Reliability estimation in Rayleigh distribution based on fuzzy lifetime ...
Int J Syst Assur Eng Manag (Oct-Dec 2014) 5(4):487–494 DOI 10.1007/s13198-013-0190-5
ORIGINAL ARTICLE
Reliability estimation in Rayleigh distribution based on fuzzy lifetime data Abbas Pak • Gholam Ali Parham • Mansour Saraj
Received: 22 March 2013 / Revised: 4 August 2013 / Published online: 27 August 2013 Ó The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2013
Abstract Some work has been done in the past on the estimation of reliability characteristics of Rayleigh distribution based on complete and censored samples. But, traditionally it is assumed that the available data are performed in exact numbers. However, in real world situations, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. In this paper, we present a Bayesian approach to estimate the parameter and reliability function of Rayleigh distribution from fuzzy lifetime data. Based on fuzzy data, the Bayes estimates can not be obtained in explicit forms; therefore, we provide two approximations, namely Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo method to compute the Bayes estimates of the parameter and reliability function. Their performance is then assessed through Monte Carlo simulations. Finally, one real data set is analyzed for illustrative purposes. Keywords Fuzzy lifetime data Rayleigh distribution Reliability estimation Bayesian estimation
A. Pak (&) G. A. Parham Department of Statistics, Shahid Chamran University of Ahvaz, Ahvaz, Iran e-mail: [email protected] G. A. Parham e-mail: [email protected] M. Saraj Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran e-mail: [email protected]
1 Introduction The Rayleigh distribution provides a population model which is useful in several areas of statistics, including life testing and reliability. The density, reliability and hazard (failure rate) functions of the Rayleigh distribution are given, respectively, by 2
f ðt; kÞ ¼ 2ktekt ; 2
t [ 0; k [ 0;
ð1Þ
RðtÞ ¼ ekt ;
t [ 0;
ð2Þ
HðtÞ ¼ 2kt;
t[0
ð3Þ
It is clear from (3) that the probability density function (p.d.f.) of Rayleigh distribution has a linearly increasing failure rate which makes it a suitable model for the lifetime of components that age rapidly with time. From now on Rayleigh distribution with parameter k will be denoted by RayleighðkÞ. Estimation of the reliability of a product requires its lifetime data. Many reliability analysis methods are based on the availability of a large amount of lifetime data. In these methods, the parameters of the lifetime distribution are assumed to be constant but unknown and sample statistics are used as the estimators of these parameters. This requires a relatively large amount of lifetime data. However, in many engineering applications, there may be very few available data points, sometimes only one or two observations. In these cases it is impossible to estimate lifetime distribution parameters with conventional reliability analysis methods. The Bayesian approach has been developed to handle such difficulties. In this approach, the parameters of the lifetime distributions are considered to be random variables themselves. This enables an engineer to systematically combine subjective judgment based on
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intuition, experience, or indirect information with observed data to obtain a balanced estimate and to update the estimate as more information and data become available. This method takes advantage of all available prior information (combining experience, subjective judgment, and available lifetime data). Thus, it can be used even when there are only a few lifetime data points. Several authors have considered Bayesian approach for the estimation purposes of the parameter and other reliability characteristics of Rayleigh distribution based on complete and censored samples. Among others, Chung (1995) obtained the best invariant estimator and the Bayes estimator of the parameter of Rayleigh distribution under entropy loss. Fernandez (2000) presented a Bayesian approach to inference in reliability studies based on doubly type-II censored data from a Rayleigh distribution. Lee et al. (2011) obtained a Bayes estimator under the Rayleigh distribution with the progressive type-II right censored sample. Dey and Maiti (2012) derived Bayes estimator of Rayleigh parameter and its associated risk based on extended Jeffrey’s prior. The above research results for Rayleigh distribution are limited to precise lifetime data. Precisely reported lifetimes are common when data comes from specially designed life tests. In such a case a failure should be precisely defined, and all tested items should be continuously monitored. However, in real situations, these test requirements might not be fulfilled. In these cases, it is sometimes impossible to obtain exact observations of lifetime. The obtained lifetime data may be ‘‘polluted’’ and imprecise most of the time. In addition, restricted by human and other resources in experiment, especially for new equipments, exceptionally long-life equipments, and non-mass-production products, for which there is no comparative reliability information available, more often than not, the lifetime is based upon subjective evaluation or rough estimate. For example, ‘‘The lifetime of a bearing is around 8.17 9 106 revolutions’’ and ‘‘The lifetime of some shaft be between 1,500 and 2,000 h, but near to 2,000 h’’ etc. are imprecise quantities relating to lifetime. The lack of precision of lifetime data can be described using fuzzy sets. The classical statistical estimation methods are not appropriate to deal with such imprecise cases. Therefore, the conventional procedures used for estimating the reliability characteristics of Rayleigh distribution will have to be adapted to the new situation. In recent years, several researchers pay attention to applying the fuzzy sets theory to reliability analysis. Wu (2004) studied the Bayesian system reliability assessment under fuzzy environments. Huang et al. (2006) proposed a new method to determine the membership function of the estimates of the parameters and the reliability function of multiparameter lifetime distributions. Viertl (2009) presented generalized parametric procedures for reliability estimation including fuzzy point estimators and generalized
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Bayesian procedures. Rotshtein (2011) presented a new method to model system reliability based on the algebra of regular algorithms and fuzzy set theory. Kumar and Yadav (2012) introduced a new algorithm to construct the membership function and non-membership function of fuzzy reliability of a system having components following different types of intuitionistic fuzzy failure rates. Wang et al. (2012) presented two system reliability analysis methods for different scenarios of reliability modeling processes. To our best knowledge, there are no reports on Bayesian estimation of the reliability characteristics of Rayleigh distribution based on fuzzy lifetime data. In this paper, we present a Bayesian approach to estimate the parameter and reliability function of Rayleigh distribution when the available data are reported in the form of fuzzy numbers. In Sect. 2, we review the fundamental notations and basic definitions of fuzzy set theory used in the paper. In Sect. 3, we introduce a generalized likelihood function based on fuzzy lifetime data. In Sect. 4, the Bayes estimates of the parameter k and reliability function RðtÞ are obtained by using the approximation forms of Lindley (1980) and Tierney and Kadane (1986) as well as a Markov Chain Monte Carlo technique under the assumption of gamma prior. A Monte Carlo simulation study is presented in Sect. 5, which provides a comparison of all estimation procedures developed in this paper and one real data set is analyzed for illustrative purposes. Finally, conclusions and recommendations are provided in Sect. 6.
2 Basic definitions of fuzzy sets Consider an experiment characterized by a probability space W ¼ ðX; F ; Ph Þ, where ðX; F Þ is a measurable space and Ph belongs to a specified family of probability measures fPh ; h 2 Hg on ðX; FÞ. A fuzzy set A~ in X is characterized by a membership function lA~ðxÞ which associates with each point x in X a real number in the interval ½0; 1, with the value of lA~ðxÞ at x representing the ‘‘grade of membership’’ of x ~ We hereafter assume that the sample space X is a set in a in A. Euclidean space (usually R) and F is the smallest Borel rfield on X. A fuzzy event in X is a fuzzy subset A~ of X, whose membership function lA~ is Borel measurable. Example 1 To evaluate the problem of psychological depression in a population, there is no exact method that can measure and express the exact value for the severity of the disease in each person and, so measurement results may be reported by means of the following imprecise observation: ‘‘The severity of disease is approximately 30–40.’’ A fuzzy approach lies in expressing the preceding observation as a fuzzy event x~ such as that defined, for instance, by the membership function represented in Fig. 1.
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489
μ ~x
f~t1 ; ~t6 g can be immediately constructed by defining 5 P lt~6 ¼ 1 lt~i ).
1
i¼1
On the other hand, according to Zadeh (1968) given the f assoexperiment W ¼ ðX; F ; Ph Þ; h 2 H, and a f.i.s. W ciated with it, each probability measure Ph on ðX; F Þ f defined as follows: induces a probability measure on W 0 10
20
30
40
50
60
x
Fig. 1 Fuzzy approach of the imprecise observation ‘‘approximately 30–40’’
The set consisting of all observable events from the experiment W determines a fuzzy information system associated with it, which is defined as follows. Definition 1 (see Tanaka et al. (1979)). A fuzzy inforf associated with the experiment W mation system (f.i.s.) W ~1 ; x ~K g of W, i.e., a set of K fuzzy is a fuzzy partition fx events on W satisfying the orthogonality condition K X
lx~k ðxÞ ¼ 1
k¼1
~k . where lx~k denotes the membership function of x We now examine a brief example illustrating the preceding concept: Example 2 An investigator is interested in analyzing the time of reaction of persons to a certain stimulus in a psychological experience. Assume that the investigator has not a mechanism of measurement sufficiently precise to determine exactly the time of reaction of persons, but rather he can only approximate them by means of the following fuzzy observations: ~t1 ¼‘‘approximately lower than 15 s’’, ~t2 ¼ ‘‘approximately 25 s’’, ~t3 ¼ ‘‘approximately 35–45 s’’, ~t4 ¼ ‘‘approximately 55 s’’, ~t5 ¼ ‘‘approximately higher than 65 s’’, which are characterized by the membership functions in Fig. 2 (Clearly, a f.i.s. T~ ¼ μ ~t
μ~t
1
μ~t
μ~t
3
2
μ~t
4
5
1
0 15
25
35
45
55
65
t
Fig. 2 Membership functions of the fuzzy observations ~t1 ; ~t2 ; ~ t3 ; ~t4 and ~t5
f induced Definition 2 The probability distribution on W f to ½0; 1 such that by Ph is the mapping P h from W Z f ~ ¼ lx~ ðxÞdPh ðxÞ; ~2 W P h ðxÞ x ð4Þ X
In order to model imprecise lifetimes, a generalization of real numbers is necessary. These lifetimes can be represented by fuzzy numbers. A fuzzy number is a subset, denoted by x~, of the set of real numbers (denoted by R) and is characterized by the so called membership function lx~ð:Þ. Fuzzy numbers satisfy the following constraints (Dubois and Prade (1980)): (1) lx~ : R ! ½0; 1 is Borel-measurable; (2) 9x0 2 R : lx~ðx0 Þ ¼ 1; (3) The so-called kcuts (0\k 1), defined as Bk ðx~Þ ¼ fx 2 R : lx~ðxÞ kg, are all closed interval, i.e., Bk ð~ xÞ ¼ ½ak ; bk ; 8k 2 ð0; 1. With the definition of a fuzzy number given above, an exact (non-fuzzy) number can be treated as a special case of a fuzzy number. For a non-fuzzy real observationx0 2 R, its corresponding membership function is lx0 ðx0 Þ ¼ 1. Among the various types of fuzzy numbers, the triangular and trapezoidal fuzzy numbers are most convenient and useful in describing fuzzy lifetime data. The triangular fuzzy number can be defined as x~ ¼ ða; b; cÞ and its membership function is defined by the following expression: 8xa > a x b; > > > > c b > > : 0 otherwise: The trapezoidal fuzzy number can be defined as x~ ¼ ða; b; c; gÞ with the membership function 8xa > a x b; > > > ba > > > < 1 bxc lx~ðxÞ ¼ g x > > c x g; > > gc > > > : 0 otherwise:
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3 Fuzzy data and likelihood function Suppose that n identical units are placed on a life test with the corresponding lifetimes X1 ; ; Xn . It is assumed that these variables are independent and identically distributed as RayleighðkÞ. Let X ¼ ðX1 ; ; Xn Þ denotes the vector of complete lifetimes. If a realization x of X was known exactly, we could obtain the complete-data likelihood function as n P 2 Lðx; hÞ ¼ ð2kxÞn e
k
xi
ð5Þ
i¼1
Now consider the problem where x is not observed precisely and only partial information about x is available in the form of a fuzzy subset x~ with the Borel measurable membership function lx~ðxÞ. In this setting, the fuzzy observation x~ can be understood as encoding the observer’s partial knowledge about the realization x of random vector X, and the membership function lx~ðxÞ is seen as a possibility distribution interpreted as a soft constraint on the unknown quantity x. The fuzzy set x~ can be considered to be generated by a two-step process: (1) A realization x is drawn from X; (2) The observer encodes his/her partial knowledge of x in the form of a possibility distribution lx~. Example 3 Consider a case study on the light emitting diodes (LEDs) manufacturing process that focuses on the lifetime of LED sources. The unknown lifetime xi of diode i may be regarded as a realization of a random variable Xi induced by random sampling from the stable process. Since the data given by luminous intensity of a particular LED inevitably have some degree of imprecision, the production engineers suggest to determine the lifetime of each diode i in the form of lower bounds ai and upper bounds ci , as well as a point estimate bi . This information may be encoded as a triangular fuzzy number x~i ¼ ðai ; bi ; ci Þ, and is interpreted as a possibility distribution related to the unknown value xi , itself a realization of the random variable Xi . Information about x may be represented by the joint possibility distribution lx~ðxÞ ¼ lx~1 ðx1 Þ lx~n ðxn Þ:
ð6Þ
Once x~ is given, and assuming the joint membership function lx~ðxÞ to be decomposable as in (6), we can obtain the observed-data likelihood function by using the expression (4) as follows: ‘ð~ x; kÞ ¼ ð2kÞn
n Z Y
2
xekx lx~i ðxÞdx:
ð7Þ
i¼1
4 Bayesian estimation
received frequent attention for statistical inference. In this section, we consider the Bayesian estimation of the unknown parameter k and reliability function RðtÞ. As a conjugate prior fork, we take the Gammaða; bÞ density with p.d.f. given by pðkÞ ¼
ba a1 kb k e ; CðaÞ
ð8Þ
k [ 0;
where a [ 0 and b [ 0. Based on this prior and the likelihood function (7), the posterior density function of k given the data can be written as follows: pðkjx~Þ ¼
pðkÞ‘ð~ x; kÞ R1
ð9Þ
:
pðkÞ‘ð~ x; kÞdk
0
Then, under a squared error loss function, the Bayes estimate of any function of k, saygðkÞ, is R1 R1 gðkÞpðkÞ‘ð~ x; kÞdk gðkÞeQðkÞ dk 0 0 EðgðkÞjx~Þ ¼ R1 ¼ R1 ; ð10Þ QðkÞ pðkÞ‘ð~ x; kÞdk e dk 0
0
where QðkÞ ¼ ln pðkÞ þ ln ‘ð~ x; kÞ qðkÞ þ L ðkÞ. Note that Eq. (10) cannot be obtained analytically. Therefore, in the following, we adopt two approximations-Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo (MCMC) method for approximating (10). 4.1 Lindley’s approximation Lindley (1980) first proposed his procedure to approximate the ratio of two integrals such as (10). For the oneparameter case, Lindley’s approximation of (10) is the form 1 1 gðkÞ þ g11 r11 þ q1 g1 r11 þ L 3 r211 g1 ; 2 2
ð11Þ
where dgðkÞ d 2 gðkÞ dqðkÞ ; g11 ¼ ; ; q1 ¼ dk dk dk2 2 1 o3 L ðkÞ o L ðkÞ L3 ¼ ; r11 ¼ : 3 ok ok2
g1 ¼
Evaluating all the expressions in (11) at the MLE of k produces the approximation g^B to (10). In our case, we have Z n X 2 log 2xekx lx~i ðxÞdx: ð12Þ L ðkÞ ¼ n log k þ i¼1
In recent decades, the Bayes viewpoint, as a powerful and valid alternative to traditional statistical perspectives, has
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Equating the partial derivative of the log-likelihood (12) with respect to k to zero, the resulting equations is:
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R 3 kx2 n lx~i ðxÞdx x e oL ðkÞ n X R ¼ ¼ 0: ok k i¼1 xekx2 lx~i ðxÞdx
491
ð13Þ
Since there is no closed form of the solution to the likelihood equation (13), an iterative numerical search can be used to obtain the MLE. In the following, we describe the Newton–Raphson method to determine the MLE of the parameter k. Newton–Raphson algorithm is a direct approach for estimating the relevant parameters in a likelihood function. In this algorithm, the solution of the likelihood equation is obtained through an iterative procedure. Let k^ðhÞ be the parameter value from the hth step. Then, at the ðh þ 1Þth step of iteration process, the updated parameter is obtained as k^ðhþ1Þ ¼ k^ðhÞ
o ok L ðkÞjh o2 L ðkÞjh ok2
ð14Þ
;
where the notation Ajh , for any partial derivative A, means the partial derivative evaluated at k^ðhÞ . The second-order derivative of the log-likelihood with respect to the parameter, required for proceeding with the Newton– Raphson method, is obtained as follows o2 n L ðkÞ ¼ 2 k 8 ok2 "R #2 9 R 5 kx2 2 n < X lx~i ðxÞdx x e x3 ekx lx~i ðxÞdx = R þ : R kx2 : xekx2 lx~i ðxÞdx lx~i ðxÞdx ; xe i¼1
The iteration process then continues until convergence, i.e., until k^ðhþ1Þ k^ðhÞ \e, for some pre-fixed e [ 0. The maximum likelihood estimate of k via Newton–Raphson ^ in this paper. algorithm is thereafter refereed as ‘‘k’’ Now, to apply Lindley’s form in (11), we first obtain n r11 ¼ 2 ^ k 8 "R #2 9 R 5 kx ^2 ^2 n < X lx~i ðxÞdx x e x3 ekx lx~i ðxÞdx = R ; þ R ^2 ^2 : xekx lx~ ðxÞdx lx~ ðxÞdx ; xekx i¼1 i
i
The approximate Bayes estimate of k, say k^B , for the squared error loss function is the posterior mean of gðkÞ ¼ k, which is by (11) as follows 1 k^B ¼ k^ þ q1 r11 þ L 3 r211 : 2
ð15Þ
where q1 ¼ a1 b. Similarly, for the posterior mean of k^ 2
RðtÞ, we have gðkÞ ¼ ekt and 1 4 1 2 ^2 kt 2 ^ RB ðtÞ ¼ e 1 þ t r11 t q1 r11 þ L3 r11 : 2 2
ð16Þ
4.2 Tierney and Kadane’s approximation Setting HðkÞ ¼ QðkÞ=n and H ðkÞ ¼ ½ln gðkÞ þ QðkÞ=n, the expression in (10) can be re-expressed as R1 nH ðkÞ e dk EðgðkÞjx~Þ ¼ 0R1 : ð17Þ nHðkÞ e dk 0
Following Tierney and Kadane (1986), Eq. (17) can be approximated as 12
f ; g^BT ðkÞ ¼ exp n½H ðk Þ HðkÞ ð18Þ f where k and k maximize H ðkÞ and HðkÞ, respectively, and f and f are minus the inverse of the second derivatives respectively. of H ðkÞ and HðkÞ at k and k, In our case, we have ( 1 HðkÞ ¼ k þ ðn þ a 1Þ log k kb n ) Z n X 2 log 2xekx lx~i ðxÞdx ; þ i¼1
where k is a constant; therefore, k can be obtained by solving the following equation: ( ) R 3 kx2 n X lx~i ðxÞdx x e oHðkÞ 1 n þ a 1 R ¼ b ¼ 0; ok n k xekx2 lx~i ðxÞdx i¼1
and (R ^2 n x7 ekx lx~i ðxÞdx 2n X ¼ 3 R ^2 k^ lx~i ðxÞdx xekx i¼1 ) R 3 kx R ^2 ^2 ð x e lx~i ðxÞdxÞð x5 ekx lx~i ðxÞdxÞ þ R ^2 lx~i ðxÞdxÞ2 ð xekx (R R 5 kx 2 ^ ^2 n X lx~i ðxÞdx x3 ekx lx~i ðxÞdx x e R þ2 R ^2 ^2 kx kx lx~i ðxÞdx lx~i ðxÞdx xe xe i¼1 "R #2 !) ^2 3 kx lx~i ðxÞdx x e : R ^2 kx lx~i ðxÞdx xe
L 3
and from the second derivative of HðkÞ, we have 2 1 o HðkÞ ; f¼ j k¼k ok2 where
o2 HðkÞ 1 nþa1 j ¼ k¼k n k2 ok2 0R "R # 2 19 2 2 n = 5 kx 3 kx X e l ðxÞdx e l ðxÞdx x x x~i x~i @R A R þ 2 2 ; xekx xekx lx~i ðxÞdx lx~i ðxÞdx i¼1
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4.3 MCMC method In this subsection, we consider Markov Chain Monte Carlo technique such as Metropolis–Hastings algorithm to draw samples from the posterior density function and then compute the Bayes estimates. Based on the gamma prior k Gammaða; bÞ and the likelihood function (7), the posterior p.d.f. of k can be written as n Z Y 2 pðkjx~Þ / knþa1 ekb xekx lx~i ðxÞdx: ð19Þ i¼1
Noting that the density function pðkjx~Þ is not known, but by experimentation, we observed that it is similar to normal distribution. So to generate random samples from pðkjx~Þ, we can use the Metropolis–Hastings algorithm with normal proposal distribution as follows: Step 1: Start with an initial guess kð0Þ . Step 2: Set h ¼ 1. Step 3: Generate kðhÞ from pðkjx~Þ using Metropolis– Hastings algorithm with the proposal distribution ^ 1Þ, where Ið:Þ is the indicator qðkÞ Iðk [ 0ÞNðk; function, as follows: ^ (a). Let m ¼ kðh1Þ . Here we set kð0Þ k. (b). Generate u from the proposal distribution q. n o jx~ÞqðmÞ (c). Let pðm; uÞ ¼ min 1; pðu pðmjx~ÞqðuÞ . (d). Accept u with probability pðm; uÞ or accept m with probability 1 pðm; uÞ. ðhÞ 2
Step 4: Compute RðhÞ ðtÞ ¼ ek t . Step 5: Set h ¼ h þ 1. Step 6: Repeat Steps 3–5, K times and obtain kðhÞ and RðhÞ ðtÞ for h ¼ 1; ; K. Now the Bayes estimates of the parameter k and reliability function RðtÞ with respect to squared error loss function become K X ^ jx~Þ ¼ 1 k^BM ¼ Eðk kðhÞ K h¼1
and
1
membership degree
Now, following the same argument with gðkÞ ¼ k and 2 gðkÞ ¼ ekt , respectively, in H ðkÞ, k^BT and R^BT in Eq. (18) can then be obtained straightforwardly.
0 0.05
0.25
0.5
0.75
1
1.5
2
3
x
Fig. 3 Fuzzy information system used to encode the simulated data
5 Numerical study In this section, we present some experimental results, mainly to observe how the different methods behave for different sample sizes. First, for fixed k ¼ 1 and different choices of n, we have generated i.i.d random samples from the Rayleigh distribution. Then each realization of x was fuzzified using the f.i.s. shown in Fig. 3, and the Bayes estimates of the parameter k and reliability function RðtÞ, at t ¼ 1, for the fuzzy sample were computed using the Lindley’s approximation (Lindley), Tierney and Kadane’s approximation (T&K), and MCMC technique. We have assumed that k has Gammaða; bÞ prior. To make the comparison meaningful, it is assumed that the priors are non-informative, and they are a ¼ b ¼ 0. Note that in this case the priors are non-proper also. Press (2001) suggested using very small non-negative values of the hyperparameters in this case, and it will make the priors proper. We have tried a ¼ b ¼ 0:0001. The results are not significantly different than the corresponding results obtained using nonproper priors, and are not reported due to space. The average values and mean squared errors of the estimates over 5,000 replications are presented in Tables 1 and 2. From the experiments, we found that using Lindley’s approximation or MCMC technique for the computation of Bayes estimates give similar estimation results, but Lindley’s approximation is computationally slower. For all the methods, it is observed that as the sample size increases, the MSEs of the estimates decrease. In terms of minimum MSEs, the performance of the T&K estimates for the parameter k and reliability function RðtÞ is generally best followed by the Lindley and MCMC estimates.
6 Application example
K 1X ^ R^BM ðtÞ ¼ EðRðtÞ jx~Þ ¼ RðhÞ ðtÞ; K h¼1
respectively.
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To demonstrate the application of proposed methods to real data, let us consider the data collected during the experiment reported by Pak et al. (2013). In this experiment, a
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493
Table 1 Average values (AV) and mean squared errors (MSE) of the Bayes estimates of k for different sample sizes n
k^BT
k^B
k^BM
AV
MSE
AV
MSE
AV
MSE
15
1.1678
0.1296
1.1622
0.1259
1.1682
0.1298
20 25
1.1639 1.1448
0.1148 0.0881
1.1608 1.1410
0.1121 0.0853
1.1638 1.1451
0.1146 0.0883
30
1.1376
0.0687
1.1334
0.0645
1.1374
0.0685
40
1.1319
0.0557
1.1285
0.0523
1.1317
0.0556
50
1.1235
0.0463
1.1203
0.0437
1.1239
0.0460
70
1.1136
0.0319
1.1114
0.0304
1.1134
0.0316
1e-47
π (λ x~ ) 5e-48
0e+00 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
λ
Table 2 Average values (AV) and mean squared errors (MSE) of the Bayes estimates of reliability function RðtÞ for different sample sizes n
R^B
R^BT
R^BM
AV
MSE
AV
MSE
AV
MSE
15
0.3306
0.0112
0.3325
0.0102
0.3308
0.0114
20
0.3311
0.0085
0.3343
0.0075
0.3314
0.0086
25
0.3328
0.0077
0.3352
0.0066
0.3332
0.0079
30
0.3341
0.0066
0.3363
0.0051
0.3346
0.0067
40
0.3357
0.0054
0.3378
0.0040
0.3354
0.0053
50
0.3364
0.0043
0.3384
0.0032
0.3365
0.0046
70
0.3386
0.0035
0.3397
0.0029
0.3382
0.0034
sample of 25 ball bearings is placed on a life test. A ball bearing may work perfectly over a certain period but be breaking for some time and finally be unusable at a certain time. So, the observed failure times of the ball bearings are reported by fuzzy numbers x~i ¼ ðai ; xi ; bi Þ, where ai ¼ 0:05xi , bi ¼ 0:03xi and the values of xi ; i ¼ 1; ; 25; are presented in Table 3. The corresponding membership functions of these fuzzy numbers are defined as
Table 3 Failures of 25 ball bearings data
Fig. 4 Plot of the posterior function pðkjx~Þ for the fuzzy sample x~
8 x ðxi ai Þ > > < ai lx~i ðxÞ ¼ x þ bi x > i > : bi
x i ai x x i ; i ¼ 1; ; 25: x i x x i þ bi ;
For these fuzzy data, the maximum likelihood estimate of k is found from (14) to be k^ ¼ 0:00014. Therefore, the Bayes estimate of k and reliability function RðtÞ; at t ¼ 1, under the Gammað2; 2Þ prior and using Lindley’s approximation become k^B ¼ 0:000158 and R^B ¼ 0:9998, respectively. Also, by using the Tierney and Kadane’s method, the Bayes estimates are k^BT ¼ 0:000143 and R^BT ¼ 0:9981. The plot of the posterior function pðkjx~Þ for these data is presented in Fig. 4. It is easily seen that this plot is quite similar to Normal distribution. Thus, we can use Metropolis–Hastings algorithm with the proposal ^ 1Þ and compute the approximate Bayes distribution Nðk; estimates of k and reliability function RðtÞ as k^BM ¼ 0:000156 and R^BM ¼ 0:9996; respectively.
i
1
2
3
4
5
6
7
8
9
10
xi
17.88
28.92
33.00
41.52
42.12
45.60
48.48
51.84
51.96
54.12
i
11
12
13
14
15
16
17
18
19
20
xi
55.56
7.80
67.80
67.80
68.64
68.64
68.88
84.12
93.12
98.64
i xi
21 105.12
22 105.84
23 127.92
24 128.04
25 173.40
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7 Conclusions Classical reliability estimation in Rayleigh distribution is based on precise lifetime data. It is usually assumed that observed lifetime data are precise real numbers. However, in real world situations, the results of the experimental performance can not always be recorded or measured precisely, but each observable event may only be identified with a fuzzy subset of the sample space. In this paper, we have discussed the Bayesian estimation procedure for the parameter and reliability function of Rayleigh distribution when the lifetime observations are fuzzy numbers. We have presented two approximations, namely Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo technique to compute the Bayes estimates. We have then carried out a simulation study to assess the performance of these procedures. Based on the results of the simulation study, we see clearly that, the T&K procedure gives the most precise parameter estimates as shown by MSEs in Tables 1 and 2. Thus, it would seem reasonable to recommend the use of the Tierney and Kadane’s approximation for estimating the unknown parameter k and reliability function RðtÞ from the Rayleigh distribution.
References Chung Y (1995) Estimation of scale parameter from Rayleigh distribution under entropy loss. J Appl Math Comput 2(1):33–40 Dey S, Maiti SS (2012) Bayesian estimation of the parameter of Rayleigh distribution under the extended Jeffrey’s prior. Electron J Appl Statist Anal 5(1):44–59
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Int J Syst Assur Eng Manag (Oct-Dec 2014) 5(4):487–494 Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New York Fernandez AJ (2000) Bayesian inference from type II doubly censored Rayleigh data. Stat Probabil Lett 48:393–399 Huang H, Zuo M, Sun Z (2006) Bayesian reliability analysis for fuzzy lifetime data. Fuzzy Sets Syst 157:1674–1686 Kumar M, Yadav SP (2012) A novel approach for analyzing fuzzy system reliability using different types of intuitionistic fuzzy failure rates of components. ISA Transactions 51:288–297 Lee W, Wu J, Hong M, Lin L, Chan R (2011) Assessing the lifetime performance index of Rayleigh products based on the Bayesian estimation under progressive type II right censored samples. J Comput Appl Math 235:1676–1688 Lindley DV (1980) Approximate Bayesian method. Trabajos de Estadistica 31:223–237 Pak A, Parham GA, Saraj M (2013) On estimation of Rayleigh scale parameter under doubly type II censoring from imprecise data. J Data Sci 11:303–320 Press SJ (2001) The subjectivity of scientists and the Bayesian approach. Wiley, New York Rotshtein AP (2011) Fuzzy-Algorithmic reliability analysis of complex systems. Cybern Syst Anal 47(6):919–931 Tanaka H, Okuda T, Asai K (1979) Fuzzy information and decision in statistical model. In: Gupta MM (ed) Advances in fuzzy sets theory and applications. North-Holland, Amsterdam, pp 303–320 Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Amer Statist Assoc 81:82–86 Viertl R (2009) On reliability estimation based on fuzzy lifetime data. J Stat Plan Infer 139:1750–1755 Wang Z, Huang HZ, Li Y, Pang Y, Xiao NC (2012) An approach to system reliability analysis with fuzzy random variables. Mech Mach Theory 52:35–46 Wu HC (2004) Bayesian system reliability assessment under fuzzy environments. Reliab Eng Syst Safe 83:277–286 Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 10:421–427
ORIGINAL ARTICLE
Reliability estimation in Rayleigh distribution based on fuzzy lifetime data Abbas Pak • Gholam Ali Parham • Mansour Saraj
Received: 22 March 2013 / Revised: 4 August 2013 / Published online: 27 August 2013 Ó The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2013
Abstract Some work has been done in the past on the estimation of reliability characteristics of Rayleigh distribution based on complete and censored samples. But, traditionally it is assumed that the available data are performed in exact numbers. However, in real world situations, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. In this paper, we present a Bayesian approach to estimate the parameter and reliability function of Rayleigh distribution from fuzzy lifetime data. Based on fuzzy data, the Bayes estimates can not be obtained in explicit forms; therefore, we provide two approximations, namely Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo method to compute the Bayes estimates of the parameter and reliability function. Their performance is then assessed through Monte Carlo simulations. Finally, one real data set is analyzed for illustrative purposes. Keywords Fuzzy lifetime data Rayleigh distribution Reliability estimation Bayesian estimation
A. Pak (&) G. A. Parham Department of Statistics, Shahid Chamran University of Ahvaz, Ahvaz, Iran e-mail: [email protected] G. A. Parham e-mail: [email protected] M. Saraj Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran e-mail: [email protected]
1 Introduction The Rayleigh distribution provides a population model which is useful in several areas of statistics, including life testing and reliability. The density, reliability and hazard (failure rate) functions of the Rayleigh distribution are given, respectively, by 2
f ðt; kÞ ¼ 2ktekt ; 2
t [ 0; k [ 0;
ð1Þ
RðtÞ ¼ ekt ;
t [ 0;
ð2Þ
HðtÞ ¼ 2kt;
t[0
ð3Þ
It is clear from (3) that the probability density function (p.d.f.) of Rayleigh distribution has a linearly increasing failure rate which makes it a suitable model for the lifetime of components that age rapidly with time. From now on Rayleigh distribution with parameter k will be denoted by RayleighðkÞ. Estimation of the reliability of a product requires its lifetime data. Many reliability analysis methods are based on the availability of a large amount of lifetime data. In these methods, the parameters of the lifetime distribution are assumed to be constant but unknown and sample statistics are used as the estimators of these parameters. This requires a relatively large amount of lifetime data. However, in many engineering applications, there may be very few available data points, sometimes only one or two observations. In these cases it is impossible to estimate lifetime distribution parameters with conventional reliability analysis methods. The Bayesian approach has been developed to handle such difficulties. In this approach, the parameters of the lifetime distributions are considered to be random variables themselves. This enables an engineer to systematically combine subjective judgment based on
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intuition, experience, or indirect information with observed data to obtain a balanced estimate and to update the estimate as more information and data become available. This method takes advantage of all available prior information (combining experience, subjective judgment, and available lifetime data). Thus, it can be used even when there are only a few lifetime data points. Several authors have considered Bayesian approach for the estimation purposes of the parameter and other reliability characteristics of Rayleigh distribution based on complete and censored samples. Among others, Chung (1995) obtained the best invariant estimator and the Bayes estimator of the parameter of Rayleigh distribution under entropy loss. Fernandez (2000) presented a Bayesian approach to inference in reliability studies based on doubly type-II censored data from a Rayleigh distribution. Lee et al. (2011) obtained a Bayes estimator under the Rayleigh distribution with the progressive type-II right censored sample. Dey and Maiti (2012) derived Bayes estimator of Rayleigh parameter and its associated risk based on extended Jeffrey’s prior. The above research results for Rayleigh distribution are limited to precise lifetime data. Precisely reported lifetimes are common when data comes from specially designed life tests. In such a case a failure should be precisely defined, and all tested items should be continuously monitored. However, in real situations, these test requirements might not be fulfilled. In these cases, it is sometimes impossible to obtain exact observations of lifetime. The obtained lifetime data may be ‘‘polluted’’ and imprecise most of the time. In addition, restricted by human and other resources in experiment, especially for new equipments, exceptionally long-life equipments, and non-mass-production products, for which there is no comparative reliability information available, more often than not, the lifetime is based upon subjective evaluation or rough estimate. For example, ‘‘The lifetime of a bearing is around 8.17 9 106 revolutions’’ and ‘‘The lifetime of some shaft be between 1,500 and 2,000 h, but near to 2,000 h’’ etc. are imprecise quantities relating to lifetime. The lack of precision of lifetime data can be described using fuzzy sets. The classical statistical estimation methods are not appropriate to deal with such imprecise cases. Therefore, the conventional procedures used for estimating the reliability characteristics of Rayleigh distribution will have to be adapted to the new situation. In recent years, several researchers pay attention to applying the fuzzy sets theory to reliability analysis. Wu (2004) studied the Bayesian system reliability assessment under fuzzy environments. Huang et al. (2006) proposed a new method to determine the membership function of the estimates of the parameters and the reliability function of multiparameter lifetime distributions. Viertl (2009) presented generalized parametric procedures for reliability estimation including fuzzy point estimators and generalized
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Bayesian procedures. Rotshtein (2011) presented a new method to model system reliability based on the algebra of regular algorithms and fuzzy set theory. Kumar and Yadav (2012) introduced a new algorithm to construct the membership function and non-membership function of fuzzy reliability of a system having components following different types of intuitionistic fuzzy failure rates. Wang et al. (2012) presented two system reliability analysis methods for different scenarios of reliability modeling processes. To our best knowledge, there are no reports on Bayesian estimation of the reliability characteristics of Rayleigh distribution based on fuzzy lifetime data. In this paper, we present a Bayesian approach to estimate the parameter and reliability function of Rayleigh distribution when the available data are reported in the form of fuzzy numbers. In Sect. 2, we review the fundamental notations and basic definitions of fuzzy set theory used in the paper. In Sect. 3, we introduce a generalized likelihood function based on fuzzy lifetime data. In Sect. 4, the Bayes estimates of the parameter k and reliability function RðtÞ are obtained by using the approximation forms of Lindley (1980) and Tierney and Kadane (1986) as well as a Markov Chain Monte Carlo technique under the assumption of gamma prior. A Monte Carlo simulation study is presented in Sect. 5, which provides a comparison of all estimation procedures developed in this paper and one real data set is analyzed for illustrative purposes. Finally, conclusions and recommendations are provided in Sect. 6.
2 Basic definitions of fuzzy sets Consider an experiment characterized by a probability space W ¼ ðX; F ; Ph Þ, where ðX; F Þ is a measurable space and Ph belongs to a specified family of probability measures fPh ; h 2 Hg on ðX; FÞ. A fuzzy set A~ in X is characterized by a membership function lA~ðxÞ which associates with each point x in X a real number in the interval ½0; 1, with the value of lA~ðxÞ at x representing the ‘‘grade of membership’’ of x ~ We hereafter assume that the sample space X is a set in a in A. Euclidean space (usually R) and F is the smallest Borel rfield on X. A fuzzy event in X is a fuzzy subset A~ of X, whose membership function lA~ is Borel measurable. Example 1 To evaluate the problem of psychological depression in a population, there is no exact method that can measure and express the exact value for the severity of the disease in each person and, so measurement results may be reported by means of the following imprecise observation: ‘‘The severity of disease is approximately 30–40.’’ A fuzzy approach lies in expressing the preceding observation as a fuzzy event x~ such as that defined, for instance, by the membership function represented in Fig. 1.
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489
μ ~x
f~t1 ; ~t6 g can be immediately constructed by defining 5 P lt~6 ¼ 1 lt~i ).
1
i¼1
On the other hand, according to Zadeh (1968) given the f assoexperiment W ¼ ðX; F ; Ph Þ; h 2 H, and a f.i.s. W ciated with it, each probability measure Ph on ðX; F Þ f defined as follows: induces a probability measure on W 0 10
20
30
40
50
60
x
Fig. 1 Fuzzy approach of the imprecise observation ‘‘approximately 30–40’’
The set consisting of all observable events from the experiment W determines a fuzzy information system associated with it, which is defined as follows. Definition 1 (see Tanaka et al. (1979)). A fuzzy inforf associated with the experiment W mation system (f.i.s.) W ~1 ; x ~K g of W, i.e., a set of K fuzzy is a fuzzy partition fx events on W satisfying the orthogonality condition K X
lx~k ðxÞ ¼ 1
k¼1
~k . where lx~k denotes the membership function of x We now examine a brief example illustrating the preceding concept: Example 2 An investigator is interested in analyzing the time of reaction of persons to a certain stimulus in a psychological experience. Assume that the investigator has not a mechanism of measurement sufficiently precise to determine exactly the time of reaction of persons, but rather he can only approximate them by means of the following fuzzy observations: ~t1 ¼‘‘approximately lower than 15 s’’, ~t2 ¼ ‘‘approximately 25 s’’, ~t3 ¼ ‘‘approximately 35–45 s’’, ~t4 ¼ ‘‘approximately 55 s’’, ~t5 ¼ ‘‘approximately higher than 65 s’’, which are characterized by the membership functions in Fig. 2 (Clearly, a f.i.s. T~ ¼ μ ~t
μ~t
1
μ~t
μ~t
3
2
μ~t
4
5
1
0 15
25
35
45
55
65
t
Fig. 2 Membership functions of the fuzzy observations ~t1 ; ~t2 ; ~ t3 ; ~t4 and ~t5
f induced Definition 2 The probability distribution on W f to ½0; 1 such that by Ph is the mapping P h from W Z f ~ ¼ lx~ ðxÞdPh ðxÞ; ~2 W P h ðxÞ x ð4Þ X
In order to model imprecise lifetimes, a generalization of real numbers is necessary. These lifetimes can be represented by fuzzy numbers. A fuzzy number is a subset, denoted by x~, of the set of real numbers (denoted by R) and is characterized by the so called membership function lx~ð:Þ. Fuzzy numbers satisfy the following constraints (Dubois and Prade (1980)): (1) lx~ : R ! ½0; 1 is Borel-measurable; (2) 9x0 2 R : lx~ðx0 Þ ¼ 1; (3) The so-called kcuts (0\k 1), defined as Bk ðx~Þ ¼ fx 2 R : lx~ðxÞ kg, are all closed interval, i.e., Bk ð~ xÞ ¼ ½ak ; bk ; 8k 2 ð0; 1. With the definition of a fuzzy number given above, an exact (non-fuzzy) number can be treated as a special case of a fuzzy number. For a non-fuzzy real observationx0 2 R, its corresponding membership function is lx0 ðx0 Þ ¼ 1. Among the various types of fuzzy numbers, the triangular and trapezoidal fuzzy numbers are most convenient and useful in describing fuzzy lifetime data. The triangular fuzzy number can be defined as x~ ¼ ða; b; cÞ and its membership function is defined by the following expression: 8xa > a x b; > > > > c b > > : 0 otherwise: The trapezoidal fuzzy number can be defined as x~ ¼ ða; b; c; gÞ with the membership function 8xa > a x b; > > > ba > > > < 1 bxc lx~ðxÞ ¼ g x > > c x g; > > gc > > > : 0 otherwise:
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3 Fuzzy data and likelihood function Suppose that n identical units are placed on a life test with the corresponding lifetimes X1 ; ; Xn . It is assumed that these variables are independent and identically distributed as RayleighðkÞ. Let X ¼ ðX1 ; ; Xn Þ denotes the vector of complete lifetimes. If a realization x of X was known exactly, we could obtain the complete-data likelihood function as n P 2 Lðx; hÞ ¼ ð2kxÞn e
k
xi
ð5Þ
i¼1
Now consider the problem where x is not observed precisely and only partial information about x is available in the form of a fuzzy subset x~ with the Borel measurable membership function lx~ðxÞ. In this setting, the fuzzy observation x~ can be understood as encoding the observer’s partial knowledge about the realization x of random vector X, and the membership function lx~ðxÞ is seen as a possibility distribution interpreted as a soft constraint on the unknown quantity x. The fuzzy set x~ can be considered to be generated by a two-step process: (1) A realization x is drawn from X; (2) The observer encodes his/her partial knowledge of x in the form of a possibility distribution lx~. Example 3 Consider a case study on the light emitting diodes (LEDs) manufacturing process that focuses on the lifetime of LED sources. The unknown lifetime xi of diode i may be regarded as a realization of a random variable Xi induced by random sampling from the stable process. Since the data given by luminous intensity of a particular LED inevitably have some degree of imprecision, the production engineers suggest to determine the lifetime of each diode i in the form of lower bounds ai and upper bounds ci , as well as a point estimate bi . This information may be encoded as a triangular fuzzy number x~i ¼ ðai ; bi ; ci Þ, and is interpreted as a possibility distribution related to the unknown value xi , itself a realization of the random variable Xi . Information about x may be represented by the joint possibility distribution lx~ðxÞ ¼ lx~1 ðx1 Þ lx~n ðxn Þ:
ð6Þ
Once x~ is given, and assuming the joint membership function lx~ðxÞ to be decomposable as in (6), we can obtain the observed-data likelihood function by using the expression (4) as follows: ‘ð~ x; kÞ ¼ ð2kÞn
n Z Y
2
xekx lx~i ðxÞdx:
ð7Þ
i¼1
4 Bayesian estimation
received frequent attention for statistical inference. In this section, we consider the Bayesian estimation of the unknown parameter k and reliability function RðtÞ. As a conjugate prior fork, we take the Gammaða; bÞ density with p.d.f. given by pðkÞ ¼
ba a1 kb k e ; CðaÞ
ð8Þ
k [ 0;
where a [ 0 and b [ 0. Based on this prior and the likelihood function (7), the posterior density function of k given the data can be written as follows: pðkjx~Þ ¼
pðkÞ‘ð~ x; kÞ R1
ð9Þ
:
pðkÞ‘ð~ x; kÞdk
0
Then, under a squared error loss function, the Bayes estimate of any function of k, saygðkÞ, is R1 R1 gðkÞpðkÞ‘ð~ x; kÞdk gðkÞeQðkÞ dk 0 0 EðgðkÞjx~Þ ¼ R1 ¼ R1 ; ð10Þ QðkÞ pðkÞ‘ð~ x; kÞdk e dk 0
0
where QðkÞ ¼ ln pðkÞ þ ln ‘ð~ x; kÞ qðkÞ þ L ðkÞ. Note that Eq. (10) cannot be obtained analytically. Therefore, in the following, we adopt two approximations-Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo (MCMC) method for approximating (10). 4.1 Lindley’s approximation Lindley (1980) first proposed his procedure to approximate the ratio of two integrals such as (10). For the oneparameter case, Lindley’s approximation of (10) is the form 1 1 gðkÞ þ g11 r11 þ q1 g1 r11 þ L 3 r211 g1 ; 2 2
ð11Þ
where dgðkÞ d 2 gðkÞ dqðkÞ ; g11 ¼ ; ; q1 ¼ dk dk dk2 2 1 o3 L ðkÞ o L ðkÞ L3 ¼ ; r11 ¼ : 3 ok ok2
g1 ¼
Evaluating all the expressions in (11) at the MLE of k produces the approximation g^B to (10). In our case, we have Z n X 2 log 2xekx lx~i ðxÞdx: ð12Þ L ðkÞ ¼ n log k þ i¼1
In recent decades, the Bayes viewpoint, as a powerful and valid alternative to traditional statistical perspectives, has
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Equating the partial derivative of the log-likelihood (12) with respect to k to zero, the resulting equations is:
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R 3 kx2 n lx~i ðxÞdx x e oL ðkÞ n X R ¼ ¼ 0: ok k i¼1 xekx2 lx~i ðxÞdx
491
ð13Þ
Since there is no closed form of the solution to the likelihood equation (13), an iterative numerical search can be used to obtain the MLE. In the following, we describe the Newton–Raphson method to determine the MLE of the parameter k. Newton–Raphson algorithm is a direct approach for estimating the relevant parameters in a likelihood function. In this algorithm, the solution of the likelihood equation is obtained through an iterative procedure. Let k^ðhÞ be the parameter value from the hth step. Then, at the ðh þ 1Þth step of iteration process, the updated parameter is obtained as k^ðhþ1Þ ¼ k^ðhÞ
o ok L ðkÞjh o2 L ðkÞjh ok2
ð14Þ
;
where the notation Ajh , for any partial derivative A, means the partial derivative evaluated at k^ðhÞ . The second-order derivative of the log-likelihood with respect to the parameter, required for proceeding with the Newton– Raphson method, is obtained as follows o2 n L ðkÞ ¼ 2 k 8 ok2 "R #2 9 R 5 kx2 2 n < X lx~i ðxÞdx x e x3 ekx lx~i ðxÞdx = R þ : R kx2 : xekx2 lx~i ðxÞdx lx~i ðxÞdx ; xe i¼1
The iteration process then continues until convergence, i.e., until k^ðhþ1Þ k^ðhÞ \e, for some pre-fixed e [ 0. The maximum likelihood estimate of k via Newton–Raphson ^ in this paper. algorithm is thereafter refereed as ‘‘k’’ Now, to apply Lindley’s form in (11), we first obtain n r11 ¼ 2 ^ k 8 "R #2 9 R 5 kx ^2 ^2 n < X lx~i ðxÞdx x e x3 ekx lx~i ðxÞdx = R ; þ R ^2 ^2 : xekx lx~ ðxÞdx lx~ ðxÞdx ; xekx i¼1 i
i
The approximate Bayes estimate of k, say k^B , for the squared error loss function is the posterior mean of gðkÞ ¼ k, which is by (11) as follows 1 k^B ¼ k^ þ q1 r11 þ L 3 r211 : 2
ð15Þ
where q1 ¼ a1 b. Similarly, for the posterior mean of k^ 2
RðtÞ, we have gðkÞ ¼ ekt and 1 4 1 2 ^2 kt 2 ^ RB ðtÞ ¼ e 1 þ t r11 t q1 r11 þ L3 r11 : 2 2
ð16Þ
4.2 Tierney and Kadane’s approximation Setting HðkÞ ¼ QðkÞ=n and H ðkÞ ¼ ½ln gðkÞ þ QðkÞ=n, the expression in (10) can be re-expressed as R1 nH ðkÞ e dk EðgðkÞjx~Þ ¼ 0R1 : ð17Þ nHðkÞ e dk 0
Following Tierney and Kadane (1986), Eq. (17) can be approximated as 12
f ; g^BT ðkÞ ¼ exp n½H ðk Þ HðkÞ ð18Þ f where k and k maximize H ðkÞ and HðkÞ, respectively, and f and f are minus the inverse of the second derivatives respectively. of H ðkÞ and HðkÞ at k and k, In our case, we have ( 1 HðkÞ ¼ k þ ðn þ a 1Þ log k kb n ) Z n X 2 log 2xekx lx~i ðxÞdx ; þ i¼1
where k is a constant; therefore, k can be obtained by solving the following equation: ( ) R 3 kx2 n X lx~i ðxÞdx x e oHðkÞ 1 n þ a 1 R ¼ b ¼ 0; ok n k xekx2 lx~i ðxÞdx i¼1
and (R ^2 n x7 ekx lx~i ðxÞdx 2n X ¼ 3 R ^2 k^ lx~i ðxÞdx xekx i¼1 ) R 3 kx R ^2 ^2 ð x e lx~i ðxÞdxÞð x5 ekx lx~i ðxÞdxÞ þ R ^2 lx~i ðxÞdxÞ2 ð xekx (R R 5 kx 2 ^ ^2 n X lx~i ðxÞdx x3 ekx lx~i ðxÞdx x e R þ2 R ^2 ^2 kx kx lx~i ðxÞdx lx~i ðxÞdx xe xe i¼1 "R #2 !) ^2 3 kx lx~i ðxÞdx x e : R ^2 kx lx~i ðxÞdx xe
L 3
and from the second derivative of HðkÞ, we have 2 1 o HðkÞ ; f¼ j k¼k ok2 where
o2 HðkÞ 1 nþa1 j ¼ k¼k n k2 ok2 0R "R # 2 19 2 2 n = 5 kx 3 kx X e l ðxÞdx e l ðxÞdx x x x~i x~i @R A R þ 2 2 ; xekx xekx lx~i ðxÞdx lx~i ðxÞdx i¼1
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4.3 MCMC method In this subsection, we consider Markov Chain Monte Carlo technique such as Metropolis–Hastings algorithm to draw samples from the posterior density function and then compute the Bayes estimates. Based on the gamma prior k Gammaða; bÞ and the likelihood function (7), the posterior p.d.f. of k can be written as n Z Y 2 pðkjx~Þ / knþa1 ekb xekx lx~i ðxÞdx: ð19Þ i¼1
Noting that the density function pðkjx~Þ is not known, but by experimentation, we observed that it is similar to normal distribution. So to generate random samples from pðkjx~Þ, we can use the Metropolis–Hastings algorithm with normal proposal distribution as follows: Step 1: Start with an initial guess kð0Þ . Step 2: Set h ¼ 1. Step 3: Generate kðhÞ from pðkjx~Þ using Metropolis– Hastings algorithm with the proposal distribution ^ 1Þ, where Ið:Þ is the indicator qðkÞ Iðk [ 0ÞNðk; function, as follows: ^ (a). Let m ¼ kðh1Þ . Here we set kð0Þ k. (b). Generate u from the proposal distribution q. n o jx~ÞqðmÞ (c). Let pðm; uÞ ¼ min 1; pðu pðmjx~ÞqðuÞ . (d). Accept u with probability pðm; uÞ or accept m with probability 1 pðm; uÞ. ðhÞ 2
Step 4: Compute RðhÞ ðtÞ ¼ ek t . Step 5: Set h ¼ h þ 1. Step 6: Repeat Steps 3–5, K times and obtain kðhÞ and RðhÞ ðtÞ for h ¼ 1; ; K. Now the Bayes estimates of the parameter k and reliability function RðtÞ with respect to squared error loss function become K X ^ jx~Þ ¼ 1 k^BM ¼ Eðk kðhÞ K h¼1
and
1
membership degree
Now, following the same argument with gðkÞ ¼ k and 2 gðkÞ ¼ ekt , respectively, in H ðkÞ, k^BT and R^BT in Eq. (18) can then be obtained straightforwardly.
0 0.05
0.25
0.5
0.75
1
1.5
2
3
x
Fig. 3 Fuzzy information system used to encode the simulated data
5 Numerical study In this section, we present some experimental results, mainly to observe how the different methods behave for different sample sizes. First, for fixed k ¼ 1 and different choices of n, we have generated i.i.d random samples from the Rayleigh distribution. Then each realization of x was fuzzified using the f.i.s. shown in Fig. 3, and the Bayes estimates of the parameter k and reliability function RðtÞ, at t ¼ 1, for the fuzzy sample were computed using the Lindley’s approximation (Lindley), Tierney and Kadane’s approximation (T&K), and MCMC technique. We have assumed that k has Gammaða; bÞ prior. To make the comparison meaningful, it is assumed that the priors are non-informative, and they are a ¼ b ¼ 0. Note that in this case the priors are non-proper also. Press (2001) suggested using very small non-negative values of the hyperparameters in this case, and it will make the priors proper. We have tried a ¼ b ¼ 0:0001. The results are not significantly different than the corresponding results obtained using nonproper priors, and are not reported due to space. The average values and mean squared errors of the estimates over 5,000 replications are presented in Tables 1 and 2. From the experiments, we found that using Lindley’s approximation or MCMC technique for the computation of Bayes estimates give similar estimation results, but Lindley’s approximation is computationally slower. For all the methods, it is observed that as the sample size increases, the MSEs of the estimates decrease. In terms of minimum MSEs, the performance of the T&K estimates for the parameter k and reliability function RðtÞ is generally best followed by the Lindley and MCMC estimates.
6 Application example
K 1X ^ R^BM ðtÞ ¼ EðRðtÞ jx~Þ ¼ RðhÞ ðtÞ; K h¼1
respectively.
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To demonstrate the application of proposed methods to real data, let us consider the data collected during the experiment reported by Pak et al. (2013). In this experiment, a
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493
Table 1 Average values (AV) and mean squared errors (MSE) of the Bayes estimates of k for different sample sizes n
k^BT
k^B
k^BM
AV
MSE
AV
MSE
AV
MSE
15
1.1678
0.1296
1.1622
0.1259
1.1682
0.1298
20 25
1.1639 1.1448
0.1148 0.0881
1.1608 1.1410
0.1121 0.0853
1.1638 1.1451
0.1146 0.0883
30
1.1376
0.0687
1.1334
0.0645
1.1374
0.0685
40
1.1319
0.0557
1.1285
0.0523
1.1317
0.0556
50
1.1235
0.0463
1.1203
0.0437
1.1239
0.0460
70
1.1136
0.0319
1.1114
0.0304
1.1134
0.0316
1e-47
π (λ x~ ) 5e-48
0e+00 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
λ
Table 2 Average values (AV) and mean squared errors (MSE) of the Bayes estimates of reliability function RðtÞ for different sample sizes n
R^B
R^BT
R^BM
AV
MSE
AV
MSE
AV
MSE
15
0.3306
0.0112
0.3325
0.0102
0.3308
0.0114
20
0.3311
0.0085
0.3343
0.0075
0.3314
0.0086
25
0.3328
0.0077
0.3352
0.0066
0.3332
0.0079
30
0.3341
0.0066
0.3363
0.0051
0.3346
0.0067
40
0.3357
0.0054
0.3378
0.0040
0.3354
0.0053
50
0.3364
0.0043
0.3384
0.0032
0.3365
0.0046
70
0.3386
0.0035
0.3397
0.0029
0.3382
0.0034
sample of 25 ball bearings is placed on a life test. A ball bearing may work perfectly over a certain period but be breaking for some time and finally be unusable at a certain time. So, the observed failure times of the ball bearings are reported by fuzzy numbers x~i ¼ ðai ; xi ; bi Þ, where ai ¼ 0:05xi , bi ¼ 0:03xi and the values of xi ; i ¼ 1; ; 25; are presented in Table 3. The corresponding membership functions of these fuzzy numbers are defined as
Table 3 Failures of 25 ball bearings data
Fig. 4 Plot of the posterior function pðkjx~Þ for the fuzzy sample x~
8 x ðxi ai Þ > > < ai lx~i ðxÞ ¼ x þ bi x > i > : bi
x i ai x x i ; i ¼ 1; ; 25: x i x x i þ bi ;
For these fuzzy data, the maximum likelihood estimate of k is found from (14) to be k^ ¼ 0:00014. Therefore, the Bayes estimate of k and reliability function RðtÞ; at t ¼ 1, under the Gammað2; 2Þ prior and using Lindley’s approximation become k^B ¼ 0:000158 and R^B ¼ 0:9998, respectively. Also, by using the Tierney and Kadane’s method, the Bayes estimates are k^BT ¼ 0:000143 and R^BT ¼ 0:9981. The plot of the posterior function pðkjx~Þ for these data is presented in Fig. 4. It is easily seen that this plot is quite similar to Normal distribution. Thus, we can use Metropolis–Hastings algorithm with the proposal ^ 1Þ and compute the approximate Bayes distribution Nðk; estimates of k and reliability function RðtÞ as k^BM ¼ 0:000156 and R^BM ¼ 0:9996; respectively.
i
1
2
3
4
5
6
7
8
9
10
xi
17.88
28.92
33.00
41.52
42.12
45.60
48.48
51.84
51.96
54.12
i
11
12
13
14
15
16
17
18
19
20
xi
55.56
7.80
67.80
67.80
68.64
68.64
68.88
84.12
93.12
98.64
i xi
21 105.12
22 105.84
23 127.92
24 128.04
25 173.40
123
494
7 Conclusions Classical reliability estimation in Rayleigh distribution is based on precise lifetime data. It is usually assumed that observed lifetime data are precise real numbers. However, in real world situations, the results of the experimental performance can not always be recorded or measured precisely, but each observable event may only be identified with a fuzzy subset of the sample space. In this paper, we have discussed the Bayesian estimation procedure for the parameter and reliability function of Rayleigh distribution when the lifetime observations are fuzzy numbers. We have presented two approximations, namely Lindley’s approximation and Tierney and Kadane’s approximation as well as a Markov Chain Monte Carlo technique to compute the Bayes estimates. We have then carried out a simulation study to assess the performance of these procedures. Based on the results of the simulation study, we see clearly that, the T&K procedure gives the most precise parameter estimates as shown by MSEs in Tables 1 and 2. Thus, it would seem reasonable to recommend the use of the Tierney and Kadane’s approximation for estimating the unknown parameter k and reliability function RðtÞ from the Rayleigh distribution.
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