑖=1 )

𝑑λ 

= [
Γ(α𝑛 − 𝑝 + 1)

(β + ∑
1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛−𝑝+1 +
Γ(α𝑛 + 2)

(β + ∑
1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛−𝑝+2] 

∗
(β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

α𝑛+2

Γ(α𝑛 + 1) [α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

 

𝐸(λ−𝑝|𝑋) =
Γ(α𝑛 − 𝑝 + 1) [α𝑛 − 𝑝 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

Γ(α𝑛 + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) [α𝑛 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

 

Finally 

λ�̂� =
1

α − 1
[

Γ(α𝑛 − 𝑝 + 1) [α𝑛 − 𝑝 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

Γ(α𝑛 + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) [α𝑛 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

]

−
1

𝑝

 

4. FINALPREMIUMSUSINGBOTHFREQUENCYANDCLAIMSEVERITY 

The tables bellow represents the final premiums that will be used in the numerical application 

 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3069 
https://internationalpubls.com 

 

TABLE1.FinalpremiumsusingInvers-

GammaLindleydistribution under Quadratic loss function for 

the claim severity 
Claimfrequencydistri
bution 

Finalpremium 

 
PoissonAkashunder 

   

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
 

∗
(𝑛 + 1)(𝑛 + 2)(𝑛 + 3) + (𝑛 + 1)(𝑡 + γ)2

(𝑡 + γ)3 + (𝑡 + γ)(𝑛 + 2)(𝑛 + 1)
∗

∗
(α𝑛 + 2) (α𝑛 + 3 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

(α − 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) (α𝑛 + 2 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

.  (11) 

Quadraticloss 
function(4) 

 

 

 
PoissonAkashunder 

  

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
 

                          ∗ [−
1

𝑎
𝑙𝑛 (

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾 + 𝑎)2

(𝑡 + 𝛾 + 𝑎)𝑛+3
∗

∗
(𝑡 + 𝛾)𝑛+3

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2
)] 

                          ∗
(α𝑛 + 2) (α𝑛 + 3 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

(α + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) (α𝑛 + 2 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

. (12) 

Linexlossfunction(5) 

 

 

 
PoissonAkashunder 

𝑃remiu𝑚𝑡+1 =

= 100
γ(γ2 + 2)

γ2 + 6
∗ [(

Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + γ)2

(𝑡 + θ)𝑛−𝑝+3
) 

                        ∗
(𝑡 + γ)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]]
]

−
1

𝑝

 

                        ∗
(α𝑛 + 2) (α𝑛 + 3 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

(α + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) (α𝑛 + 2 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 )

. (13) 

Entropylossfunction(
7) 

 
TABLE2.FinalpremiumsusingInvers-
GammaLindleydistribution under Linex loss function for the 
claim severity 

Claimfrequencydistri
bution 

Finalpremium 

 

 

Poisson Akash 
under 

Quadraticlossfuncti
on(4) 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
θ(θ2 + 2)

𝜃2
 

                               +6
(𝑛 + 1)(𝑛 + 2)(𝑛 + 3) + (𝑛 + 1)(𝑡 + γ)2

(𝑡 + γ)3 + (𝑡 + γ)(𝑛 + 2)(𝑛 + 1)
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3070 
https://internationalpubls.com 

                                ∗
1

α − 1
−

−
1

𝑐
ln [(

β + ∑
1

𝑥𝑖

𝑛
𝑖=1

𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)

α𝑛+2

(
α𝑛 + 1 + 𝑐 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1

α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)].  (14) 

 

 

 

Poisson Akash 
under 

Linexlossfunction(
5) 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
 

                          ∗ [−
1

𝑎
𝑙𝑛 (

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾 + 𝑎)2

(𝑡 + 𝛾 + 𝑎)𝑛+3
∗

∗
(𝑡 + 𝛾)𝑛+3

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2
)] 

                          ∗
1

α − 1
−

−
1

𝑐
ln [(

β + ∑
1

𝑥𝑖

𝑛
𝑖=1

𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)

α𝑛+2

(
α𝑛 + 1 + 𝑐 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1

α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)].   (15) 

 

 

 

 

Poisson Akash 

under Entropy loss 
function (7) 

𝑃remiu𝑚𝑡+1 =

= 100
γ(γ2 + 2)

γ2 + 6
∗

∗ [(
Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + γ)2

(𝑡 + θ)𝑛−𝑝+3
) 

                        ∗
(𝑡 + γ)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]]
]

−
1

𝑝

 

              ∗
1

α − 1
−

−
1

𝑐
ln [(

β + ∑
1

𝑥𝑖

𝑛
𝑖=1

𝑐 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)

α𝑛+2

(
α𝑛 + 1 + 𝑐 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1

α𝑛 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1

)] . (16) 

 

 

 
TABLE3.FinalpremiumsusingInvers-GammaLindleydistribution under 
Entropy loss function for the claim severity 

Claimfrequencydistri
bution 

Finalpremium 

 

 

Poisson Akash 

under 
Quadraticlossfuncti

on(4) 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
θ(θ2 + 2)

𝜃2
 

                               +6
(𝑛 + 1)(𝑛 + 2)(𝑛 + 3) + (𝑛 + 1)(𝑡 + γ)2

(𝑡 + γ)3 + (𝑡 + γ)(𝑛 + 2)(𝑛 + 1)
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3071 
https://internationalpubls.com 

                            ∗

∗
1

α − 1
[

Γ(α𝑛 − 𝑝 + 1) [α𝑛 − 𝑝 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

Γ(α𝑛 + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) [α𝑛 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

]

−
1

𝑝

. (17) 

 

 

 

Poisson Akash 

under 
Linexlossfunction(

5) 

𝑃𝑟𝑒𝑚𝑖𝑢𝑚𝑡+1 = 100
γ(γ2 + 2)

γ2 + 6
 

                          ∗ [−
1

𝑎
𝑙𝑛 (

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾 + 𝑎)2

(𝑡 + 𝛾 + 𝑎)𝑛+3
∗

∗
(𝑡 + 𝛾)𝑛+3

(𝑛 + 2)(𝑛 + 1) + (𝑡 + 𝛾)2
)] 

                          ∗

∗
1

α − 1
[

Γ(α𝑛 − 𝑝 + 1) [α𝑛 − 𝑝 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

Γ(α𝑛 + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) [α𝑛 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

]

−
1

𝑝

.   (18) 

 

 

 

 

Poisson Akash 

under Entropy loss 
function (7) 

𝑃remiu𝑚𝑡+1 =

= 100
γ(γ2 + 2)

γ2 + 6
∗

∗ [(
Γ(𝑛 − 𝑝 + 3) + Γ(𝑛 − 𝑝 + 1)(𝑡 + γ)2

(𝑡 + θ)𝑛−𝑝+3
) 

                        ∗
(𝑡 + γ)𝑛+3

[Γ(𝑛 + 1)[(𝑛 + 2)(𝑛 + 1) + (𝑡 + γ)2]]
]

−
1

𝑝

 

∗
1

α − 1
[

Γ(α𝑛 − 𝑝 + 1) [α𝑛 − 𝑝 + 1 + β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ]

Γ(α𝑛 + 1) (β + ∑
1

𝑥𝑖

𝑛
𝑖=1 ) [α𝑛 + 1 + β + ∑

1

𝑥𝑖

𝑛
𝑖=1 ]

]

−
1

𝑝

. (19) 

 

 

5. NUMERICALAPPLICATION 

For the numerical analysis, we utilized a real data set to evaluate the premiums 
determined solely by the number of claims and those calculated usingboth claim 
frequency and severity.We employed the different loss functions aforementioned. We 
focused on the data set related to one-year car insurance 
policiesissuedin2004or2005.Thisdatasetcanbefoundonthewebsiteof the Faculty of 
Business and Economics at Macquarie University in Sydney, 
Australia.Forfurtherreference,see[3].Theportfolio comprises a total of 67,856 policies, 
with 4,624 of them having at least one claim recorded.Among these policies, 4,333 
submitted one claim, 271 filed two 
claims,18presentedthreeclaims,and2submittedfourclaims. 

5.1. Premiums using only claim frequency.The following tables [4-8] represents the 

premiums based only on the claim frequency and using different loss functions (Quadratic, 

Linex (a=1.1 and a=-0.3), Entropy (p=0.2 and p=-0.2)). 

TABLE4.PremiumsusingPoisson-

Akashwithγ=14.0125 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3072 
https://internationalpubls.com 

t 
0 

Claimnumber 

1 2 3 4 

1 93.10335 187.7328 283.7319 380.8901 478.9653 

2 87.10775 175.4826 265.0100 355.5334 446.8701 

3 81.84590 164.7550 248.6416 333.3880 418.8542 

4 77.18976 155.2800 234.2057 313.8769 394.1853 

5 73.03964 146.8484 221.3763 296.5537 372.2957 

6 69.31671 139.2954 209.8972 281.0674 352.7389 

7 65.95779 132.4893 199.5640 267.1384 335.1592 

 
 
TABLE5.Premiums using Poisson-
Akash under linex with γ = 14.0125 
anda =1.1 

 

 

 

 
 

 
 
 
 
 
TABLE6.Premiums using Poisson-Akash under 
linex with γ = 14.0125 et a =−0.3 
 
 
 
 
 
 
 
 
 
 
 
TABLE7.PremiumsusingPoisson-Akashunderentropywith 

γ=14.0125etp=−0.2 

t 
Claimnumber 

0 1 2 3 4 

1 60.54909 151.9319 246.6078 342.9572 440.4815 

2 56.67214 142.0671 230.4023 320.1982 411.0317 

3 53.26630 133.4216 216.2259 300.3148 385.3215 

4 50.25001 125.7803 203.7168 282.7908 362.6792 

5 47.55958 118.9764 192.5945 267.2267 342.5842 

6 45.14460 112.8783 182.6386 253.3087 324.6273 

7 42.96458 107.3807 173.6732 240.7868 308.4824 

 
TABLE8.PremiumsusingPoisson-Akashunderentropywith 

γ=14.0125etp=+0.2 

t 
Claimnumber 

0 1 2 3 4 
1 89.74638 180.8718 273.2438 366.6832 460.9817 
2 84.16501 169.4815 255.8516 343.1409 431.1913 
3 79.24454 159.4601 240.5731 322.4814 405.0633 
4 74.87319 150.5725 227.0418 304.2025 381.9597 
5 70.96317 142.6349 214.9718 287.9125 361.3823 
6 67.44461 135.5014 204.1363 273.3010 342.9363 
7 64.26111 129.0547 194.3536 260.1196 326.3050 

 

t 
Claimnumber 

0 1 2 3 4 
1 94.07916 189.7294 286.7865 385.0298 484.2065 
2 87.95956 177.2214 267.6655 359.1284 451.4195 
3 82.59612 166.2833 250.9721 336.5396 422.8403 
4 77.85570 156.6341 236.2678 316.6628 397.7068 
5 73.63484 148.0569 223.2142 299.0345 375.4296 
6 69.85195 140.3807 211.5459 283.2909 355.5462 
7 66.44175 133.4695 201.0515 269.1430 337.6885 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3073 
https://internationalpubls.com 

t 
Claimnumber 

0 1 2 3 4 

1 43.30648 133.28359 227.5163 323.5812 420.8996 

2 40.54181 124.65254 212.5987 302.1461 392.7958 

3 38.11183 117.08492 199.5447 283.4153 368.2582 

4 35.95882 110.39385 188.0226 266.9037 346.6460 

5 34.03769 104.43409 177.7753 252.2361 327.4629 

6 32.31269 99.09100 168.6004 239.1175 310.3188 

7 30.75509 94.27286 160.3365 227.3127 294.9030 
 
 

• Discussion.We observe that the premiums using the linex loss 
function in the under estimation are the highest for good drivers but 
also with bad drivers. The tables show that both good and bad drivers 
have lower premiums when employing the entropy loss function. 

5.2. Premiums using both frequency and claim severity. 

5.2.1. Premiums using Inverse-Gamma Lindley under Quadratic loss function for the 

claim severity. This first part shows the tables [9-13] that represents the premiums 

using the Invers-Gamma Lindley distribution (IGL) under the quadratic loss function 

for the severity claim. For the claim frequency we will use Poisson- Akash distribution 

(PA) under different loss functions (Quadratic, Linex (a=1.1 and a=-0.3) and Entropy 

(p=0.2 and p=-0.2)).These tables were calculated using formulas (11), (12) and (13) 

from table 1. 

 
TABLE9. Premiums using PA under quadratic loss 
function and IGL under quadratic loss function 
with γ=14.0125, α=1.08, β=0.001766 

t 
Claimnumber 

0 1 2 3 4 

1 958.7095 873.0867 1318.0627 1898.941 2592.533 

2 896.9712 816.1151 1231.0910 1772.524 2418.809 

3 842.7886 766.2240 1155.0526 1662.118 2267.165 

4 794.8431 722.1587 1087.9913 1564.844 2133.638 

5 752.1083 682.9460 1028.3929 1478.479 2015.154 

6 713.7722 647.8194 975.0670 1401.271 1909.298 

7 679.1846 616.1665 927.0647 1331.828 1814.142 

TABLE10.Premiums using PA under linex loss 
function and IGL under quadratic loss function 
with γ=14.0125, α=1.08, β=0.001766, a=1.1 

t 
Claimnumber 

0 1 2 3 4 

1 924.1418 841.1782 1269.3408 1828.112 2495.192 

2 866.6690 788.2056 1188.5459 1710.741 2333.943 

3 816.0017 741.5993 1117.5704 1607.742 2192.518 

4 770.9887 700.2660 1054.7118 1516.612 2067.464 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3074 
https://internationalpubls.com 

5 730.7263 663.3507 998.6409 1435.398 1956.083 

6 694.4947 630.1748 948.3052 1362.552 1856.238 

7 661.7134 600.1930 902.8603 1296.835 1766.217 

TABLE11.Premiums using PA under linex loss 
function and IGL under quadratic loss function 
with γ=14.0125, α=1.08, β=0.001766, a=-0.3 

t 
Claimnumber 

0 1 2 3 4 

1 968.7576 882.3723 1332.253 1919.579 2620.903 

2 905.7425 824.2015 1243.427 1790.447 2443.434 

3 850.5138 773.3316 1165.878 1677.830 2288.741 

4 801.7004 728.4566 1097.571 1578.734 2152.699 

5 758.2371 688.5663 1036.931 1490.847 2032.117 

6 719.2837 652.8667 982.726 1412.356 1924.493 

7 684.1680 620.7250 933.975 1341.822 1827.833 

 
TABLE12. Premiums using PA under Entropy 
loss function and IGL under quadratic loss 
function with γ=14.0125, α=1.08, β=0.001766, 
p=-0.2 

t 
Claimnumber 

0 1 2 3 4 

1 89.91753 562.9445 1419.3163 2661.842 4294.249 

2 84.16011 526.3932 1326.0473 2485.199 4007.143 

3 79.10233 494.3594 1244.4574 2330.875 3756.495 

4 74.62302 466.0466 1172.4629 2194.864 3535.755 

5 70.62765 440.8366 1108.4504 2074.064 3339.849 

6 67.04132 418.2415 1051.1506 1966.041 3164.787 

7 63.80391 397.8714 999.5514 1868.852 3007.392 

 
TABLE13. Premiums using PA under Entropy loss 
function and IGL under quadratic loss function 
with γ=14.0125, α=1.08, β=0.001766, p=0.2 

t 
Claimnumber 

0 1 2 3 4 

1 64.31164 493.8480 1309.4378 2511.456 4103.346 

2 60.20601 461.8679 1223.5813 2345.089 3829.361 

3 56.59740 433.8280 1148.4512 2199.711 3590.144 

4 53.40011 409.0360 1082.1373 2071.558 3379.447 

5 50.54717 386.9536 1020.1601 1957.716 3192.432 

6 47.98548 367.1562 970.3553 1855.896 3025.294 

7 45.67238 349.3038 922.7940 1764.274 2875.006 

 

• Discussion.We note that the premiums using the entropy loss function are the 

most punitive with bad drivers and the lightest with good drivers in these 

tables. In addition to attracting drivers and requiring them to behave well in 

order to receive some very interesting premiums, this can assist insurers in 

establishing balance and solvability within the insurance companies. We 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3075 
https://internationalpubls.com 

observe that premiums using linex loss function in the over estimation are the 

most severe with bad drivers, and those in the under estimation are the most 

severe with good drivers. 

 

5.3.2. PremiumsusingInverse-

GammaLindleyunderlinexlossfunctionfortheclaimseverity.The premiums 

employing the Invers-Gamma Lindley distribution under the linex loss function 

for the severity claim are provided in this section's tables [14–16]. The Poisson-

Akash distribution under several loss functions (Quadratic, Linex (a=1.1 and 

a=-0.3), and Entropy (p=0.2 and p=-0.2)) will be used to determine the claim 

frequency. Table 2's formulas (14), (15), and (16) were used to create the tables 

which follow. 
TABLE14.Premiums using Poisson-Akash under Quadratic loss 
function and Inverse-Gamma Lindley Linex loss function with 
γ=14.0125, α=1.08, β=0.001766 et a=1.1 

t 
Claimnumber 

0 1 2 3 4 

1 831.8621 2293.440 4652.594 7848.582 11874.183 

2 778.2924 2143.785 4345.595 7326.086 11078.501 

3 731.2787 2012.731 4077.189 6869.759 10383.949 

4 689.6770 1896.979 3840.471 6467.716 9772.374 

5 652.5964 1793.975 3630.097 6110.755 9229.703 

6 619.3326 1701.703 3441.863 5791.645 8744.865 

7 589.3213 1618.557 3272.421 5504.627 8309.040 

 
TABLE15. Premiums using Poisson-Akash under Entropy loss 
function and Inverse-Gamma Lindley Linex loss function with 
γ=14.0125,α=1.08,β=0.001766,a=1.1etp=-0.2 

t 
Claimnumber 

0 1 2 3 4 

1 564.8014 1856.730 3993.547 6943.714 10703.656 

2 528.6372 1736.175 3731.115 6482.922 9988.027 

3 496.8676 1630.520 3501.545 6080.351 9363.274 

4 468.7316 1537.137 3298.973 5725.549 8813.068 

5 443.6353 1453.988 3118.860 5410.430 8324.761 

6 421.1084 1379.464 2957.635 5128.638 7888.409 

7 400.7732 1312.278 2812.450 4875.112 7496.091 

TABLE16. Premiums using Poisson-Akash under Entropy 
loss function and Inverse-Gamma Lindley Linex loss 
function with γ=14.0125, α=1.08, β=0.001766, a=1.1 et 
p=0.2 

t 
Claimnumber 

0 1 2 3 4 

1 403.9624 1628.833 3684.381 6551.416 10227.818 

2 378.1736 1523.355 3422.806 6117.428 9544.896 

3 355.5068 1430.872 3201.411 5738.193 8948.636 

4 339.4235 1349.102 3044.823 5403.891 8423.462 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3076 
https://internationalpubls.com 

5 317.5033 1276.269 2878.878 5106.922 7957.314 

6 301.4125 1210.972 2730.301 4841.313 7540.715 

7 286.8832 1152.090 2596.477 4602.308 7166.113 

 

• Discussion.According to these tables, premiums that use the entropy loss 

function are the gentler with all drivers, whereas those that use the 

quadratic loss function are the harshest with both good and bad drivers. 

 

5.3.3. Premiums using Inverse-Gamma Lindley under Entropy loss 

functionfortheclaimseverity.This part introduces the tables [17-22] which 

represents the premiums using the Invers-Gamma Lindley distribution 

undertheentropylossfunctionfortheseverityclaim.Poisson-Akash distribution under 

different loss functions (Quadratic, Linex (a=1.1 and a=-0.3) and Entropy (p=0.2 

and p=-0.2)) will be employed fortheclaimfrequency.The formulas (17), (17) and 

(18) from table 3 were used to create the tables bellow. 
TABLE17.Premiums using Poisson-Akash under quadratic loss 
functionandInverse-GammaLindleyunderentropylossfunction 
with γ=14.0125, α=1.08, β=0.001766, p= -1.1 

t 
Claimnumber 

0 1 2 3 4 

1 551.1474 556.9714 860.9861 1255.7779 1728.308 

2 515.6550 520.6273 804.1743 1172.1782 1612.496 

3 484.5062 488.8001 754.5044 1099.1657 1511.402 

4 456.9431 460.6893 710.6986 1034.8385 1422.386 

5 432.3755 435.6742 671.7677 977.7245 1343.400 

6 410.3367 413.2658 636.9341 926.6668 1272.831 

7 390.4528 393.0733 605.5780 880.7437 1209.395 

 

 
TABLE18.Premiums using Poisson-Akash under linex loss 
function and Inverse-Gamma Lindley under entropy loss function 
withγ=14.0125,α=1.08,β=0.001766,a=1.1,p=-1.1 

t 
Claimnumber 

0 1 2 3 4 

1 531.2749 536.6159 829.1599 1208.9383 1663.416 

2 498.2347 502.8229 776.3830 1131.3205 1555.920 

3 469.1068 473.0912 730.0203 1063.2070 1461.639 

4 443.2296 446.7232 688.9597 1002.9424 1378.272 

5 420.0833 423.1737 652.3331 949.2348 1304.020 

6 399.2543 402.0097 619.4528 901.0616 1237.459 

7 380.4086 382.8833 589.7672 857.6029 1177.446 

 
TABLE19.Premiums using Poisson-Akash under linex loss 
function and Inverse-Gamma Lindley under entropy loss 
function withγ=14.0125,α=1.08,β=0.001766,a=-0.3,p=-1.1 

Claimnumber 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3077 
https://internationalpubls.com 

t 
0 1 2 3 4 

1 556.9239 562.8950 870.2552 1269.4263 1747.221 

2 520.6975 525.7859 812.2324 1184.0306 1628.9912 

3 488.9473 493.3343 761.5761 1109.5563 1525.1786 

4 460.8853 464.7070 716.9560 1044.0235 1435.7094 

5 435.8989 439.2596 677.3449 985.9903 1352.9708 

6 413.5051 416.4857 641.9372 933.9975 1282.9960 

7 393.3177 396.0918 610.0919 887.3526 1218.5522 

 
TABLE20. Premiums using Poisson-Akash under entropy 
loss function and Inverse-Gamma Lindley under entropy loss 
function with γ=14.0125, α=1.08, β=0.001766, s = -1.1, p= 
-1.1 

T 
Claimnumber 

0 1 2 3 4 

1 574.6295 569.9507 874.8278 1271.1963 1745.464 

2 537.5994 532.7376 817.0750 1186.5394 1628.472 

3 505.1048 500.1524 766.5856 1112.6062 1526.351 

4 476.3537 471.3745 722.0595 1047.4702 1436.431 

5 450.7294 445.7675 682.4906 989.6401 1356.644 

6 427.7445 422.8320 647.0879 937.9439 1285.361 

7 407.0082 402.1624 615.2208 891.4481 1221.286 

 

 

 

 

 
TABLE21. Premiums using Poisson-Akash under entropy 
loss function and Inverse-Gamma Lindley under entropy loss 
function with γ=14.0125, α=1.08, β=0.001766, s= -0.2, p= 
-1.1 

t 
Claimnumber 

0 1 2 3 4 

1 358.4347 450.7562 748.3329 1130.7149 1589.443 

2 335.4842 421.4892 699.1569 1055.6795 1483.175 

3 315.3225 395.8393 656.1387 990.1247 1390.402 

4 297.4668 373.1690 618.1797 932.3488 1308.699 

5 281.5402 352.9830 584.4292 881.0347 1236.188 

6 267.2442 334.8908 554.2180 835.1478 1171.392 

7 254.3390 318.5802 527.0123 793.8635 1113.135 

 
TABLE22. Premiums using Poisson-Akash under entropy loss 
function and Inverse-Gamma Lindley under entropy loss function with 
γ=14.0125, α=1.08, β=0.001766, s= 0.2, p= -1.1 

t 
Claimnumber 

0 1 2 3 4 

1 256.3630 395.4298 690.3996 1066.8330 1518.783 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3078 
https://internationalpubls.com 

2 239.9969 369.8230 645.1319 996.1624 1417.373 

3 225.6120 347.3711 605.5196 934.4077 1328.831 

4 212.8668 327.5199 570.5557 879.9699 1250.845 

5 201.4942 309.8383 539.4600 831.6114 1181.624 

6 191.2827 293.9862 511.6187 788.3598 1119.761 

7 182.0620 279.6916 486.5421 749.4401 1064.134 

 

• Discussion.Referring to the tables, the premiums with the entropy loss 

function with "p=-1.1" and those with the linex loss function in the 

underestimation are nearly identical; these premiums are highest for poor 

drivers. The results also demonstrate that premiums with good and bad 

drivers are lower when applying the entropy loss function with "p=-0.2" 

and "p=0.2." 

 
6. Conclusion 

This study presented the Bayesian bonus-malus premiums for claim frequency and claim 
severity. Using the Bayesian approach, we employed the Poisson-Akash distribution for claim 
frequency and the Invers-Gamma Lindley distribution for claim severity.  
Different loss functions (quadratic, linex, and entropy) were additionally introduced in this 
work. Using the R software and an actual car insurance dataset, we computed the premiums 
for the frequency only and for both the frequency and the severity of the claim. We established 
the premiums using the several loss functions. 

We found from the numerical application that premiums that use distinct loss functions for 
the frequency and severity of claims provide greater outcomes than those that apply the same 
loss function. This numerical research demonstrated that using various loss functions might 
provide flexible outcomes that could be very beneficial to the insurer. The insurer has a wide 
range of choices thanks to the wide variety of these outcomes. The insurance company selects 
the loss function based on its goal, offering excellent drivers the highest rewards or penalizing 
bad drivers most severely, deciding how to draw clients and make them attractive offers, 
maintaining the organization's solvability, and maintaining equilibrium. 

 
REFERENCES 
 

[1] H. Buhlmann. Experience rating and credibility. ASTIN Bulletin: The 
Journal of the 
IAA, 4(3):199–207, 1967. 

[2] C. Clemente, G. R. Guerreiro, and J. M. Bravo. Modelling motor insurance 
claim frequency 
and severity using gradient boosting. Risks, 11(9):163, 2023. 

[3] P. De Jong and G. Z. Heller. Generalized linear models for insurance data. 
Cambridge 
Books, 2008. 

[4] E. Gomez-Deniz. Bivariate credibility bonus–malus premiums 
distinguishing betweentwo types of claims. Insurance: Mathematics and 
Economics, 70:117–124, 2016. 

[5] E. Gomez-Deniz, A. Hern´andez-Bastida, and M. P. Fern´andez-S´anchez. 
Computing credibility 
bonus-malus premiums using the total claim amount distribution. Hacettepe 
Journal 
of Mathematics and Statistics, 43(6):1047–1061, 2014. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3079 
https://internationalpubls.com 

[6] T. Herzog. Introduction to credibility theory (winsted, connecticut, actex 
publications 
inc, 1996. 

[7] A. Moumeesri and T. Pongsart. Bonus-malus premiums based on claim 
frequency and 
the size of claims. Risks, 10(9):181, 2022. 

[8] M. M. Nassar and F. H. Eissa. Bayesian estimation for the exponentiated 
weibull model. 
Communications in Statistics-Theory and Methods, 33(10):2343–2362, 
2004. 

[9] I. Ouchen, A. Sadoun, F. Metiri, and M. R. Remita. On bayesian bonus-
malus premium 
under linex loss function with applications. Studies in Engineering and 
Exact Sciences, 
5(2):e6226, 2024. 

[10] J. Rojo. On the admissibility of c X + d with respect to the linex loss 
function. Communications 
in Statistics-Theory and Methods, (16):3745–3748, 1989. 

[11] R. Shanker. On poisson-akash distribution and its applications. 
Biometrics&Biostatistics 
International Journal, 3(6), 2016. 

[12] H. Varian. A bayesian approach to real estate assessment. North-Holland 
Pub. Co., pages 
195–208, 1975. 

[13] A. Zellner. Bayesian estimation and prediction using asymmetric loss 
functions. Journal 
of the American Statistical Association, 81(394):446–451, 1986.