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RESEARCH ARTICLE

A Reliability Test of a Complex System Based on Empirical Likelihood Yan Zhou1☯, Liya Fu2☯, Jun Zhang3, Yongchang Hui2* 1 College of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, Shenzhen, China, 2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China, 3 College of Mathematics and Computational Science, Institute of Statistical Sciences, Shen Zhen-Hong Kong Joint Research Center for Applied Statistical Sciences, Shenzhen University, Shenzhen, China

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☯ These authors contributed equally to this work. * [email protected]

Abstract OPEN ACCESS Citation: Zhou Y, Fu L, Zhang J, Hui Y (2016) A Reliability Test of a Complex System Based on Empirical Likelihood. PLoS ONE 11(10): e0163557. doi:10.1371/journal.pone.0163557

To analyze the reliability of a complex system described by minimal paths, an empirical likelihood method is proposed to solve the reliability test problem when the subsystem distributions are unknown. Furthermore, we provide a reliability test statistic of the complex system and extract the limit distribution of the test statistic. Therefore, we can obtain the confidence interval for reliability and make statistical inferences. The simulation studies also demonstrate the theorem results.

Editor: Tao Lu, State University of New York, UNITED STATES Received: July 3, 2016 Accepted: September 11, 2016 Published: October 19, 2016 Copyright: © 2016 Zhou et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper. The data in our simulation studies are random numbers which are generated by computer, not real data. Funding: Y. Zhou’s research was supported by the Tianyuan Fund for Mathematics (No. 11526143), the Doctor Start Fund of Guangdong Province (No. 85118-000043) and The Natural Science Foundation of SZU (No. 836-00008303). L. Fu’s research was supported by The National Science Foundation of China (No. 11201365 and No. 11301408) and the Doctoral Programs Foundation of Ministry of Education of China (No. 2012020112005). Yongchang Hui’s research was

Introduction In practice, little information can be directly derived from a complex system (CS). What we usually obtain is information about subsystems. Therefore, system reliability, which is based on subsystem data, is a very important research topic and has been a concern for a long time. However, due to the complexity of the system, life distribution, subsystem reliability, and diversity of the data distribution, there are many difficulties studying of a CS. In recent years, many researchers have provided various methods for calculating and estimating system reliability under the assumption that the distribution of the subsystems (family) is known. For example, for a series system with two subsystems and with binomially distributed pass-fail failure data, Buehler [1] proposed a model and derived the exact lower confidence limit of the system reliability. Rosenblatt [2] proposed an approximate method for system reliability confidence limits based on the asymptotic normality of a U-statistic. Weaver [3] derived a simple and accurate ordering method to calculate the system reliability confidence limits, but it requires the same sample size for all subsystems. Rice and Moore [4] presented a Monte Carlo model for binomial distributed data that is valid for a zero failure case. Coit [5] provided a method which does not require any parametric assumptions for component reliability or time to failure. Tian [6] summarized and compared the advantages and limitations of the parametric methods.

PLOS ONE | DOI:10.1371/journal.pone.0163557 October 19, 2016

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A Reliability Test of Complex System

supported by The National Science Foundation of China (No. 11401461). This study was also supported in part by Fundamental Research Funds for the Central Universities (No. 2015gjhz15). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist.

However, only a few studies have investigated the reliability test of a CS. The main reasons are that the life distributions of the subsystems may vary, and it is difficult to find out which one it is. It is not easy to find a test statistic and its asymptotic distribution because of the complexity of the system structures. In recent studies, regarding the series system in which the subsystems’ life distribution is an index distribution, Yu et al. [7] considered the system failure rate of testing and gave an accurate unbiased test. Based on the method proposed by Yu et al. [7], Li [8] further considered subsystem lives following different distributions for the series system and completed the corresponding test by approximately transforming the non-index distribution to index distribution. In a normal situation, it is difficult to determine the life distribution of a subsystem, and the structure of the system may be very complicated. In this paper, we assume the following: (i) A complex system is described by minimal paths, and these minimal paths are known; (ii) The subsystems of the complex system are independent, and the distribution of life is unknown. Under these two assumptions, we provide a reliability test statistic for a complex system using the empirical likelihood method [9, 10]. Furthermore, we also extract the limit distribution of the test statistic. Therefore, we can obtain the confidence interval and make statistical inferences for the system reliability based on the limit distribution. This paper is organized as follows. In Section 2, we describe the reliability test problem of the complex system. In Section 3, we provide a test statistic and derive its asymptotic distribution. In Section 4, we carry out simulation studies for a bridge system. Finally, we draw several conclusions. The proof of the asymptotic distribution is given in the Appendix.

Materials and Methods Notations The life of a complex system is Z = max1jk minr2Oj Zr, where k is the number of minimal paths, Oj is the jth minimal path, and Zr is the life of the rth subsystem. The specific expression of complex system reliability has been derived in [11, 12]. For convenience, we first define some operations and give some notions. Definition 2.1: Let α = (a1, a2, . . ., am)T and β = (b1, b2, . . ., bm)T be two column vectors with m elements. Define α  β if ai  bi, i = 1, 2, . . ., m; ( 1 ai ¼ 1; bi ¼ 0 T Define α \ β = (δ1, δ2, . . ., δm) , where di ¼ , i = 1, 2, . . ., m; 0 others Define α  β = (γ1, γ2, . . ., γm)T, where γ1 = a1 + b1 and γi = max{ai, bi} for i = 2, . . ., m. Definition 2.2: Let A = (α1, α2, . . ., αk) be an m × k matrix, and Aj = (α1 \ αj, α2 \ αj, . . ., αj−1 \ αj) for a given j, where j = 2, . . ., k. If αt1 \ αj  αt \ αj for all t1 < t  j, then we cut out the column αt \ αj of Aj. Denote the gained matrix by Aj , which is an m × nj matrix, where Pnj  nj  j − 1 and j = 2, 3, . . ., k. Define VðAj Þ ¼ max 1sm t¼1 Aj ðs; tÞ, where Aj ðs; tÞ is the s row and the t column element of Aj . Definition 2.3: Let C = (c1, c2, . . ., ck) be an m × k matrix. For a fixed index j, j = 1, 2, . . ., k, ~ j ¼ ðfi i ...i Þ be an define fi1i2, . . ., ij = ci1  ci2  . . .  cij, where 1  i1 < i2 < . . . < ij  k. Let C 1 2 j m  Ckj matrix whose column vectors are made of all vectors fi1 i2. . .ij, where Ckj is the combination number of j in k.

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A Reliability Test of Complex System

Definition 2.4: Let α = (a1, a2, . . ., am)T and R = (r1, r2, . . ., rm)T be vectors with m elements, and D = (d1, d2, . . ., dk) be an m × k matrix. Define Ra ¼

m Y

riai and RD ¼

i¼1

k X

Rdj :

j¼1

Based on these definitions, the reliability function for a system can be calculated with the following theorem [11]. Theorem 2.5: Let Fr(t) be a life distribution function for an independent subsystem Sr, and Rr(t) = 1 − Fr(t) is the reliability of the rth subsystem, r = 1, 2, . . ., m. The minimal path matrix of a complex system is Am×k, and F1,A(t) is the life function for the complex system. C1,A(t) = 1 − F1,A(t) is the reliability function. Define  T 1 C¼ ¼ ðc1 ; c2 ; . . . ; ck Þ; A where 1 is a vector of 1s, Thus, k X

C1;A ðtÞ ¼

~ C~ j ; RðtÞ

j¼1 T

~ ¼ f 1; R1 ðtÞ; R2 ðtÞ; . . . ; Rm ðtÞg . where RðtÞ In view of the characteristics of the minimal path matrix, the calculation procedure can be cut short; Thus, we can obtain the following corollary. Corollary: According to Zhang et al. [11], suppose that there are ri zeroes in the ith row of A in Theorem 2.5, where i = 1, . . ., m. Let l = max1  i  m ri. Then l X

C1;A ðtÞ ¼

~ C~ j ðtÞ R

j¼1

k X

j

ð 1Þ Ckj

j¼lþ1

m Y

Ri ðtÞ: i¼1

A complex system reliability test based on the empirical likelihood To infer the reliability lower confidence limit or the confidence limit of a CS using subsystem data, we can construct the following hypothesis tests: For a given t, test whether C1,A(t) is not less than C0 or not; that is, ð1Þ H0 : C1;A ðtÞ ¼ C0 vs H1 : C1;A ðtÞ 6¼ C0 ; ð2Þ H0 : C1;A ðtÞ  C0 vs H1 : C1;A ðtÞ < C0 : We use empirical likelihood (EL), which is a nonparametric method introduced by Owen [9, 10, 13], to test the two hypotheses. Here, we first give a short introduction to empirical likelihood. The definitions of the empirical distribution function and the empirical accumulate function are given as follows. Definition 3.1: Let X1 ; X2 ; . . . ; Xn 2 R be independently identically distributed, then the empirical cumulative distribution function of X1, X2, . . ., Xn is Fn ðxÞ ¼

n 1X I ; n i¼1 ðXi xÞ

for −1 < x < 1.

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A Reliability Test of Complex System

Definition 3.2: Let X1 ; X2 ; . . . ; Xn 2 R, be independent and with a common cumulative distribution F, the nonparametric likelihood of the F is LðFÞ ¼ Pni¼1 ðFðXi Þ

FðXi ÞÞ:

Define RðFÞ ¼

LðFÞ : LðFn Þ

Then the empirical likelihood ratio statistic is defined to be ( ) n n X X n t) to replace Rr(t), where Zir is the observed lifetime of the rth subsystem of the ith sample. For any t > 0, we have 8 9 j C tÞ Š r;l g j¼1 l¼1 j

¼

C k X k X j ~c j ð 1Þ Pmr¼1 ½Rr ðtފ r;l j¼1 l¼1

¼

PLOS ONE | DOI:10.1371/journal.pone.0163557 October 19, 2016

C1;A ðtÞ:

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A Reliability Test of Complex System

Under the null hypothesis, we obtain n X wi ½C1;A ðZi ; tÞ

C0 Š ¼ 0:

i¼1

Therefore, a statistic for the two-sided test based on the EL is given as follows: ( ) n n n Y X X