- Research
- Open Access

# Analysis of order statistics of uncertain variables

- Yuan Gao
^{1}, - Rong Gao
^{2}Email author and - Lixing Yang
^{1}

**3**:1

https://doi.org/10.1186/s40467-014-0025-1

© Gao et al.; licensee Springer. 2015

**Received: **13 October 2014

**Accepted: **23 November 2014

**Published: **14 January 2015

## Abstract

In this paper, a concept of order uncertain variable is introduced as a main tool in analysis of k-out-of-n system with uncertain lifetimes. Then, by the operational law of uncertain variable, the uncertainty distribution of order uncertain variable is deduced. At last, the k-out-of-n system is studied, and the uncertainty distribution of k-out-of-n system lifetime is given.

## Keywords

- Uncertain variable
- Uncertainty distribution
- Order statistics of uncertain variables
- Operational law
- k-out-of-n system
- System reliability

## Introduction

The real world is filled with indeterminacy. In fact, randomness and fuzziness are two cases of indeterminacy, behind which probability theory and credibility theory have been proposed. However, some practical systems behave neither random nor fuzzy, so it cannot be modeled precisely by probability theory or credibility theory. In order to deal with this type of uncertainty, Liu [6] founded an uncertainty theory in 2007, which is a branch of mathematics based on normality, duality, subadditivity, and product axioms. Since then, based on uncertainty theory, significant work both in theory and engineering has been done by many researchers. As an application of uncertainty theory, Liu [7] proposed a spectrum of uncertain programming which is a type of mathematical programming involving uncertain variables and applied uncertain programming to system reliability design, facility location problem, vehicle routing problem, project scheduling problem, finance, and so on. In addition, Li and Liu [5] presented uncertain logic in which the truth value is defined as the uncertain measure that the proposition is true. Furthermore, uncertain inference was pioneered by Liu [8] as a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Other researchers also have done a lot of theoretical work in uncertainty theory, such as Chen [1], Chen and Liu [2], Gao [3], Gao et al. [4], Liu [9], Peng [11], Peng and Chen [12], Qin and Kar [13], Wang and Peng [14], Yang [15], Yao [16], You [17] and Zhu [18], etc.

Problems of k-out-of-n system play an important role in system reliability design. Up to now, k-out-of-n system with random lifetimes has been researched in many books related to system reliability design. This paper aims to give some analysis of k-out-of-n system with uncertain lifetimes. Based on the uncertainty theory, this paper gives a new concept of order uncertain variable, which is used as a tool to analyze k-out-of-n system with uncertain lifetimes in this paper. It is similar to the order statistics (Balakrishnan and Cohen [10]) in mathematical statistics. However, order uncertain variable just requires that the initial uncertain variables are independent. Besides, the uncertainty distribution of order statistics of uncertain variables has a much simpler form than the ones of random variables.

The remainder of this paper is organized as follows. In Section “Preliminary”, basic concepts and properties regarding uncertain variables are reviewed. In Section “Order uncertain variable”, the concept of order uncertain variable is given. In Section “Uncertainty distribution of order uncertain variable”, the uncertainty distribution of order uncertain variable is deduced. In Section “k-out-of-n system with uncertain lifetimes”, the order uncertain variable is used as a tool to analyze k-out-of-n system with uncertain lifetimes.

## Preliminary

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

Let *Γ* be a nonempty set, and
a *σ*-algebra over *Γ*. Each element
is assigned a number
. In order to ensure that the number
has certain mathematical properties, Liu [6] presented the following four axioms:

###
**Definition**
**1**.

(Liu [6]) The set function is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.

The triplet
is called an *uncertainty space*. Furthermore, the product uncertain measure on the product *σ*-algebra
is defined by Liu [6] as follows:

###
**Axiom**
**4**.

(Product Axiom) Let
be uncertainty spaces for *i*=1,2,⋯ The product uncertain measure
is an uncertain measure satisfying
where *Λ*
_{
i
} are arbitrarily chosen events from
for *i*=1,2,⋯, respectively.

###
**Definition**
**2**.

*ξ*from an uncertainty space to the set of real numbers, i.e., for any Borel set B of real numbers, the set

###
**Definition**
**3**.

(Liu [6]) The uncertainty distribution *Φ*:ℜ→[0,1] of an uncertain variable *ξ* is defined by

###
**Definition**
**4**.

(Liu [8]) The uncertain variables *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
} are said to be independent if
for any Borel sets *B*
_{1},*B*
_{2},⋯,*B*
_{
n
} of real numbers.

More generally, the independence of uncertain variables was given by Liu [9].

###
**Theorem**
**1**.

(Liu [9]) The uncertain variables *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
} are said to be independent if and only if

###
**Definition**
**5**.

(Liu [6]) Let *ξ* be an uncertain variable. Then, the expected value of *ξ* is defined by
provided that at least one of the two integrals is finite.

From Remark 1.6 in Liu ([6]), the following theorem is given without proof.

###
**Theorem**
**2**.

(Liu [8]) Let *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
} be independent uncertain variables, and \(f : \mathfrak {R}^{n} \rightarrow \mathfrak {R}\) a measurable function. Then, *ξ*=*f*(*ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
}) is an uncertain variable such that
where *B*,*B*
_{1},*B*
_{2},⋯,*B*
_{
n
} are Borel sets of real numbers.

## Order uncertain variable

###
**Definition**
**6**.

*f*

_{ i }:ℜ

^{ n }→ℜ,

*i*=1,2,⋯,

*n*be a series of measurable functions. We call

*f*

_{1},

*f*

_{2},⋯,

*f*

_{ n }the order functions of ℜ

^{ n }if for any (

*x*

_{1},

*x*

_{2},⋯,

*x*

_{ n })∈ℜ

^{ n },

*x*

^{(1)},

*x*

^{(2)},⋯,

*x*

^{(n)}} is the the rearrangement of {

*x*

_{1},

*x*

_{2},⋯,

*x*

_{ n }} in ascending order of magnitude, that is,

*x*

^{(1)}≤

*x*

^{(2)}≤⋯≤

*x*

^{(n)}.

###
**Definition**
**7**.

*ξ*

_{1},

*ξ*

_{2},⋯,

*ξ*

_{ n }be independent uncertain variables, and

*f*

_{1},

*f*

_{2},⋯,

*f*

_{ n }order functions of ℜ

^{ n }. Define

Then, *ξ*
^{(1)},*ξ*
^{(2)},⋯,*ξ*
^{(n)} are called order uncertain variables of *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
}, and each *ξ*
^{(i)} is called the *i*th order uncertain variable of *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
}.

*n*=2 and the relationship between

*ξ*

_{ i }and

*ξ*

^{(i)}(

*i*=1,2) is

*n*=3 the relationship among the respective uncertain variables is

The most frequently encountered functions of order uncertain variable are *ξ*
^{(1)} and *ξ*
^{(n)}. Obviously, \(\xi ^{(1)}=\min \limits _{1\leq k\leq n} \xi _{k}\), and \(\xi ^{(n)}=\max \limits _{1\leq k\leq n} \xi _{k}\).

## Uncertainty distribution of order uncertain variable

In this section, the uncertainty distribution of the *k*th order uncertain variable for *k*=1,2,⋯,*n* will be given.

###
**Theorem**
**3**.

*ξ*

_{1},

*ξ*

_{2},⋯,

*ξ*

_{ n }be independent uncertain variables with distributions

*Φ*

_{1}(

*x*),

*Φ*

_{2}(

*x*),⋯,

*Φ*

_{ n }(

*x*), respectively. The uncertainty distribution of the

*k*th (

*k*=1,2,⋯,

*n*) order uncertain variable

*ξ*

^{(k)}is

where *f*
_{
k
}:ℜ^{
n
}→ℜ,*k*=1,2,⋯,*n* are order functions of ℜ^{
n
}.

###
*Proof*.

*x*∈ℜ, define a series of new variables

*i*=1,2,⋯,

*n*. Then,

*η*

_{1}(

*x*),

*η*

_{2}(

*x*),⋯,

*η*

_{ n }(

*x*) are independent, and for each

*i*. According to the definitions of uncertain variable and order uncertain variable, by Lemma 1, we have where

*f*

_{ k }:ℜ

^{ n }→ℜ,

*k*=1,2,⋯,

*n*are order functions of ℜ

^{ n }. The theorem is proved. □

###
**Corollary**
**1**.

Let *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
} be independent uncertain variables with the same uncertainty distribution *Φ*(*x*), and let *ξ*
^{(1)},*ξ*
^{(2)},⋯,*ξ*
^{(n)} be the corresponding order uncertain variables. Then, for any *k*=1,2,⋯,*n*, the *k*th order uncertain variable has uncertainty distribution *Φ*(*x*).

###
*Proof*.

It follows from Theorem 2 immediately. □

###
**Corollary**
**2**.

Let *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
} be independent uncertain variables with the same uncertainty distribution *Φ*(*x*), and let *ξ*
^{(1)},*ξ*
^{(2)},⋯,*ξ*
^{(n)} be the corresponding order uncertain variables. Then, for any *k*=1,2,⋯,*n*, the *k*th order uncertain variable has expected value *E*[*ξ*
_{1}] provided that *E*[*ξ*
_{1}] exists.

###
*Proof*.

It follows from Corollary 1 and Definition 4 immediately. □

## k-out-of-n system with uncertain lifetimes

System reliability design with uncertain lifetimes was first studied by Liu [7]. The k-out-of-n system is a significant kind of problem in system reliability design. In this section, order uncertain variable is used as a tool in analysis of k-out-of-n system with uncertain lifetimes.

A system of *n* components is called a k-out-of-n system if it remains operational only if at least *k* components continue to function. Assume that the *i*th element of this component has independent lifetime *ξ*
_{
i
} with uncertainty distribution *Φ*
_{
i
}(*x*), *i*=1,2,⋯,*n*, respectively. Then, what is the distribution *Φ*(*x*) of this system lifetime *η*?

Let *ξ*
^{(1)},*ξ*
^{(2)},⋯,*ξ*
^{(n)} be the order uncertain variables of *ξ*
_{1},*ξ*
_{2},⋯,*ξ*
_{
n
}. Then, for any *x*>0,*x*∈ℜ, by Theorem 2
where *f*
_{
k
}:ℜ^{
n
}→ℜ,*k*=1,2,⋯,*n* are order functions of ℜ^{
n
}. Thus, we get the uncertainty distribution of *η*.

*k*=1 and

*k*=

*n*correspond respectively to parallel system and series system (Figure 1). According to the formula (Equation ??), the uncertainty distribution

*Φ*

_{ p }(

*x*) of parallel system lifetime

*η*

_{ p }is and the uncertainty distribution

*Φ*

_{ s }(

*x*) of series system lifetime

*η*

_{ s }is

## Conclusion

In order to analyze k-out-of-n system with uncertain lifetimes, we presented a new concept of order uncertain variable and gave uncertainty distribution of order uncertain variable in this paper. By using order uncertain variable, the uncertainty distribution of k-out-of-n system lifetime was deduced.

## Declarations

### Acknowledgments

The research was supported by the Fundamental Research Funds for the central universities (No. 2014JBM155) and the State Key Laboratory of Rail Traffic Control and Safety (No. RCS2014ZQ001 and No. RCS2014ZT22, Beijing Jiaotong University).

## Authors’ Affiliations

## References

- Chen, X, Dai, W: Maximum entropy principle for uncertain variables. Int. J. Fuzzy Syst. 13(3), 232–236 (2011).MathSciNetGoogle Scholar
- Chen, X, Liu, B: Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization Decis. Making. 9(1), 69–81 (2010).MATHMathSciNetGoogle Scholar
- Gao, X: Some properties of continuous uncertain measure. Int. J. Uncertainty Fuzziness Knowledge-Based Syst. 17(3), 419–426 (2009).View ArticleMATHGoogle Scholar
- Gao, Y, Yang, L, Li, S, Kar, S: On distribution function of the diameter in uncertain graph. Inf. Sci. 296, 61–74 (2015).View ArticleMathSciNetGoogle Scholar
- Li, X, Liu, B: Hybrid logic and uncertain logic. J. Uncertain Syst. 3(2), 83–94 (2009).Google Scholar
- Liu, B: Uncertainty Theory. 2nd ed. Springer-Verlag, Berlin (2007).MATHGoogle Scholar
- Liu, B: Theory and Practice of Uncertain Programming. 2nd ed. Springer-Verlag, Berlin (2009).View ArticleMATHGoogle Scholar
- Liu, B: Some research problem in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009).Google Scholar
- Liu, B: Polyrectangular theorem and independence of uncertain vectors. J. Uncertainty Anal. Appl. 1, 9 (2013). doi:10.1186/2195-5468-1-9.View ArticleGoogle Scholar
- Balakrishnan, N, Cohen, AC: Order statistics and inference: estimation methods. Academic Press, Inc., San Diego (1991). ISBN 0120769484, 9780120769483.MATHGoogle Scholar
- Peng, Z, Iwamura, K: Some properties of product uncertain measure. J. Uncertain Syst. 6(4), 263–269 (2012).MathSciNetGoogle Scholar
- Peng, Z, Chen, X: Uncertain systems are universal approximators. J. Uncertainty Anal. Appl. 2, 13 (2014). doi:10.1186/2195-5468-2-13.View ArticleGoogle Scholar
- Qin, Z, Kar, S: Single-period inventory problem under uncertain environment. Appl. Math. Comput. 219(18), 9630–9638 (2013).View ArticleMATHMathSciNetGoogle Scholar
- Wang, X, Peng, Z: Method of moments for estimating uncertainty distributions. J. Uncertainty Anal. Appl. 2, 5 (2014). doi:10.1186/2195-5468-2-5.View ArticleGoogle Scholar
- Yang, L, Liu, P, Li, S, Gao, Y, Ralescu, D: Reduction methods of type-2 uncertain variables and their applications to solid transportation problem. Inf. Sci. 291, 204–237 (2015).View ArticleMathSciNetGoogle Scholar
- Yao, K: Extreme values and integral of solution of uncertain differencial equation. J. Uncertainty Anal. Appl. 1, Article, 2 (2013).Google Scholar
- You, C: Some convergence theorems of uncertain sequences. Math. Comput. Modell. 49(3-4), 482–487 (2009).View ArticleMATHGoogle Scholar
- Zhu, Y: Uncertain optimal control with application to a portfolio selection model. Cybernet. Syst. 41(7), 535–547 (2010).View ArticleMATHGoogle Scholar

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