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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.12 No.1 pp.46-50

An Individual Risk Model and Its Uncertainty Distribution

Li Ren*
Dongchang College of Liaocheng University, Liaocheng, China
Received: September 5, 2012, Revised: January 2, 2013, Accepted: March 5, 2013


In insurance statistics, the sum of homogeneous insurance is often needed. The sum is called individual risk modelwhich is a fundamental model in risk analysis for insurance. This paper first presents an individual risk model basedon the uncertainty theory. Then its uncertainty distribution is provided. Finally, its arithmetic is shown by a numericalexample.

12-1-08_46-50_ Li Ren.pdf277.8KB


It is well known that there are many types of uncertainty in real life. A. N. Kolmogoroff (1933) presented the probability theory dealing with a type of uncertainty that an event will happen (called randomness). Besides, Zadeh (1965) initializedthe fuzzy set theory via membership function that aimed to study the uncertainty called fuzzyness. The probability theory and the fuzzy set theory have been widely accepted to study uncertainness and have also been applied to many aspects of our life. 

However, there are still many other uncertainness that can not be described by the probability theory and the fuzzy set theory. In order to rationally deal with the degree of belief, an uncertainty theory was introduced by Liu (2007). The uncertainty theory is a branch of axiomatic mathematics based on normality, duality, subadditivity and product axioms. And it was refined by Liu (2010a). Nowadays, the uncertainty theory has become a branch of axiomatic mathematics for modeling human uncertainty. Uncertain statistics is an important branch for uncertainty theory. It was introduced by Liu (2010a). Chen and Ralescu (2012) stated that the questionnaire survey may successfully acquire the expert’s experimental data. Wang et al., (2012) recast the Delphi method as a process to determine the uncertainty distributions. Liu (2010a) suggested the principle of least squares method to propose an uncertainty distribution. Furthermore, the uncertainty theory has been applied to uncertain programming (Liu, 2009a; Gao, 2011, 2012; Meng and Zhang, 2013; Peng and Yao, 2011; Rong, 2011; Sheng and Yao, 2012; Zhang and Chen, 2012; Zhang and Meng, 2013), uncertain risk analysis (Liu, 2010b), uncer-tain logic (Chen et al., 2012; Li and Liu, 2009; Zhang, 2011; Zhang and Peng, n.d.) uncertain process (Yao and Li, 2012), and others (Dai and Chen, 2012; Chen and Ralescu, 2011; Gao, 2009; Zhang et al., 2013; Zhu, 2010).  

In order to study dynamic systems in uncertainty theory, Liu (2009b) introduced uncertain process which is essentially a sequence of uncertain variables indexed by time. There is an important type of uncertain process called renewal process, which is an uncertain process in which events occur continuously and independently of one another in uncertain times. Yao and Li (2012) studied the uncertain alternating renewal process. Based on the uncertain renewal process, Yao and Ralescu (2013) investigated the age replacement policy in uncertain environment. Liu (2013) presented an uncertain insurance model by assuming that the claim is an uncertain renewal reward process, and proved a formula for calculating ruin index. Different from it, this paper aims to study insurance by uncertain statistics. We will build a flow shop scheduling model in the uncertain environment based on the uncertainty theory. It is well known that in insurance statistics the sum of homogeneous insurance is often needed, and the sum is called individual risk model, which is a fundamental model in risk analysis for insurance. These problems are studied by many scholars (Kass et al., 2001; Peng, 2009; Wen and Wang, 2006). 

The paper is organized as follows. Section 2 recalls several concepts of uncertain variables. Section 3 introduces an individual risk model and its uncertainty distribution. Section 4 gives a numerical example. Finally, a brief summary is included.  


In this section, we introduce some foundational concepts and properties of uncertainty theory, which will be used throughout this paper.  

Let Γ be a nonempty set, and L a σ -algebra over Γ. Each element Λ∈L is called an event.  

Definition 1 (Liu, 2007). The set function M is called an uncertain measure if it satisfies the following three axioms:

Axiom 1 (Normality). M{Γ} =1 ;

Axiom 2 (Self-Duality). M{Λ} + M{Λc} =1 for any event Λ ;

Axiom 3 (Countable Subadditivity). For every countable sequence of events {Λi}, we have   


Liu (2009b) presented the product measure axiom of uncertainty theory in 2009 as follows: 

Axiom 4 (Product Measure Axiom). Let Γk be nonempty sets on which Mk are uncertain measures, k =1, 2, …, n, respectively. Then the product uncertain measure M is an uncertain measure on the product σ -algebra Ł1 × Ł2 × … × Łn satisfying 


That is, for each event Λ∈Ł, we have 


Definition 2 (Liu, 2010a). An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set 


is an event. 

Definition 3 (Liu, 2010a). The uncertainty distribution Φ of an uncertain variable ξ is defined by 


for any real number x. 

Definition 4 (Liu, 2010a). An uncertain variable ξ is called linear if it has a linear uncertainty distribution 


denoted by Ł(a,b) where a and b are real numbers with a < b. 

Definition 5 (Liu, 2010a). An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution 


denoted by Z(a, b, c)where a, b, c are real numbers with a < b < c. 

Definition 6 (Liu, 2010a). An uncertain variable ξ is called normal if it has a normal uncertainty distribution 


denoted by N(e,σ ) where e and σ are real numbers with σ > 0. 

Definition 7 (Liu, 2010a). An uncertain variable ξ is called lognormal if ln ξ is a normal uncertainty variable N(e,σ). In other words, a lognormal uncertain variable has an uncertainty distribution 


denoted by LOGN(e,σ ) where e and σ are real numbers with σ > 0.

Definition 8 (Liu, 2010a). The uncertain variables X1, X2, …, Xm are said to be independent if 


for any Borel sets B1, B2, …, Bm of real numbers. 

Definition 9 (Liu, 2010a). An uncertainty distribution Φ is said to be regular if its inverse function Φ−1(α) exists and is unique for each α ∈(0, 1). 

Theorem 1 (Liu, 2010a). Let ξ1, ξ2, …, ξn be independent uncertain variables with uncertainty distributions Φ1, Φ2, …, Φn, respectively. If f is a strictly increasing function, then 


is an uncertain variable with inverse uncertainty distribution 


Theorem 2 (Liu, 2010a). Let ξ1, ξ2, …, ξn be independent uncertain variables with uncertainty distributions Φ1, Φ2, …, Φn, respectively. If f is a strictly increasing function, then 


is an uncertain variable with inverse uncertainty distribution 



In this section, we will present an individual risk model based on the uncertainty theory. It is a fundamental model in risk analysis for insurance. Assume that in a stated time (such as one year’s time), the insurer has done business with n customers, and every customer bought N policies. It is well known that in an insurance company, if customers have some prescriptive losses, then the insurer should shell out the payment. It is evident that the payment is an uncertain variable. Thus, we have the the following definition: 

Definition 10. In a stated time, the total indemnity for the insurer is given by 


which is called individual risk model, where ξij are the payment for the jth insurance contract of ith insured, and nj customers subscribe the jth insurance contract. 

The quantity of the insurance contracts is supplied by the insurer. 

Commonly, it is hard to obtain the accurate distribution of S. In order to study the distribution of S and obtain its properties, some hypotheses should be given. 

1) The uncertainty distribution ϕij of every uncertain variable ξij are as follows: 


where 0 is the minimum payment, b is the maximum payment and p(x) is a strictly increasing function. 

2) Uncertain variables ξij are independent.

3) Every policy settles a claim once at most.

4) The homogeneous insurances have characteristic of the same insurance marks. From the thought of uncertainty theory, this kind of homogeneity shows that the payment variables ξij of the same kind insurance policy are iid uncertain variables. This means 


Based on the above hypothesis, we give the distribution of S in the following. 

Theorem 3. In the individual risk model with the limit of N = 1, then S =  is an uncertain variable with the uncertainty distribution  


and its inverse uncertainty distribution 


where ξ1, ξ2, …, ξn are independent uncertain variables with the common uncertainty distribution ϕ(x). 

Proof. Note that ξ1, ξ2, …, ξn are independent uncertain variables with the common uncertainty distribution ϕ(α). By Theorem 2, we have 


It is evident that  and


Counterevidence. Suppose Φ(x) ≠. If Φ(x) > , then there exists x1, x2, …, xn such that x1 + x2 + … + xn = x and ϕ(xi) >, i= 1, 2, …, n. Since ϕ(x) is a increasing function, x2 >  this is inconsistent with x1 + x2 + … + xn = x. Thus 


Since S = and it increase on each ξi, i ∈ 1, 2, …, n, by Theorem 1 we have 


The theorem is proved. 

Theorem 4. In the above individual risk model, S =  is an uncertain variable with the uncertainty distribution 


and its inverse uncertainty distribution is 


where Φj(xj) = M{ ξij < xj}, i∈{1, 2, …, nj}, 0 < j ≤ N. 

Proof. Let 


Note that ξ1j, ξ2j, …, ξnjj are independent uncertain variables with the common uncertainty distribution ϕ (x). Since Φj(x) = M{ξij < xj}, i ∈{1, 2, …, nj}, 0 < j ≤ N, by Theorem 3, we have Φj(xj) =  Since S =  = by Theorem 2, we have 


By Theorem 3 and Theorem 1, inverse uncertainty distribution of S =  =  is


The theorem is proved. 


Assume that the insurers have sold four kinds of policies to a fellowship with 1,000 members, and the distributions of these policies are as follows, 


Then by the distribution of it we can get the uncertainty distribution for the total payments lacked in the coming year as follow: 



This paper provided an individual risk model via uncertainty theory and gave its uncertainty distribution. The model and its arithmetic were shown by a numerical example. 


This work was supported by National Natural Science Foundation of China (Grant No. 61273044). 


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