Estimating the parameter of Exponential distribution under Type-II censoring from fuzzy data

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1 Jounal of Statistical Theoy and Applications, Vol. 15, No. 2 (June 2016), Estimating the paamete of Exponential distibution unde Type-II censoing fom fuzzy data Nayeeh Baghei Khoolenjani Depatment of Statistics, Univesity of Isfahan, Isfahan, , Ian n.b.khoolenjani@gmail.com Fateme Shahsanaie Depatment of Statistics, Behbahan Khatam Alanbia Univesity of Thechnology, Behbahan, Ian shahsanaei@bkatu.ac.i Received 23 Octobe 2015 Accepted 23 Novembe 2015 Statistical analysis of lifetime distibutions unde Type-II censoing scheme is based on pecise lifetime data. Howeve, in eal situations all obsevations and measuements of Type-II censoing scheme ae not pecise numbes but moe o less non-pecise, also called fuzzy. This pape deals with the estimation of exponential mean paamete unde Type-II censoing scheme when the lifetime obsevations ae fuzzy and ae assumed to be elated to undelying cisp ealization of a andom sample. We use Newton-Raphson algoithm as well as EM algoithm to detemine the maximum likelihood estimate (MLE) of paamete. We also obtain the estimate, via moment method and Bayesian pocedue, of the unknown paamete. In addition, a new numeical method fo paamete estimation is povided. Monte Calo simulations ae pefomed to investigate pefomance of the diffeent methods. Finally, an example is pesented in ode to illustate the methods of infeence discussed hee. Keywods: Type-II censoing; Fuzzy lifetime data; Exponential distibution; Maximum likelihood estimation; Bayes estimato Mathematics Subject Classification: 62N02, 62N86 1. Intoduction In life testing and eliability studies, the expeimente may not always obtain complete infomation on failue times fo all expeimental units. Data obtained fom such expeiments ae called censoed data. One of the most common censoing scheme is Type-II (failue) censoing, whee the life testing expeiment will be teminated upon the -th ( is pe-fixed) failue. This scheme is often adopted fo toxicology expeiments and life testing applications by enginees as it has been poven Coesponding autho 181

2 N.B. Khoolenjani and F. Shahsanaie to save time and money. Seveal authos have addessed infeential issues based on Type-II censoed samples; see fo example, [13], [2], [19], [3], [9], [16], [11], [17], and [10]. Thei eseach esults fo estimating paametes of diffeent lifetime distibutions unde Type-II censoing ae limited to pecise data. Howeve, in eal situations all obsevations and measuements of continuous vaiables ae not pecise numbes but moe o less non-pecise. Fo instance, the lifetime obsevations may be epoted as impecise quantities such as: about 1000h, appoximately 1400h, almost between 1000h and 1200h, essentially less than 1200h, and so on. This impecision is diffeent fom vaiability and eos. The best up-to-date mathematical model fo this impecision ae so-called non-pecise (fuzzy) numbes. Classical statistical pocedues ae not appopiate to deal with such impecise cases. Theefoe, we need suitable statistical methodology to handle these data as well. In ecent yeas, seveal eseaches pay attention to applying the fuzzy sets to estimation theoy. Huang et al. [8] poposed a new method to detemine the membeship function of the estimates of the paametes and the eliability function of multipaamete lifetime distibutions. Coppi et al. [4] pesented some applications of fuzzy techniques in statistical analysis. Pak et al. [14], [15] conducted a seies of studies to develop the infeential pocedues fo the lifetime distibutions on the basis of fuzzy numbes. In this pape, ou objective is to devise the methods fo paamete estimation egading a life-test fom which the Type-II censoed data ae epoted in the fom of fuzzy numbes. We analyze the data unde the assumptions that the lifetimes of the test units ae independent identically distibuted exponential andom vaiables. In Section 2, we fist pesent in geate detail the poblem addessed in this pape. Some peliminay concepts about fuzzy numbes is included in this section. In Section 3, we intoduce a genealization of the likelihood function based on Type-II censoing fom fuzzy data and then discuss Newton-Raphson algoithm as well as EM algoithm to detemine the MLE of the unknown paamete. We then study the estimate via moment method and estimate via Bayesian pocedue fo the mean paamete. A new paamete estimation method, called computational appoach estimation (CAE), is also established. In Section 4, simulation study is caied out to assess the pefomance of the poposed methods and a eal data fom the life testing expeiment is analyzed fo illustative puposes. 2. Poblem desciption Conside a eliability expeiment in which n identical units ae placed on a life-test. Let X 1,...,X n denote the lifetimes of these expeimental units. We assume that these vaiables ae independent and identically distibuted as Exponential E(θ), with pobability density function (pdf) f (x;θ) = 1 θ exp( x ),x > 0,θ > 0. (2.1) θ Let X 1:n X 2:n... X n:n denote the coesponding odeed lifetimes. Suppose the expeimente decides to cay out the life-test until the time of the th failue, then the data aising fom such a life-test would be of the fom X 1:n X 2:n... X :n with the emaining n lifetimes being moe than X :n. This situation is efeed to as Type-II censoing. We also denote the obseved values of such a Type-II ight censoed sample by x 1:n,...,x :n. Based on these obsevations, the likelihood function is given by: n! L(θ;x) = (n )!θ exp( 182 x i:n + (n )x :n ). θ

3 Thus the maximum likelihood estimato of θ can be obtained by Estimating the paamete of Exponential distibution unde Type-II censoing ˆθ = 1 ( x i:n + (n )x :n ). Howeve, in many fields of application, it is sometimes impossible to obtain exact obsevations of lifetime. The obtained lifetime data may be impecise most of the time. Fo instance, conside a life-testing expeiment in which n identical ball beaings ae placed on test, and we ae inteested in the lifetime of these ball beaings. In pactice, howeve, measuing the lifetime of a ball beaing may not yield an exact esult. A ball beaing may wok pefectly ove a cetain peiod but be baking fo some time, and finally be unusable at a cetain time. So, the numbe of evolutions to failue (in millions) fo ball beaings may epoted by means of the following impecise quantities: appoximately lowe than 45, appoximately 50 to 70, appoximately 75, appoximately 80, appoximately 90 to 100, appoximately highe than 120, and so on. In ode to model impecise lifetimes, a genealization of eal numbes is necessay. These lifetimes can be epesented by fuzzy numbes. A fuzzy numbe is a subset, denoted by x, of the set of eal numbes (denoted by R) and is chaacteized by the so called membeship function µ x (.). Fuzzy numbes satisfy the following constaints (see [5]): (1) µ x : R [0,1] is Boel-measuable; (2) x 0 R : µ x (x 0 ) = 1; (3) The so-called λ cuts (0 < λ 1), defined as B λ ( x) = {x R : µ x (x) λ}, ae all closed inteval, i.e., B λ ( x) = [a λ,b λ ], λ (0,1]. Among the vaious types of fuzzy numbes, LR-type fuzzy numbes (the tiangula and tapezoidal fuzzy numbes ae special cases of the LR-type fuzzy numbes) ae most convenient and useful in descibing fuzzy lifetime data. Theefoe, we shall focus on the set of LR-type fuzzy numbes. Definition 2.1. (see [21] pp.62). Let L (and R) be deceasing, shape functions fom R + to [0,1] with L(0) = 1; L(x) < 1 fo all x > 0; L(x) > 0 fo all x < 1; L(1) = 0 o (L(x) > 0 fo all x and L(+ ) = 0). Then a fuzzy numbe x is called of LR-type if fo m,α > 0, β > 0 in R, { L( m x µ x (x) = α ) x m R( x m β ) x m, whee m is called the mean value of x and α and β ae called the left and ight speads, espectively. Symbolically, the LR-type fuzzy numbe is denoted by x = (α,m,β). Definition 2.2. Suppose that x i = (α i,m i,β i ), i = 1,...,n, be the LR-type fuzzy numbes. The fuzzy mean value of these numbes can be obtained as n n 1 x = (ᾱ, m, β), with ᾱ = i, m = n α 1 i and n m β = 1 n 3. Data, likelihood and paamete estimations n β i. (2.2) In this section, we will fist intoduce a genealization of the likelihood function based on Type- II censoing when the lifetime obsevations ae epoted in the fom of LR-type fuzzy numbes. 183

4 N.B. Khoolenjani and F. Shahsanaie The maximum likelihood estimate of the mean paamete will then be obtained using the Newton- Raphson as well as the EM algoithm in Section 3.2. The estimation, via moment method, Bayesian pocedue and CAE method, of the unknown paamete will be discussed in Sections 3.3, 3.4 and 3.5, espectively Fuzzy lifetime data and the likelihood function Suppose that n independent units ae put on a test and that the lifetime distibution of each unit is given by (2.1). Now conside the poblem whee unde the Type-II censoing scheme, failue times ae not obseved pecisely and only patial infomation about them ae available in the fom of fuzzy numbes x i = (α i,m i,β i ), i = 1,...,, with thei coesponding membeship functions µ x1 (x 1 ),..., µ x (x ). Let the maximum value of the means of these fuzzy numbes to be m (). The lifetime of n suviving units, which ae emoved fom the test afte the mth failue, can be encoded as fuzzy numbes z +1,..., z n with the membeship functions { 0 z m() µ z j (z) =, j = + 1,...,n. 1 z > m () The fuzzy data w = ( x 1,..., x, z +1,..., z n ) is thus the vecto of obseved lifetimes. The coesponding obseved-data log-likelihood function can be obtained, using Zadeh s definition of the pobability of a fuzzy event (Zadeh [20]), as L O ( w;θ) = 1 ( log θ exp x ) θ = logθ + µ xi (x)dx + (n )log m () 1 θ exp( z θ ) dz ( exp x ) µ xi (x)dx (n ) m () θ θ. (3.1) 3.2. Maximum likelihood estimation The maximum likelihood estimate of the paamete θ can be obtained by maximizing the likelihood function (3.1). By taking the deivative of the log-likelihood (3.1) with espect to θ, we have θ L O( w;θ) = x θ + exp ( θ x ) 2 θ µ xi (x)dx ( ) exp x θ µ xi (x)dx + (n )m () θ 2 (3.2) To achieve estimation via ML method, it is not easy to solve the equation θ L O( w;θ) = 0, diectly. In the following, Theoem 1 discusses the existence and uniqueness of the MLE of θ. Theoem 3.1. Unde the Type-II censoing, the MLE of the mean paamete θ of an Exponential population exists and is unique. Poof. Let λ = 1 θ. Due to the invaiance popety of MLEs, we will show the existence and uniqueness of the MLE of λ instead of θ. The log-likelihood function L = L O ( w;λ) based on a Type-II 184

5 censoed sample can be witten as L = logλ + Estimating the paamete of Exponential distibution unde Type-II censoing log Diffeentiating of (3.3) with espect to λ yields L λ = λ exp( λx) µ xi (x)dx (n )λm () (3.3) xexp( λx) µ xi (x)dx exp( λx) µ xi (x)dx (n )m (). (3.4) Let g(λ) denote the function on the ight-hand side of the expession in (3.4). Then it is easily seen that lim g(λ) =. λ 0 Since the two tems (n )m () and xexp( λx) µ xi (x)dx in (3.4) ae both positive, we have lim g(λ) = lim λ λ [ λ < lim λ λ = 0, ] xexp( λx) µ xi (x)dx exp( λx) µ xi (x)dx (n )m () λ (0, ). Theefoe, the equation g(λ) = 0 has at least one oot in (0, ). To pove that the oot is unique, we conside the fist deivative of g, g (λ), given by Now let Then g (λ) can be witten as g (λ) = λ 2 + w i (λ) = 2 λ 2 log s(λ) = exp( λx), g (λ) = λ 2 + exp( λx) µ xi (x)dx. (3.5) exp( λx) µ xi (x)dx, 2 θ 2 logw i(θ). It is clealy that s(λ) is a log-concave function of λ, and this implies that w i (λ), i = 1,...,, ae also log-concave in λ (see Pekopa-Leindle inequality in the Appendix A). It follows that g is a stictly deceasing function w..t. λ and hence the equation g(λ) = 0 has exactly one solution. Since thee is no closed fom of the solution to the likelihood equation (3.2), an iteative numeical seach can be used to obtain the MLE. In the following, we descibe the Newton-Raphson method and the EM algoithm fo detemining the MLE of the paamete θ. 185

6 N.B. Khoolenjani and F. Shahsanaie Newton-Raphson algoithm Newton-Raphson algoithm is a diect appoach fo estimating the elevant paametes in a likelihood function. In this algoithm, the solution of the likelihood equation is obtained though an iteative pocedue. Let θ (h) be the paamete value fom the h th step. Then, at the (h+1) th step of iteation pocess, the updated paamete is obtained as ˆθ (h+1) = θ (h) θ L O( w;θ) h 2 θ 2 L O ( w;θ) h (3.6) whee the notation A h, fo any patial deivative A, means the patial deivative evaluated at θ (h). The second-ode deivative of the log-likelihood with espect to the paamete, equied fo poceeding with the Newton-Raphson method, is obtained as follows. 2 θ 2 L O( w;θ) = θ 2 2(n )m () [ θ 3 ( x 2 )exp 2x ( x ) ] θ + { 4 θ 3 θ µ xi (x)dx ( ) exp x θ µ xi (x)dx [ x exp ( x ) ] 2 θ 2 θ µ xi (x)dx ( ) exp x }. (3.7) θ µ xi (x)dx The iteation pocess then continues until convegence, i.e., until θ (h+1) θ (h) < ε, fo some pefixed ε > 0. It should be pointed out that the second-ode deivative of the log-likelihood is equied at evey iteation in the Newton-Raphson method. Sometimes the calculation of the deivatives based on censoed fuzzy data ae complicated. Anothe viable altenative to the Newton-Raphson algoithm is the well-known EM algoithm. In the following, we discuss how that can be used to detemine the MLE in this case. Once the maximum likelihood estimate of θ is obtained, we can use the asymptotic nomality of the MLEs to constuct the appoximate confidence inteval. It is known that the asymptotic distibution of the MLE of θ is given by ( ˆθ θ) N(0,Va(θ)). Hee, the asymptotic vaiance Va(θ) can be appoximated by Va(θ) 1 I( ˆθ), whee I( ˆθ) denotes the obseved Fishe infomation given by I( ˆθ) = 2 θ 2 L O( w;θ) θ= ˆθ. Remak 3.1. In the above poofs, we have exchanged integation and diffeentiation in (3.2), (3.4), and (3.7). Stict justification of these exchanges equies application of the dominated convegence theoem. To use this theoem, we note that the functions that we encounte in these elations can be eadily veified to be bounded by Lebesgue integable functions. Hence, fo the cases that we 186

7 Estimating the paamete of Exponential distibution unde Type-II censoing consideed, the dominated convegence theoem can be eadily applied to exchange the ode of integation and diffeentiation EM algoithm Expectation Maximization (EM) algoithm has emeged as a vey impotant tool fo estimating the paametes involved in a model, especially when the available data ae incomplete. Since the obseved fuzzy data w can be seen as an incomplete specification of a complete data, the EM algoithm is applicable to obtain the maximum likelihood estimate of the paamete. Fist of all, denote the lifetime of the failed and censoed units by X = (X 1,...,X ) and Z = (Z +1,...,Z n ), espectively. The combination of (X,Z) = W foms the complete lifetimes and the coesponding log-likelihood function is denoted by L(W;θ). In the following, we use the EM algoithm to detemine the MLE of θ. The log-likelihood function based on the complete lifetime W becomes L(W;θ) = nlogθ 1 θ [ x i + n z j ]. (3.8) j=+1 To pefom the E-step of the EM algoithm, we need to compute the conditional expectation of the complete-data log likelihood conditionally on the obseved data w, using the cuent fit θ (h) of the paamete θ: whee E θ (h) (L(W;θ) w) = nlogθ 1 θ E θ (h)(x i x i ) = [ E θ (h)(x i x i ) + ) xexp ( x µ (θ (h) ) xi (x)dx ), i = 1,...,, exp ( x µ (θ (h) ) xi (x)dx ] n E θ (h)(z j z j ), (3.9) j=+1 E θ (h)(z j z j ) = m () + θ (h), j = + 1,...,n. The M-step then consists in finding θ (h+1) which maximizes E θ (h)(logl(w;θ) w). This is easily achieved by solving the likelihood equation. Fom we get ˆθ (h+1) = 1 n θ E θ (h)(logl(w;θ) w) = 0, [ ( E θ (h)(x i x i ) + (n ) m () + θ (h))]. (3.10) The MLE of θ can be obtained by epeating the E- and M-steps until the diffeence L O ( w;θ (h+1) ) L O ( w;θ (h) ) becomes smalle than some abitay small amount. Due to the esult of the Theoem 187

8 N.B. Khoolenjani and F. Shahsanaie 3.1, it is then evident that the EM algoithm and the Newton Raphson method would convege to the same value. The maximum likelihood estimate of θ via EM algoithm is theeafte efeeed as ˆθ EM in this pape Method of moments Let X be andom vaiable which has the Exponential distibution with pdf given by (2.1). It is well-known that the kth moment of the Exponential model with paamete θ is E(X k ) = θ k Γ(k + 1). (3.11) Equating the fist sample moment to the coesponding population moment, the following equation can be used to find the estimate of moment method. θ = 1 n [ xexp( x θ )µ x i (x)dx exp( x θ )µ x i (x)dx + (n )( m () + θ ) ]. (3.12) Since the closed fom of the solution to Eq. (3.12) could not be obtained, an iteative numeical pocess to obtain the paamete estimate is descibed as follows: 1. Let the initial estimate of θ, say θ (0) with h = In the (h + 1)th iteation, we compute new estimate of θ, say θ (h+1) fom: θ (h+1) = 1 n [ xexp( x )µ θ (h) xi (x)dx ( exp( x + (n ) m )µ θ (h) xi (x)dx () + θ (h))]. (3.13) 3. Checking convegence, if the convegence occus then the cuent θ (h+1) is the estimate of θ by the method of moments; Othewise set h = h + 1 and go to Step 2. The esultant estimate of θ is theeafte efeeed as ˆθ MM in this pape Bayesian estimation In this subsection we conside the Bayes estimation of the unknown paamete. We fist epaameteize the model as follows λ = 1 θ. As conjugate pio fo λ, we take the Gamma(a,b) density with pdf given by π(λ) = ba Γ(a) λ a 1 exp( λb), λ > 0, (3.14) whee a > 0 and b > 0. Based on the new paametization, the posteio density function of λ given the data can be witten as follows. π(λ)l( w;λ) π(λ w), (3.15) π(λ)l( w;λ)dλ 0 188

9 Estimating the paamete of Exponential distibution unde Type-II censoing whee ] l( w;λ) = λ exp [ (n )λm () exp( λx)µ xi (x)dx is the likelihood function based on the Type-II censoed sample. Then, unde a squaed eo loss, the Bayes estimate of any function of λ, say g(λ), is g(λ)π(λ)l( w;λ)dλ 0 E(g(λ) w) = π(λ)l( w;λ)dλ 0 g(λ)e Q(λ) dλ 0 = 0 eq(λ) dλ, (3.16) whee Q(λ) = lnπ(λ) + lnl( w;λ) ρ(λ) + L(λ). Since the Eq. (18) can not be obtained analytically, we adopt Lindley s appoximation to compute the Bayes estimate of θ. Lindley [12] fist poposed his pocedue to appoximate the atio of two integals such as (3.16). Fo the one-paamete case, Lindley s appoximation of (3.16) is the fom g(λ) g 11τ 11 + ρ 1 g 1 τ L 3τ 2 11g 1, (3.17) whee g 1 = dg(λ) dλ,g 11 = d2 g(λ) dλ 2,ρ 1 = dρ(λ) dλ L 3 = 3 [ L(λ) λ 3, τ 11 = 2 ] 1 L(λ) λ 2. Evaluating all the expessions in (3.17) at the MLE of λ poduces the appoximation ĝ B to (3.16). In ou case, we have L(λ) = logλ + log exp( λx)µ xi (x)dx (n )λm (). The MLE of λ is the solution of the equation L(λ) λ = 0. Now, to apply Lindley s fom in (3.17), we fist obtain τ 11 = ˆλ 2 + [ ] 2 x 2 exp( ˆλx)µ xi (x)dx xexp( ˆλx)µ exp( ˆλx)µ xi (x)dx xi (x)dx exp( ˆλx)µ xi (x)dx 189

10 N.B. Khoolenjani and F. Shahsanaie and L 3 = 2 ˆλ 3 x 3 exp( ˆλx)µ xi (x)dx { exp( ˆλx)µ xi (x)dx [ x 2 exp( ˆλx)µ xi (x)dx][ ] xexp( ˆλx)µ xi (x)dx [ 2 } exp( ˆλx)µ xi (x)dx] + 2 xexp( ˆλx)µ xi (x)dx { exp( ˆλx)µ xi (x)dx x 2 exp( ˆλx)µ xi (x)dx xexp( ˆλx)µ ( exp( ˆλx)µ xi (x)dx [ xi (x)dx exp( ˆλx)µ xi (x)dx ]2 )} The appoximate Bayes of θ, say ˆθ B, fo the squaed eo loss function is the posteio mean of g(λ) = 1 λ, which is by (3.17) as follows. whee ρ 1 = a 1 ˆλ b. ˆθ B = 1ˆλ + τ 11 ˆλ 3 2ρ 1τ 11 + L 3 τ ˆλ Computational appoach estimation method In this subsection we popose a new paamete estimation pocedue called CAE. Although the maximum likelihood estimate obtained in the peceding section is pefeable, its computation equies epeated evaluation of E- and M-steps until convegence occus. On the othe hand, the CAE method povides not only the computational ease but also easonable mean squaed eo. This finding is futhe discussed in Section 4. Suppose x i = (α i,m i,β i ), i = 1,...,, be the obseved fuzzy lifetimes unde Type-II censoing fom exponential distibution with unknown paamete θ. Gzegozewski and Hyniewicz [7] consideed the genealization of exponential model which admits vagueness in lifetimes. They obtained a fuzzy estimato of the mean lifetime θ in the pesence of vague lifetimes. Howeve, in most applications, cisp esults ae equied instead of fuzzy ones. So, we popose the following computational appoach to obtain a cisp value as an estimate of θ. Step 1: Ode the means of fuzzy numbes as m (1) < m (2) <... < m (). Step 2: Obtain the fuzzy mean value, say x, of the fuzzy numbes by using (2.2). Step 3: Convet the fuzzy numbe x into a eal value by using the cente of gavity defuzzification technique (see in the Appendix B). Denote this defuzzified value by x. Step 4: The new estimate of θ is then computed as: θ = x + ( n 1)m (). (3.18) The esultant estimate of θ via this appoach is theeafte efeeed as ˆθ CA in this pape. 190

11 4. Numeical Study Estimating the paamete of Exponential distibution unde Type-II censoing In this section, we pesent a Monte Calo simulation study and one example to illustate the methods of infeence developed in this pape. Fist, fo fixed θ = 1 and diffeent choices of n and, we have geneated Type-II censoed samples fom Exponential distibution, using the method poposed by Aggawala and Balakishnan [1], as follows. (1) Geneate Z i fom U(0,1) fo i = 1,...,. 1 a (2) Set V i = Z i i, a i = i + n, i = 1,...,. (3) Set U i = 1 V i+1 V i+2...v, i = 1,...,. (4) Thus, X i = F 1 (U i ), i = 1,...,, is the desied Type-II censoed sample fom the Exponential distibution. Then each ealization of x was fuzzified using the method poposed by Pak et al. [15], and the estimate of θ fo the fuzzy sample wee computed using the fou methods pesented in the peceding section. We have used the initial estimate of θ fo iteative pogesses of Newton-Raphson method, EM algoithm and moment method to be θ (0) = 1 x i. Fo computing the Bayes estimato, we have assumed that λ has Gamma(a, b) pio. To make the compaison meaningful, it is assumed that the pio is non-infomative, and they ae a = b = 0. The aveage values and mean squaed eos of the estimates, the aveage numbe of iteations needed fo achieving convegence, based on 1000 eplication ae pesented in Table 1. In viewing the table, we find that using Newton-aphson o EM algoithm fo the computation of MLE give simila estimation esults, but Newton-Raphson method needs smalle iteations fo convegence. The pefomance of MLE and MME in tems of AVs and MSEs ae quite simila to each othe. Among the fou pocedues developed hee, the CAE method poduces estimates with smallest MSE and then followed by MLE and MME and then the Bayes pocedue. In all the cases, it is obseved that fo fixed n as m inceases, the MSEs of the estimates decease as expected. Since these fou pocedues fo estimating the mean paamete θ of Exponential distibution have diffeent featues, we let uses to decide based on thei pefeences. Example 1. To demonstate the application of poposed methods, let us conside the following life-testing expeiment in which n = 23 identical batteies ae placed on test. The unknown lifetime x i of battey i may be egaded as a ealization of a andom vaiable X i, induced by andom sampling fom a total population of batteies, which is distibuted as Exponential by an unknown mean paamete of θ. A tested battey may be consideed as failed, o -stictly speaking- as nonconfoming, when at least one value of its paametes falls beyond specification limits. In pactice, howeve, we don t have the possibility to measue all paametes and ae not able to define pecisely the moment of a failue. So, the obseved lifetimes (in 100h) ae epoted in the fom of lowe and uppe bounds, as well as a point estimate which ae as follows. DATA SET: (2.90, 3.63, 3.99),(5.24, 6.55, 7.20),(6.56, 8.20, 9.02),(7.14, 8.93, 9.82), (11.60, 14.51, 15.96),(12.14, 15.18, 16.69),(12.65, 15.82, 17.40),(13.24, 16.56, 18.21), (13.67, 17.09, 18.79),(13.88, 17.36, 19.09),(15.64, 19.56, 21.51),(17.05, 21.32, 23.45), (17.40, 21.76, 23.93),(17.80, 22.26, 24.48),(19.01, 23.77, 26.14),(19.34, 24.18, 26.59), (23.13, 28.92, 31.81),(23.34, 29.18, 32.09),(26.07, 32.59, 35.84),(30.29, 37.87, 41.65), (43.97, 54.97, 60.46),(48.09, 60.12, 66.13),(73.48, 91.86, 98.04). 191

12 N.B. Khoolenjani and F. Shahsanaie Table 1. Aveages values and mean squaed eos of the estimates, aveage numbe of iteations (AI) needed fo convegence. n ˆθ NR ˆθ EM ˆθ MM ˆθ B ˆθ CA AI NR AI EM AI MM (0.305) (0.304) (0.305) (0.331) (0.300) (0.211) (0.210) (0.211) (0.221) (0.202) (0.142) (0.141) (0.142) (0.156) (0.114) (0.238) (0.237) (0.238) (0.246) (0.235) (0.110) (0.110) (0.110) (0.119) (0.104) (0.100) (0.100) (0.100) (0.117) (0.086) (0.118) (0.117) (0.118) (0.126) (0.115) (0.066) (0.065) (0.066) (0.072) (0.057) (0.063) (0.063) (0.063) (0.070) (0.051) (0.089) (0.088) (0.089) (0.095) (0.086) (0.058) (0.058) (0.058) (0.062) (0.052) (0.036) (0.036) (0.036) (0.039) (0.034) In ou appoach, each tiple is modeled by a tiangula fuzzy numbe x i, and is intepeted as a possibility distibution elated to an unknown value x i, itself a ealization of a andom vaiable X i. Randomness aises fom the selection of objects fom the total population of batteies. In contast, fuzziness aises fom the limited ability of the obseve to descibe the moment of a failue using numbes, which is not influenced by andom factos. Suppose that the last 7 lifetimes out of n inspection items ae missing, then we can only obtain a Type-II censoed sample (2.90, 3.63, 3.99),(5.24, 6.55, 7.20),(6.56, 8.20, 9.02),(7.14, 8.93, 9.82), (11.60, 14.51, 15.96),(12.14, 15.18, 16.69),(12.65, 15.82, 17.40),(13.24, 16.56, 18.21), (13.67, 17.09, 18.79),(13.88, 17.36, 19.09),(15.64, 19.56, 21.51),(17.05, 21.32, 23.45), (17.40, 21.76, 23.93),(17.80, 22.26, 24.48),(19.01, 23.77, 26.14),(19.34, 24.18, 26.59). 192

13 Estimating the paamete of Exponential distibution unde Type-II censoing Fom these data, and using the stating value θ (0) = 16.04, which is the estimate of the paamete computed ove the centes of each fuzzy numbes, the final MLE of θ is found fom Newton- Raphson and EM algoithm to be ˆθ NR = and ˆθ EM = 26.28, espectively. Based on Lindley s appoximation, the Bayes estimate of θ unde squaed eo loss function becomes ˆθ B = Also the estimates fom the CAE and moment method ae ˆθ CA = and ˆθ MM = 26.29, espectively. 5. Conclusions Although the poblem of estimation of exponential mean paamete based on the censoed data has been studied extensively, taditionally it is assumed that the data available ae pefomed in exact numbes. In eal wold situations, howeve, we deal with non-pecise (fuzzy) data. Theefoe, the conventional pocedues used fo estimating the unknown paamete will have to be adopted to the new situation. In this pape we have poposed diffeent pocedues fo estimating the exponential distibution paamete unde Type-II censoing when the lifetime obsevations ae fuzzy numbes. They ae maximum likelihood estimation, estimation of method moments, Bayesian estimation and computational appoach estimation method. We have then caied out a simulation study to assess the pefomance of all these pocedues. Based on the esults of the simulation study, we see clealy that, maximum likelihood estimato (via any of the two pocesses mentioned in the pape) and MME give the same paamete estimates as shown by AV and MSE in Table 1. The CAE has slightly bette MSE compaed with all the estimatos consideed hee. In all the cases, it is obseved that as the effective sample size inceases the pefomances in tems of MSE become bette. The study of the applicability of the poposed appoach in estimating the paametes of exponential distibution unde the othe censoing schemes such as Hybid type-ii and Hybid pogessive type- II censoing ae possible topics fo futhe eseach. Appendix A. Pekopa-Leindle inequality In mathematics, the Pekopa-Leindle inequality is an integal inequality closely elated to the evese Young s inequality, the Bunn-Minkowski inequality and a numbe of othe impotant and classical inequalities in analysis. Statement of the inequality (see [6]): Let 0 < λ < 1 and let f,g and h be non-negative integable functions defined on R n satisfying h((1 λ)x + λy) f (x) 1 λ g(y) λ fo all x,y R n. Then ( ) 1 λ ( ) λ h(x)dx f (x)dx g(x)dx. R n R n R n Appendix B. Cente of gavity defuzzification method Defuzzyfication is the opposite pocess to the essence of idea of fuzzy sets. Moeove, defuzzyfication is the last step on fuzzy contol system and fuzzy easoning system. Finally, defuzzyfication opeation educes, fuzzy numbe to a single, cisp, numeical value, esult caies the best infomation and makes kind of synthesis about this fuzzy numbe. A common and useful defuzzification 193

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