Bayesian Reliability Estimation of Inverted Exponential Distribution under Progressive Type-II Censored Data

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1 J. Stat. Appl. Pro. 3, No. 3, (2014) 317 Journal of Statitic Application & Probability An International Journal Bayeian Reliability Etiation of Inverted Exponential Ditribution under Progreive Type-II Cenored Data Sanjay Kuar Singh, Ueh Singh, Abhianyu Singh Yadav and Pradeep K. Vihwakara Departent of Statitic and DST-CIMS, BHU Varanai , India Received: 6 May 2014, Revied: 4 Jun. 2014, Accepted: 6 Jun Publihed online: 1 Nov Abtract: The preent tudy deal with the etiation procedure for the paraeter, reliability and hazard function of the inverted exponential ditribution under progreive Type-II cenored data. For etiation purpoe, we have conidered both claical and Bayeian ethod of etiation. The Baye etiate of the paraeter, reliability and hazard function are calculated under yetric and ayetric lo function. We have alo coputed the 95% ayptotic and highet poterior denity (HPD) interval of the paraeter. Further, Monte Carlo iulation technique ha been ued to copare the perforance of the Baye etiator with correponding axiu likelihood etiator under both the lo function. Finally, we analyed one real data et for illutrative purpoe of the tudy. Keyword: Inverted exponential ditribution, Progreive cenoring, Bayeian inference, Iportance apling, Metropoli-Hating algorith. 1 Introduction The exponential ditribution i the ot popular ditribution for lifetie data analyi becaue of it iplicity and atheatical feaibility. The applicability of thi ditribution i retricted due to it contant failure rate but in realitic phenoenon, it ee to be eaningle when failure rate i non-contant. Therefore, a nuber of lifetie odel uch a Weibull, gaa, generalized exponential etc. have been propoed a a life tie odel which pae the variou type of failure rate i.e. onotonically increaing or decreaing failure rate function and it reduce to exponential ditribution for particular choice of hape paraeter. However, non-onotonicity of the hazard rate ha alo been oberved in any ituation. For exaple, in the coure of the tudy of ortality aociated with oe of the dieae, the hazard rate initially increae with tie and reache a peak after oe finite period of tie and then decline lowly, ee Singh et al. (2013). Thu, need to analye uch a data whoe hazard rate wa non-onotonically realized. Then, in view of thi, Inverted exponential ditribution (IED) ha been propoed a a lifetie odel by Lin et al. (1989). Lin et al. (1989), Prakah (2009), Singh et al. (2013) have advocated the ue of IED a an appropriate odel for thi atiation. Prakah (2009) have obtained Maxiu Likelihood etiate (MLE), confidence liit and uniforly iniu variance unbiaed etiator for the paraeter and reliability function of IED with coplete aple. Dey (2007) provide the etiation of the paraeter of IED by auing the paraeter involved in the odel a a rando variable. G. Prakah (2009) ha dicued the propertie of the variou etiator for the IED and alo preented the etiation of the paraeter, baed on lower record value. Recently, Singh et al. (2012) propoe Baye etiator of the paraeter and reliability function for the ae under the general entropy lo function for coplete, Type-I and Type-II cenored aple. The probability denity function (pdf) and cuulative ditribution function (cdf) of inverted exponential ditribution (IED) are given a; f(x,)= x 2 e /x ;x 0, > 0, (1) repectively. F(x,)=e /x ;x 0, > 0 (2) Correponding author e-ail: aybhu10@gail.co

2 3 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data Fig. 1: Figure repreent the hape of reliability and hazard function for different choice of the hape and cale paraeter. 1.1 Reliability Function and Hazard Function The reliability function and hazard function of the IED for pecified tie t are given by the following equation; R(t)=1 e /t ;,t > 0, (3) e /t H(t)= t 2 (1 e /t ;,t > 0 (4) ) repectively. The plot of the reliability and hazard function are preented below. Fro graph we ee that the hazard function ha a non-onotonic hape which increae initially, reache a axiu and drop lowly. The other advantage of thi odel i that, it i the pecial cae of the inverted gaa ditribution with known hape paraeter. 1.2 Progreive Cenoring and Likelihood Function The proble of cenoring in life tie data analyi i obviou becaue of tie and other circutance. The ot coon cenoring chee are Type-I and Type-II cenoring chee. In Type-I cenoring chee, the life teting experient will be terinated at a prefixed tie T, and in Type-II cenoring, the life teting experient will be terinated after oberving the r t h failure. Therefore, Type-I and Type-II cenoring chee are alo called a tie and failure cenoring chee repectively. However, the conventional Type-I and Type-II cenoring chee do not have the flexibility of allowing reoval of the unit during the experient. Due to thi reaon, a ore general cenoring chee called progreive Type- II (PT-II) cenoring chee ha been introduced by Cohen (1963). PT-II cenoring chee can be abbreviated a follow; Suppoe n unit are placed on experient and failure are going to be oberved. When the firt failure i oberved, R 1 of the urviving unit are randoly elected and reoved. At the econd oberved failure, R 2 of the urviving unit are randoly elected and reoved. Thi experient terinate at

3 J. Stat. Appl. Pro. 3, No. 3, (2014) / the tie when the t h failure i oberved and the reaining R = n R 1 R 2 urviving unit are all reoved. The tatitical inference on the paraeter of life tie ditribution under progreive Type-II cenoring ha been tudied by everal author uch a Cohen (1963), Vivero and Balakrihnan (1994), Aggarwala (2001) and Krihna and Kuar (2011). Thu, the likelihood function in the cae of PT-II cenoring chee i given by Balakrihnan and Sandhu (1995) a follow; where C i the contant given a follow; L(x )= C f(,){1 F( )} R i, (5) C=n(n R 1 1),,(n R 1 R 2 R 1 +1). Baed on uch a progreive Type-II right cenored aple, the likelihood function uing (1) and (2) can be expreed a; L (x ) Then, the log-likelihood function can be written a; L=lnL (x ) ln x 2 i { Ri. e / 1 e i} /x (6) 2 ln( )+ R i ln(1 e / ). (7) It i iportant to ention here that, the ot of the author have dicued the variou etiation procedure baed on Type-I or Type cenoring chee but no one ha paid attention about the etiation of the paraeter under Progreive cenoring, which i very popular and applicable cenoring chee now a day. Thu, our ai of thi paper i to conider the etiation of the unknown paraeter, reliability function and hazard function of IED uing PT-II cenoring chee under quared error lo function (SELF) and general entropy lo function (GELF). It i oberved that, the MLE of the unknown paraeter cannot be obtained in nice cloed for. Therefore, Newton- Raphon (N-R) ethod ha been ipleented to obtain the MLE. It i alo oberved that the Baye etiator are not in explicit for. Thu, aong exiting variou approxiation technique, one of the ot popular Markov Chain Monte Carlo (MCMC) technique ha been ued to obtain the Baye etiator baed on poterior aple. Monte Carlo iulation are conducted to copare the perforance of the claical etiator with correponding Baye etiator obtained in both inforative and non-inforative et-up. Further, we have alo contructed 95% approxiate confidence interval and highet poterior denity (HPD) credible interval for the paraeter. The ret of the article i organized a follow: Introductory part of the paper ha been covered in ection and ubection of 1. In Section 2, we derived the axiu likelihood etiator (MLE) of the paraeter, reliability and hazard function and obtained Fiher inforation for contructing 95% approxiate confidence interval. In ection 3, we obtained the expreion for Baye etiator under the non-inforative prior and two inforative prior uing SELF and GELF. Monte Carlo iulation reult and the analyi of data et are preented in ection 4. A real data illutration ha been dicued in ection 5 and finally in Section 6, we conclude the paper. 2 Maxiu Likelihood Etiation Now in order to obtain the axiu likelihood etiator of the paraeter, we have to axiize the above equation (7) w.r.t. the paraeter. Therefore, if we aue that i unknown then the MLE ( ˆ M ) of can be obtained by olving the following equation. ( ) 1 R i e + / (1 e /x = 0. (8) i) Fro the above equation, we oberved that the MLE of can not be obtained analytically. Therefore, we have ued Newton-Raphon ethod to obtain MLE of the paraeter. Now for pecified value of t the MLE of the reliability function ( ˆR M ) and hazard function(ĥ M ) have been obtained by uing invariance property. i.e. ˆR M = 1 e ˆ M/t, Ĥ M = ˆ Me ˆ M/t t 2 (1 e ˆ M/t )

4 320 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data 2.1 Ayptotic Confidence Interval To contruct ayptotic confidence interval we need the oberved Fiher inforation. Therefore, the oberved Fiher inforation i given by; ( I( ˆ)= E 2 ) L 2 = ˆ alo the ayptotic variance of i given by Var( ˆ)= 1 I( ˆ). But, the exact atheatical expreion for the above expectation i not exit. Therefore, by uing the concept of large aple theory the 100(1 λ)% confidence interval of i given by; [ ˆ L, ˆ U ]= ˆ M z λ/2 Var( ˆM ). 3 Bayeian Etiation In thi ection, we have obtained the Baye etiate for unknown paraeter, reliability function R(t) and the hazard rate function h(t). For etiating thee quantitie, two lo function have been taken into conideration, which are defined a; Squared error lo function (SELF): L S (, ˆ)=( ˆ ) 2, ( ) ˆ δ ( ) ˆ General entropy lo function (GELF):L G δ ln 1 ;δ 0. In each cae, ˆ repreent the etiate of unknown paraeter and δ i the hape paraeter of GELF which reflect the departure fro yetry. The Baye etiate of with repect to the L S and L G lo function i obtained fro it poterior ditribution a; ˆ S ={E ( x)} { } 1/δ ˆ G = E ( δ x) provided the expectation of the above quantity ut exit. 3.1 Baye Etiator of the Paraeter, Reliability Function and Hazard Function under Gaa Prior (Prior 1) In thi ubection, we conider the Baye procedure to derive the point etiate of the paraeter, reliability function R(t) and hazard function H(t) baed on PT-II cenored data. In Bayeian analyi, the paraeter of interet i to be conidered a a rando variable and follow oe prior ditribution. Here, we aue that, paraeter having gaa(a, b) denity i.e. π 1 () a 1 e b ; > 0 where, a, b are the hyper paraeter aue to be known. The above conidered prior i ore applicable in the ene that, it i ore flexible and having different variety of prior ditribution which ay be the reaon behind it popularity. Therefore, baed on the above prior, the poterior ditribution of i given a; p 1 ( x) +a 1 e x b i U(,,R i ) Gaa [ +a,b+ ( i)] 1 x U(,,R i ), (9) where U(,,R i )= 1 { } 1 e /x Ri i x 2. i Now, the Baye etiator of the paraeter, reliability function and hazard function under SELF and GELF are given a; +a e x b i U(,,R i )d =0 ˆ BS1 = +a 1 e (3.11) U(,,R i )d =0 b

5 J. Stat. Appl. Pro. 3, No. 3, (2014) / ˆR BS1 = Ĥ BS1 = ˆR BG1 = =0 =0 ˆ BG1 = Ĥ BG1 = =0 =0 ( 1 e /t ) +a 1 e =0 +a 1 e x b i x b i ( ) e /t t 2 (1 e /t +a 1 e ) =0 =0 =0 +a 1 e +a δ 1 e +a 1 e x b i x b i x b i { 1 e /t } δ +a 1 e =0 +a 1 e U(,,R i )d U(,,R i )d x b i U(,,R i )d U(,,R i )d U(,,R i )d x b i x b i U(,,R i )d U(,,R i )d U(,,R i )d { } δ e /t t 2 (1 e /t +a 1 e x b i U(,,R i )d ) +a 1 e U(,,R i )d =0 b (3.12) (3.13) (3.14) (3.15) (3.16) 3.2 Baye Etiator of the Paraeter, Reliability Function and Hazard Function under Inverted Gaa Prior (Prior 2) In thi ubection, we have obtained the Baye etiate under the conideration of inverted gaa prior IG(c,d) baed on PT-II cenored data. Inverted gaa prior i alo a flexible and conjugate prior for thi ditribution in coplete aple. Keeping thee point in ind we have conidered it. It wa alo conidered by everal author ee Prakah (2009) and Singh et al. (2011). Therefore, here we aue that, paraeter ha inverted gaa denity. i.e. π 2 () c 1 e d/ ; > 0,c,d > 0, where c, d are the hyper paraeter aue to be known. Therefore, baed on the above prior, the poterior ditribution of will be; p 2 ( x) c 1 e x d/ i U(,,R i ). (10) Therefore, the Baye etiator of the paraeter, reliability function and hazard function under SELF and GELF are given a; ˆ BS2 = =0 =0 c e c 1 e x d i x d i U(,,R i )d U(,,R i )d (3.2.1)

6 322 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data Ĥ BS2 = ˆR BG2 = Ĥ BG2 = ˆR BS2 = =0 ˆ BG2 = =0 =0 =0 ( 1 e /t ) c e =0 c 1 e x d/ i x d/ i ( ) e /t t 2 (1 e /t c 1 e ) =0 =0 =0 c 1 e c δ 1 e c 1 e x d/ i x d i x d i { 1 e /t } δ c 1 e =0 c 1 e U(,,R i )d U(,,R i )d x d/ i U(,,R i )d U(,,R i )d U(,,R i )d x d/ i x d i U(,,R i )d U(,,R i )d U(,,R i )d { } δ e /t t 2 (1 e /t c 1 e x d/ i U(,,R i )d ) c 1 e U(,,R i )d =0 d/ (3.2.2) (3.2.3) (3.2.4) (3.2.5) (3.2.6) 3.3 Baye Etiator of the Paraeter, Reliability Function and Hazard Function under Non-Inforative Prior (Prior 0) The election of prior ditribution i often baed on the type of prior inforation available to u. When we have little or no inforation about the paraeter, a non-inforative prior hould be ued. Jeffrey prior i one of the general cla of non-inforative prior. The iportant feature of thi prior i that it i not affected by the retriction of the paraeter pace. Several author have given a general jutification for uing Jeffrey prior for an exponential faily by howing that a proper poterior i produced. It otivated u to conider non-inforative prior for the paraeter. The prior of ay be taken a; π 3 () 1 ; > 0 Therefore, baed on the above prior, the poterior ditribution of will be; p 3 ( x) 1 e x i U(,,R i ) Therefore, the Baye etiator of the paraeter, reliability function and hazard function under above two lo function are expreed a; ˆ BS0 = =0 =0 e 1 e U(,,R i )d U(,,R i )d (3.3.1)

7 J. Stat. Appl. Pro. 3, No. 3, (2014) / ˆR BS0 = Ĥ BS0 = ˆR BG0 = =0 =0 ˆ BG0 = Ĥ BG0 = =0 =0 { 1 e /t } 1 e =0 1 e x i { } e /t t 2 (1 e /t 1 e ) =0 =0 =0 1 e δ 1 e 1 e { 1 e /t } δ 1 e =0 1 e U(,,R i )d U(,,R i )d U(,,R i )d U(,,R i )d U(,,R i )d xi U(,,R i )d U(,,R i )d U(,,R i )d { } δ e /t t 2 (1 e /t 1 e x i U(,,R i )d ) 1 e x i U(,,R i )d =0 (3.3.2) (3.3.3) (3.3.4) (3.3.5) (3.3.6) 3.4 Markov Chain Monte Carlo Method Fro previou ection 3.1, 3.2 and 3.3 we oberved that analytical olution of all conidered etiator are not poible. Therefore, MCMC ethod i ued to reolve uch type of ituation. MCMC ethod i the one of the bet ethod for obtaining the approxiate olution of the poterior expectation. After extracting or iulating the poterior aple, we ay eaily obtain the etiate of the paraeter and reliability characteritic. To obtain the olution of the poterior expectation, we have conidered the Iportance apling ethod and Metropoli-Hating algorith of MCMC ethod under Prior 1 and Prior 2 repectively. For ore detail about MCMC ethod, ee, Sith and Robert (1993), Upadhyay et al. (2001) and Singh et al. (2013). Iportance Sapling Method: To ipleent the iportance apling ethod under the conideration of Prior 1, the poterior ditribution of i given a; [ ( i)] 1 p 1 ( x) Gaa +a,b+ x U(,,R i ) Therefore, the following tep are taken to extract the aple fro the above poterior ditribution. generate 1 fro Gaa(.,.) repeat tep 1 to generate 1, 2,, Now, the Baye etiate of τ() with repect to the SELF and GELF will be; ˆτ() BS1 = 1 τ()u(,,r i )d U(,,R i )d

8 324 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data and ˆτ() BG1 = 1 τ() δ U(,,R i )d U(,,R i )d repectively. The 100(1 λ)% HPD interval for the paraeter uing the ethod of Chen and Shao (1990). One ay alo refer to Kundu and Pradhan (2009a) for a review on thi ethod. Metropoli-Hating Algorith: In cae of Prior 2 no any tandard ditribution i found for iulating the aple fro it repective poterior ditribution. Therefore ot uitable ethod for extracting the poterior aple i Metropoli under Gibb algorith are taken into conideration. The M-H under Gibb algorith conit the following tep; et the initial value of ay 0 Set l=1 Generate poterior aple for fro (10). Repeat tep 2, for all l = 1,2,3, and obtained 1, 2,, After obtaining the poterior aple, the Baye etiate of the paraeter, reliability function and hazard function under SELF are the ean of the poterior aple. Therefore, we have, and Baye etiator under GELF are given by; ˆR GS2 = ˆR BS2 = 1 Ĥ BS2 = 1 ˆ GS2 = { 1 1 Ĥ GS2 = ˆ BS2 = 1 l=1 l=1 { 1 l=1 [ l=1 l l=1 {1 e l/t } l e l/t t 2 (1 e l/t ) l=1 δ l } {1 e l/t } δ} l e l/t t 2 (1 e l/t ) ] δ After extracting the poterior aple we can eaily contruct the HPD credible interval for. Therefore, for thi purpoe order 1, 2,..., a 1 < 2 < <. Then 100(1 λ)% credible interval of i ( 1, [(1 λ)+1] ),,( [λ], ). Here[x] denote the greatet integer le than or equal to x. Then, the HPD credible interval i that interval which ha the hortet length. Further, the Baye etiate under non-inforative prior can be obtained by etting the value of hyper paraeter i to be zero in any of the above algorith.

9 J. Stat. Appl. Pro. 3, No. 3, (2014) / Algorith for Saple Generation under PT-II Uing the algorith of Balakrihnan and Agrawala (1995), we have ued the following tep to generate a PT-II cenored aple fro the IED. Specify the value of n, and. Generate iid rando nuber u 1,u 2,,u fro U(0,1). Set ξ i = ln(1 u i ), o that ξ i are iid tandard exponential variate. For given cenoring chee R=(R 1,R 2,,R ), et y 1 = ξ 1 / and for i=2,3,,, φ i = φ i 1 + n i 1 R j i+1 j=1 Now(φ 1,φ 2,...,φ ) are the progreive Type II cenored aple fro tandard exponential ditribution with cenoring chee R=(R 1,R 2,,R ). Set ψ i = 1 exp( φ i ), { o that } ψ i for a progreive Type II cenored aple fro U(0,1). Set = F 1 (ψ i )=. lnψ Now,(x 1,x 2,...,x ) are the PT-II right cenored aple fro IED() with a cenoring chee R=(R 1,R 2,,R ). ξ i 5 Siulation Study and Coparion In thi ection, we invetigate the perforance of the Baye etiator with correponding ML etiator baed on 1000 replication. In order to perfor the coparion tudy following tep taken into account. 1. For pecified value of n, and, generate PT-II cenored aple uing the algorith dicued in previou ection. 2. For different cenoring chee n i taken a 30 and the value of ha been choen in uch a way that the obervation are cenored a 10%, 20%, 40%, 50% and 60% for fixed value of = The ML etiator of the paraeter, reliability function and hazard function and correponding ayptotic confidence interval have been coputed. 4. For Bayeian analyi, we have aued that paraeter ha gaa and inverted gaa prior and Jeffrey prior. The value of hyper paraeter are taken a (a=c=4 and b=d=2). 5. Iportance apling ethod and Metropoli-Hating algorith of MCMC technique have been ued to extract poterior aple under Prior 1 and Prior 2 repectively. 6. On the bai of iulated poterior aple, we have obtained the Baye etiator of the paraeter, reliability and hazard function under the auption of the above prior uing SELF and GELF. 7. In iulation tudy under GELF, only one choice of lo paraeter δ i conidered. δ = 0.5 (for over etiation) and δ = 0.5 (for under etiation). 8. We have alo contructed 95% highet poterior denity (HPD) interval for the paraeter. 9. The perforance of the etiator have been reported under SELF and GELF both, ee, Table [2-12]. In order to chooe different cenoring chee a repreent that the nuber a i repeated tie, ee Table 1. Fro thi extenive tudy, we ay conclude the following; i. The rik of the Baye etiator i leat a copared to the rik of the ML etiator. ii. The rik of the etiator decreae a the percentage of i increae. iii. Fro the Table 2 and 3, we oberved that, the rik of the Baye etiator of the paraeter, reliability function and hazard function under Prior 1 i aller than the rik of the etiator under Prior 2. iv. Fro Table 3, we noticed that the length of HPD interval i aller than the length of ayptotic confidence interval. Further, we have alo oberved that the average confidence length in cae of Prior 1 i iniu a copared Prior 2. v. Fro the Table 4, 5 and 6, we ee that, the Baye etiator under GELF have aller rik when δ = 0.5 a copared to the δ = 0.5. vi. Fro the Table 7, we oberved that, the HPD length and rik of the Baye etiator under Prior 1 and Prior 2 i iniu a copared of Prior 0. vii. Fro the Table 8, we oberved that the rik of the etiator under GELF i iniu for δ = 0.5 a copared to thoe of the etiator for δ = 0.5. It ean that under etiation i ore eriou then over etiation.

10 326 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data 6 Real Data Analyi In thi ection, we propoe data analyi to illutrate our propoed ethodology baed on two data et. The conidered data et ha been ued by everal author when life tie follow inverted exponential ditribution. Data Set-I: We have generated a rando aple of ize 50 fro IED with paraeter = 3 and chooe different cenoring chee. The iulated data i preented a follow; 0.4, 0.77, 0.78, 1.03, 1.11, 1.16, 1.82, 1.83, 1.86, 1.9, 1.96, 2.02, 2.79, 2.84, 2.85, 3.4, 3.71, 3.75, 3.82, 4.03, 4.2, 4.29, 4.29, 4.42, 4.57, 4.74, 4.79, 5.01, 5.03, 5.26, 5.61, 5.7, 6.44, 6.98, 8.09, 8.42, 8.82, 9.28, 9.49, 10.38, 11.49, 11.65, 14.86, 22.39, 24.29, 27.63, 36.28, 49.38, 96.77, Data Set-II: 12,15,22,24,24,32,32,33,34,38,38,43,44,48,52,53,54,54,55,56,57,58,58,59,60,60,60,60,61, 62,63,65,65,67,68,70,70,72,73,75,76,76,81,83,84,85,87,91,95,96,98,99,109,110,121,127, 129,131,143,146,146,175,175,211,233,258,258,263,297,341,341,376. The data et-ii wa initially propoed by Bjerkedal (1960) and ued by Kundu and Howlader (2010) for Bayeian etiation and prediction of the invere Weibull ditribution under Type-II cenored data. Baye etiator of the paraeter and reliability function of inverted exponential ditribution under the general entropy lo function baed on coplete, Type-I and Type-II cenored aple have dicued by Singh et al. (2012). Recently, Mahehwari et al. (2014) have conidered thi data et and derived the Baye etiation procedure under hybrid cenoring chee. Here, we have alo ued thi data et under progreive Type-II cenoring chee. For thi purpoe, we have taken different cenoring chee, ee Table [8-11]. Baed on thee two data et, we have calculated the ML and Baye etiator of the paraeter, reliability function, hazard function and alo provided interval etiate for different cenoring chee and lo paraeter δ[( 1.5, 1.5),( 1.0, 1.0),( 0.5, 0.5)]. Table 1: Table repreent the different cenoring chee which are conidered for iulation. n r1 r2 r3 r4 12 3*6,0*6 0*5,6*3,0*4 0*2,2*9,0 0*6,3*6 15 3*5,0*10 1*15 0*6,5*3,0*6 0*10,3*5 30 2*6,0*12 0*8,4*3,0*7 0*3,1*12,0*3 0*12,2*6 24 1*6,0* 0*11,2*3,0*10 0*9,1*6,0*9 0*,1*6 27 3,0*26 0*12,1*3,0*12 0*13,3,0*13 0*26,3 30 0*30 7 Concluding Reark In thi paper, we propoed Bayeian and axiu Likelihood etiation for the paraeter, reliability function and hazard function under PT-II cenoring chee uing different prior inforation and lo function. Therefore, fro thi extenive tudy of the reult of iulation, we oberved that the Baye etiator with an inforative prior perfor well in all conidered cae.. Therefore, we conclude that the Baye etiator with an inforative prior ay be ued particularly when oe a priori inforation about the paraeter i known. However, if no a priori inforation about the paraeter i available, MLE ay be recoended for their ue.

11 J. Stat. Appl. Pro. 3, No. 3, (2014) / Table 2: Rik of the paraeter and reliability function R under SELF uing Prior 1 and Prior 2 for fixed value of n=30. RI ˆ M ˆ BS1 ˆ BS2 ˆ BG1 ˆ BG2 ˆR M ˆR BS1 ˆR BS2 ˆR BG1 ˆR BG1 r r r r r r r r r r r r r r r r r r r r r Table 3: Rik of the hazard function H under SELF and interval etiate of the paraeter uing Prior 1 and Prior 2. Ayptotic Interval HPD (Prior 1) HPD (Prior 2) RI Ĥ M Ĥ BS1 Ĥ BS2 Ĥ BG1 Ĥ BG2 ˆ L ˆ U Length ˆ L ˆ U Length ˆ L ˆ U Length r r r r r r r r r r r r r r r r r r r r r

12 328 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data Table 4: Rik of the paraeter under GELF for pecified value of lo paraeter δ uing Prior 1 and Prior 2 repectively. RI ˆ M ˆ BS1 δ=0.5 δ=-0.5 ˆ BS2 ˆ BG1 ˆ BG2 ˆ M ˆ BS1 ˆ BS2 ˆ BG1 ˆ BG2 r r r r r r r r r r r r r r r r r r r r r Table 5: Rik of the reliability function under GELF for pecified value of lo paraeter δ uing Prior 1 and Prior 2 repectively. RI ˆR M ˆR BS1 δ=0.5 δ=-0.5 ˆR BS2 ˆR BG1 ˆR BG2 ˆR M ˆR BS1 ˆR BS2 ˆR BG1 ˆR BG2 r r r r r r r r r r r r r r r r r r r r r

13 J. Stat. Appl. Pro. 3, No. 3, (2014) / Table 6: Rik of the hazard function under GELF for pecified value of lo paraeter δ uing Prior 1 and Prior 2 repectively. RI Ĥ M Ĥ BS1 δ=0.5 δ=-0.5 Ĥ BS2 Ĥ BG1 Ĥ BG2 Ĥ M Ĥ BS1 Ĥ BS2 Ĥ BG1 Ĥ BG2 r r r r r r r r r r r r r r r r r r r r r Table 7: Rik of the paraeter, reliability function, hazard function and correponding interval etiate under Prior 0 RI MLE Rik Under SELF Ayptotic interval ˆ M ˆR M Ĥ M ˆ BS0 ˆ BG0 ˆR BS0 ˆR BG0 Ĥ BS0 Ĥ BG0 ˆ L ˆ U Length r r r r r r r r r

14 330 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data Table 8: Table repreent the rik of the etiator under GELF uing Prior 0. RI ˆ M δ=0.5 δ=-0.5 ˆ BS0 ˆ BG0 ˆ M ˆ BS0 ˆ BG0 r r r r r r r r r Reliability Function RI ˆR M ˆR BS0 ˆR BG0 ˆR M ˆR BS0 ˆR BG0 r r r r r r r r r Hazard Function RI Ĥ M Ĥ BS0 Ĥ BG0 Ĥ M Ĥ BS0 Ĥ BG0 r r r r r r r r r Table 9: Table repreent the etiate of the etiator under SELF and correponding interval etiate for Data Set I when actual reliability and hazard function at T=8 are and repectively Schee Interval Etiate of the paraeter Etiate Under SELF MLE Ayptotic interval HPD Interval Jeffrey Prior Jeffrey Prior ˆ L L ˆ U Length Length ˆ M ˆR M Ĥ M ˆ BS0 ˆR BS0 Ĥ BS0 ˆ L ˆ U 5*2,0* *19,5*2,0* *38,5* *5,0* *10,5*5,0* *20,5* *5,0* *5,7*5,0* *10,7*

15 J. Stat. Appl. Pro. 3, No. 3, (2014) / Table 10: Etiate of the paraeter, reliability function and hazard function under GELF for different variation of lo paraeter δ, cenoring chee and for fixed value of n=50 when data I i conidered where actual reliability and hazard function at T=8 are and repectively. Etiate Under GELF δ Reoval Under etiation(-ve δ) Over etiation(+ve δ) ˆ BG0 ˆR BG0 Ĥ BG0 ˆ BG0 ˆR BG0 Ĥ BG0 5*2,0* (-1.5,1.5) 0*19,5*2,0* *38,5* *2,0* (-1.0,1.0) 0*19,5*2,0* *38,5* *2,0* (-0.5,0.5) 0*19,5*2,0* *38,5* *5,0* (-1.5,1.5) 0*10,5*5,0* *20,5* *5,0* (-1.0,1.0) 0*10,5*5,0* *20,5* *5,0* (-0.5,0.5) 0*10,5*5,0* *20,5* *5,0* (-1.5,1.5) 0*5,7*5,0* *10,7* *5,0* (-1.0,1.0) 0*5,7*5,0* *10,7* *5,0* (-0.5,0.5) 0*5,7*5,0* *10,7* Table 11: Table repreent the etiate of the etiator under SELF and correponding interval etiate for Data Set II. Schee Interval Etiate of the paraeter Etiate Under SELF MLE Ayptotic interval HPD Interval Jeffrey Prior Jeffrey Prior ˆ L ˆ U Length Length ˆ M ˆR M Ĥ M ˆ BS0 ˆR BS0 Ĥ BS0 ˆ L ˆ U 72 0* *11,0* *22,0* *8,0*14,1*6,0*14,1* *16,0* *32,0* *4,0*8,2*4,0*8,2*4,0*8,2* *13,0* ,0*8,13*2,0*8, *7,13*2,0*10, Reference [1] Cohen A. C. (1963) Progreively cenored aple in life teting. Technoetric 5: [2] Balakrihnan N, Sandhu R. A.(1995) A iple iulation algorith for generating progreive Type II cenored aple. A Stat 49(2): [3] Aggarwala R. (2001) Progreive cenoring. In: Balakrihnan N., Rao CR (ed) Handbook of tatitic 20: advance in reliability. [4] Varian H. (1975) A Bayeian approach to real etate aeent. North Holland, Aterda

16 332 S. K. Singh et. al. : Bayeian Reliability Etiation of IED under PT-II Cenored Data Table 12: Etiate of the paraeter under GELF for different variation of lo paraeter δ, cenoring chee and for fixed value of n=50 when data II i conidered. Etiate Under GELF δ Schee Jeffrey Prior (Under etiation) Jeffrey Prior (Over etiation) ˆ BG0 ˆR BG0 Ĥ BG0 ˆ BG0 ˆR BG0 Ĥ BG0 (-1.5,1.5) (-1.0,1.0) 0* (-0.5,0.5) *11,0* (-1.5,1.5) 1*22,0* *8,0*14,1*6,0*14,1* *11,0* (-1.0,1.0) 1*22,0* *8,0*14,1*6,0*14,1* *11,0* (-0.5,0.5) 1*22,0* *8,0*14,1*6,0*14,1* *16,0* ! (-1.5,1.5) 1*32,0* *4,0*8,2*4,0*8,2*4,0*8,2* *16,0* (-1.0,1.0) 1*32,0* *4,0*8,2*4,0*8,2*4,0*8,2* *16,0* (-0.5,0.5) 1*32,0* *4,0*8,2*4,0*8,2*4,0*8,2* *13,0* (-1.5,1.5) 13,0*8,13*2,0*8, *7,13*2,0*10, *13,0* (-1.0,1.0) 13,0*8,13*2,0*8, *7,13*2,0*10, *13,0* (-0.5,0.5) 13,0*8,13*2,0*8, *7,13*2,0*10, [5] Vivero R., Balakrihnan N. (1994) Interval etiation of life characteritic fro progreively Cenored data. Technoetric 36: [6] Dey S Inverted exponential ditribution a a life ditribution odel fro a Bayeian viewpoint, Data Sci. J. 6: [7] Lin C., Duran B. S. and Lewi T. O Inverted gaa a a life ditribution. Micro electron. Reliab. 29 (4): [8] Prakah G Soe etiation procedure for the inverted exponential ditribution. The South Pacific Journal of Natural Science, 27: [9] Singh S. K., Singh U. and Kuar D Baye etiator of the reliability function and paraeter of inverted exponential ditribution uing inforative and non-inforative prior. Journal of Statitical Coputation and Siulation, 83(12): [10] T. Bjerkedal, Acquiition of reitance in guinea pig infected with different doe of virulent tubercle bacilli, A. J. Hyg. 72 (1960), pp [11] A. F. M. Sith and G. O. Robert. Bayeian coputation via the Gibb apler and related Markov Chain Monte Carlo ethod. Journal of the Royal Statitical Society: Serie B, 55:3-23, [12] S. K. Upadhyay, N. Vaihta, and A. F. M. Sith. Baye inference in life teting and reliability via Markov Chain Monte Carlo iulation. Sankhya A, 63:15-40, [13] Singh S. K., Singh Ueh, and Shara V. K., Bayeian prediction of future obervation fro invere Weibull ditribution baed on type-ii hybrid cenored aple, International Journal of Advanced Statitic and Probability, vol. 1, pp , [14] Chen, M.-H., Shao, Q.-M., Monte Carlo etiation of Bayeian credible and HPD interval. Journal of Coputational and Graphical Statitic 8, [15] Kundu, D., Pradhan, B., Etiating the paraeter of the generalized exponential ditribution in preence of hybrid cenoring. Counication in Statitic-Theory and Method 38,

17 J. Stat. Appl. Pro. 3, No. 3, (2014) / [16] Pundir P. S., Singh B. P. and Mahehwari S. On hybrid cenored inverted exponential ditribution, International Journal of Current Reearch, Vol. 6, Iue, 01, pp , January, [17] H. Krihna and Kapil Kuar, Reliability etiation in Lindley ditribution with progreively type II right cenored aple, Matheatic and Coputer in Siulation 82 (2011) [] S. K. Singh, Ueh Singh and V. K. Shara, Bayeian Etiation and Prediction for Flexible Weibull Model under Type-II Cenoring Schee,Journal of Probability and Statitic Volue 2013 (2013), Article ID