 Open Journal of Statistics, 2012, 2, 356-367 http://dx.doi.org/10.4236/ojs.2012.23044 Published Online July 2012 (http://www.SciRP.org/journal/ojs) Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm Tahani A. Abushal1, Areej M. Al-Zaydi2 1Department of Mathematics, Umm Al-Qura University, Makkah Al-Mukarramah, KSA 2Department of Mathematics, Taif University, Taif, KSA Email: tabushal@yahoo.com, aree.m.z@hotmail.com Received May 15, 2012; revised June 16, 2012; accepted June 30, 2012 ABSTRACT This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values. Numerical example is presented in the methods proposed in this paper. Keywords: Mixture Distributions; Rayleigh Distribution; Generalized Order Statistics; Record Values; MCMC 1. Introduction The concept of generalized order statistics GOS was in- troduced by  as random variables having certain joint density function, which includes as a special case the joint density functions of many models of ordered ran- dom variables, such as ordinary order statistics, ordinary record values, progressive Type-II censored order statis- tics and sequential order statistics, among others. The GOS have been considered extensively by many authors, some of them are [2-18]. In life testing, reliability and quality control problems, mixed failure populations are sometimes encountered. Mixture distributions comprise a finite or infinite number of components, possibly of different distributional types, that can describe different features of data. In recent years, the finite mixture of life distributions have to be of con- siderable interest in terms of their practical applications in a variety of disciplines such as physics, biology, geology, medicine, engineering and economics, among others. Some of the most important references that discussed dif- ferent types of mixtures of distributions are [19-25]. Let the random variable follows Rayleigh lifetime model, its probability density function (PDF), cumulative distribution function (CDF) and reliability function (RF) are given below: T 0,0,t 0,0,t 2,tRt e2,ht te21,tHt e2t (1) (2) (3) Also, the hazard rate function (HRF) 2,tt (4) where ....hR The cumulative distribution function (CDF), denoted by Ht, of a finite mixture of k components, denoted by ,1,,jHtj k 1,kjjj is given by HtpHt1, ,jk (5) where, for 0p1.kp the mixing proportions j and 1jj2kp1pp The case of , in (5), is practical importance and so, we shall restrict our study to this case. In such case, the population consist of two sub-popula- tions, mixed with proportions 1 and 21. In this paper, the components are assumed to be Rayleigh distribution whose PDF, CDF, RF and HRF are given, respectively, by 112 2htph tpht (6) 112 2,HtpHtpHt (7) 112 2,RtpR tpRt (8) ,thtRt1, 2j (9) where, for , the mixing proportions jp are such Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI357 01that 12jand, 1ppp,,jHtRtjj are given from (1)-(3) after using htj instead of. Several authors have predicted future order statistics and records from homogeneous and heterogeneous popu- lations that can be represented by single component dis- tribution and finite mixtures of distributions, respectively. For more details, see [9,10,26]. Recently, a few of authors utilized the GOS’s in Bayes- ian inference. Such authors are [7-9,18]. Bayesian infer- ences based on finite mixture distribution have been dis- cussed by several authors such that: [23,24,27-33]. For Bayesian approach, the performance depends on the form of the prior distribution and the loss function assumed. A wide variety of loss functions have been de- veloped in the literature to describe various types of loss structures. The balanced loss function was suggested by .  introduced an extended class of the balanced loss function of the form  0,,10,,L,,    ., (10) where is a suitable positive weight function and  is an arbitrary loss function when estimate- ing  by . The parameter 0 is a chosen prior estimator of 0, obtained for instance from the crite- rion of maximum likelihood (ML), least squares or unbi- asedness among others. They give a general Bayesian connection between the case of and 0,0,kkr where . 01,,mmnSuppose that 1;,, 2;,,;,,,,,nmk nmkrnmTT T1,,,rrRm m1rtt  R11 11r are the first (out of ) GOS drawn from the mixture of two Ray- leigh MTR distribution. The likelihood function (LF) is given in , for by m 11imrriiRt ht,rt11,.rriiiniCkMiiRt 1,rirrt ht10,iiniMip1,,LtC R (11) where , is the parameter space, and tt1 (12) where and are given, respectively, by (5) and (7). htThe purpose of this paper is to obtained the maximum likelihood prediction (point and interval) and the Bayes prediction (point and interval) in the case of one-sample scheme and two-sample scheme. The point predictors are obtained based on balanced square error loss (BSEL) function and the balanced LINEX (BLINEX) loss func- tion. We used ML to estimate the parameters, and j of the MTR distribution based on GOS. The conju- gate prior is assumed to carry out the Bayesian analysis. The results are specialized to the upper record values. The rest of the article is organized as follows. Section 2 deals with the derivation of the maximum likelihood estimators of the involved parameters. Sections 3 and 4, deals with studying the maximum likelihood (point and interval) and the Bayes prediction (point and interval) in the case of one-sample scheme and two-sample scheme. In Section 5, the numerical computations results are presented and the concluding remarks. 2. Maximum Likelihood Estimation (MLE) Substituting (6), (7) in (11), the LF takes the form    1112 21112 2111122.irmriiiriiirrLtpRt pRtph tphtpR tpRt (13) Take the logarithm of (13), we have    111221112 211122ln ln+ln1ln ,riiiiriiirrrlLtmpRtpRtph tphtpR tpRt  ,1pppp (14) where 12 p. Differentiating (14) with re- spect to the parameters and j and equating to zero gives the following likelihood equations   1**1111**110,10,1, 2rriiirriirjjijiijriji rjrilmt ttplpttmt tj1, 2j (15)  where, for        121 2*2*2,,,,1iii iiiiiji ijiji jiiiji ijht htRtRtttht Rtht tRtttht Rttt (16) Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI358 pEquations (15) do not yield explicit solutions for and j, and have to be solved numerically to obtain the ML estimates of the three parameters. New- ton-Raphson iteration is employed to solve (15). 1, 2j,,TTnRemark: The parameters of the components are as- sumed to be distinct, so that the mixture is identifiable. For the concept of identifiability of finite mixtures and examples, see [19,36,37]. 3. Prediction in Case of One-Sample Scheme Based on the informative 1; ,,;,,nmk rnmk GOS’s from the MTR distribution, for the remaining unobserved future () components, let ;, ,rsnmkT, s = r + 1, r + 2, denote the future lifetime of the ,nths component to fail, 1snr, the maximum Likelihood pre- diction (point (MLPP) and interval (MLPI)), Bayesian prediction (point (BPP) and interval (BPI)) can be ob- tained. The conditional PDF of ;, ,ssnmkTT given that the components that had already failed is ;, ,rrnmkTT 1*111111ln ln1!,1,(1) 1!srsrsrr skksrs111,1,srsrmrssrsRtRtRtht mmssr rkkttRt RtsrRtRtht mCRtmsrC1m (17) In the case when , substituting (6) and (7) in (17), the conditional PDF takes the form     *11112 211112 21112 2lnln ]sjkssrrsrssktpRt pRtpRtpR tpRtpR tpRt2 2112 2,.krrsssrpRtphh tttt p 1m(18) And in the case when , substituting (6) and (7) in (17), the conditional PDF takes the form *211112 211 2112 2sj s11221211112 2.,srsrrmrrsrmsssssrttttttt (19) In the following, we considered two cases: the first is when the mixing proportion p is known and the second is when the two parameters ktpRtpRpR tpRpR tpRpR tpRph tph and p are assumed to be unknown. 3.1. Prediction When p Is Known In this section we estimate 1 and 2, assuming that the mixing proportion, 1p and 2p are known. 3.np r1.1. Maximum Likelihood Prediction Maximum likelihood predictio can be obtain using (18) and (19) by replacing the shape arametes 1 and 2 by 1ˆML and 2ˆML which is obtained from (15). 1) Interval prediction The MLPI for any future observation st, s = r + 1, r + 2,, n can be obtained by   *212ˆˆ,dssML MLkt t*112ˆˆPr,d ,1,sssML Lvtk tm,1.Mtmv t (20) A 1 100% MLPI (L,U) of the futu observ- tionre a st iby snear s given olving the following two nonliequations P) 2str1, Pr(.2stLtt tUt (21) 2) Point prediction The MLPP for any future observation ,st s = +2,,n can be obtained by replacing the shape pa-rameters 1r + 1, r by ˆ1ML2 and 2 and ˆML which, ob- tained from (15)  112*212,d,1,ˆˆ,d,1ss sML MLvssML MLvEtkt tmkt tm*ˆˆˆsMLt. (22) 3.1.2. Bayesian Prediction When the mixing proportion, p is known. Let the para- meters j,1, 2j have a gamma prior distribution with PDF  11π,,, 0.Γ()jj jjjjjjjjje  (23) These are chosen since they are the conjugate priors for the individual parameters. The joint prior density function of 12, is given by 1π,21112 221πππ,jjjjjje (24) where 1, 2, 0, ,0.jjjj Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI359 It then follows, from (13) and (24), that thven by e joint posterior density function is gi 211*11111222πjjjjjijjmitA eR tpR  1111 2 211112 2]rriiriiirrptph tphtpR tpRt (25) where 1πd.ALt1 (26) The Bayes predictive density function can be obtained using (18), (19) and (25) as follow:   *1*1120d, 1,d,ssjssjQtttkt mQtttkt m1) Interval prediction iction interval, for the future observation **110**ππ1. (27) Bayesian pred;,,,snmkT 1,2, ,srr n can be computed by ap- proximated 1sQtt using the MCM[17,24], using the form *C algorithm, see 1*1*isis1,*dsrjijsitktt (28) ktQttwhere  is the number of generated parameters and ij, i1, 2,3,,. They are generated from the poste- rio) using Gibbs sampler and Me- 1 100% BPI (L,r density function (25troA polis-Hastings techniques, for more details see .  U) of the future observation st is given byg the following two nonlinear equa- tions solvin*1*1ritd1,2disjsiLisjskt tkt t  (29) *1*1d.2drisjsiUisjsitkt tkt t (30) Numerical methods are generally necessary to solve the above two equgiations to obtain L and U for a ven . 2) Point prediction a) BPP for the future observation st based on BSEL function can be obtained using  1,sEtt (31) where sBS sMLtt sMLt is the ML prediction for future obser- vation thest which can be obtained using (22) and sEtt can be obtained using  *Qd.1rsssstb) BPP for the future observation Etttttt (32) stBLINX loss functio based on n can be obtained using 1ln 1,sML sat atsBLte etaE  (33) where sMLt is the ML prediction for thevation future obser- st which can be obtained using (22) and t csatEean be obtained using *1d.sratsatssteQ ttt (34) 3.2. Prediction When p and θj Are Unknown We Ee then both of the two parameters thmixing proportion jp and ,1,2jj, are assumed to be unknown. 3.2.1. Maximum Likelihood Prediction Maximum likelihood prediction can be obtain using (18) ) byand (19 replacing the parameters p, 1 and 2 by  1ˆ,ˆMLMLp and 2ˆML which we obtained using (15). 1) Interval prediction The MLPI for any future observation ,st1, srr 2,, n can be obtained by   *212ˆˆˆ,, d,1.vssML ML MLvktp tm*112ˆˆˆPr,,d ,ss sML MLMLtt ktptm1,   (35) A 1 100% MLPI (L,U) of the future observa- tion st is given by solving the following two nonliner Equations (1). 2) Piction a2oint prede observation ,The MLPP for any futurst 1, srr 2,,n can be obtained by replacing p, 1the shape pa-rameters  and 2 by ˆ,ˆ 1MLML and 2ˆpMLwhich, obtained from (15).   *112*212ˆˆˆˆ,,d, 1,ˆˆˆ,, d,1.sss ssMLML MLMLtss sML MLMLttEttktp tmtkt ptm (36) 3.2.2. Bayesian Prediction Let jp and ,1,2jj, are independent random vari- ab,Betabb and for1, 2,les such that 12~pjj to foon ith PDF llow an inverted gamma prior distributiwCopyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI360   ,,,jj (37)A joint prio110.Γjjjjjjjjjjjpe  r density function of 12,,p is then given by  21321π,,j1212111122ππππ1jjjje 1,pbb jpppj1(38) wher 12 10p p and for 1,20,jje  ,, 0.jjjb Using the likelihood function (13) and the prior density function (38), the posterior density function will be in the form    *211112 21112 211221π,ijjmriiiriir riptepR tpRtph tphtpR tpRt 9) 1,r2112 11121 22jjjjjbbAp p (3where 12πd.ALt (40) The Bayes prediction density function of TT,, ,snmk can be obtained, ssee , by 10*2**210**122π,,π,,ssjsjQttptkt pptkt p00dd,1,dd,1.p mp m (41 1) Interval prediction ction interval, for the future observ)ation Bayesian predi;,,,snmkT 1,2, ,srr n can be computed by ap- proximated sQtt using the MCMC algorithm, see , using the form *2*1*,,,diisjiii*21rssjsitQtt (42) 1,2,3,,kt pkt p t where ,,iijpi are generated from the ing Gibbs sampler an1 100% BPI (L,U) pos- d Me- terior density ftropolis-Hastinof the futuunction (39) usgs techniques. A re observation st is given by solving the fol- s lowationing two nonlinear equ**,d1,2,diisj siisj sktp ttp t11riLitk, (43) *k11*,d,2,driisj siUiiisj sttp tktp t (44) Numerical methods are generally necessary to solve ve two equations the aboto obtain L and U for a given . 1) Point prediction BPP for the future observation st based on BSEL function can be obtained using  1,ssBSsMLtt Ett  (45) where sMLt is the ML prediction for thevation future obser- st which can be obtained using (36) and sEtt can be obtained using  *2Qdr.ssstEtttttt2) BPP for the future observation s (46) st based on BLINX loss function can be obtained using 1,sML slnne Eet1at atstBLa (47) where sMLt is the ML prediction fr the future obser-vation ost which can be obtained using (36) and sEe t can be obtained using at*ssat at2.drsstEeteQttt (48) 4. Prediction in Case of Two-Sample Scheme Based on the informative 1;,,2;,,,nmknmkTT ;,,,,r nmkTdrawn  GOS from the MTR distribution and let 1NYY, where ;,, ,1,2,,,0, 0iiNMKYYiNM K be a se- cond independent generalized ordered random sample (of size N) of future observations from the samtribution. We want to predict any future (unobbYe dis- served) GOS ;,, ,1,2,,,bNMKYb N in the future sample of size . The PDF of ,1bYbN given the vector of paramters Ne, is:  *1110,1,bbjMbbbbjbjGyRy hyRyM  11[ln , 1,KbbbbRy RyhyM  (49) j where 11jbjb  and 1jM jKN Substituting from (6) and (7) in (49), we have:  *11122221110bbb bbbbjbjGyp yphypR y11112 2,1,bjMbRypRy phpR yM  (50) Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI361  *211122111ln ,bKbbbGypRypRy phpR ypRyM2211221,bbbbyphy ediction When nown 4.Maximum likelihood prediction in using (50) by replacing the shparam1 (51) 4.1. PrP Is K1.1. Maximum Likelihood Prediction caape n be obtaeters and (51)  and 2 by 1ˆML and 2ˆML 1) Interval prediction The MLPI for any futu,1bybNre observation  can be obtained by   *112*212ˆˆPr ,ˆˆsbML MLvML MLvtGyGy td ,1,,d,1.bbbyMyM1100% MLPI (L,U) of the future observa- is given by solving the following two nonlinear (52) A tion ybequations  Pr1, PryLt yUt22bbthe Mr anyutuvatiob can be ob- t sram1 t (53) 2) Point prediction TLPP fo fre ohapbsere panyeters ained by replacing the and 2 by 1ˆML and 2ˆML   212,d,1.bbML MLyy M4.1.2. Baye*1120*0ˆˆ,d,1,ˆˆbbMLbb bML MLbyG yy MyG sian PrediThe predictive dens,1bYbN is given by: ˆyEyt (54) ction ity function of *1πd, 0,b by ty (55) **0byGtwhere for 1M and 1m   21***10211112 2111112 21122111 22111220πdd.ijjjjrbbmrjiijiriir ribbjbbtG yepRtpRtph tphtpR tpRtphyphy 111 2 2bbbpRypRy 1jMbbjpR ypR yyt (56) Also, when 1M and 1m      221*101211112 2111112 211221111 2211 2211122πdlnd.jjjjrbrjiijiriir ribbb bKbbyttG yepRtpRtph tphtpR tpRtpRypRy phyphypR ypRy b2**b  (57) 1) Interval prediction ction ifor the future observation bYBayesian predinterval, ,1 bN, compcan beuted using (56) and (57) which can be approximated using MCMC algorithm by the form *1**10dibjibibj biGyyt Gy y (58) where ,1,2,3,,iji are generated from the post- erior density function (25) using Gibbs sampler and Metropolis-Hastings techniques. A 1 100% BPI (L,U) of the future observation by is given by solving the following two nonlequatioinear ns *11*0d1,2dibj biLiibj bGy yGy y, (59) *11*0d,2dibjbiUiibj bGy yGy y (60) Numerical methods such as Newton-Raphson are gen- ecessary to solveerally n the above two nonlinear Equa- tions (59) and (60), to obtain L and U for a given . 2) Pn a) BPP for the future observation by based on BSEL function caoint prediction be obtained using  ˆ1,bBS bMLbyy Eyt (61) where ˆbMLy is the ML prediction for the future obser- vation by which can be obtained using (54) and bEyt can be obtained using  *0d.bbbbEytyyt y (62) b) BPP for the future observation by based on BLINX loss function can be obtained using Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI362 ˆ1ln 1,bML bayye Eet (63) bBLaywingahere ˆbMLy is the ML prediction for the future obser- vation by which can be obtained us (54) and bayEe t can be obtained using *.dbbay ayEeteyty (64) 4.2.1. Maximum Likelihood Prediction Maximum likelihood prediction can be oand (51) by replacing thete, 0bb4.2. Prediction When p and θi Are Unknownbtain using (50) e paramrs p1 and 2 by  1ˆ,ˆMLMLp and 2ˆML 1) Interval prediction The maximum likelihood Interval prediction (MLIP) for any futu,1bybN can be by re observation obtained   2121,ˆd,1.bML ML MLvp yMva- ear *112*ˆˆˆPr,,d ,ˆˆ,,sb bMLML MLvtGy pyMGy tb(65) A 1100% MLPI (L,U) of the future obsertion by is given by solving the following two nonlinequations  Pr1, Pr22bbyLtt yUtt   (6) 62) Point prediction The MLPP for any future observation ,1bybN can be obtained by replacing the parameters p, 1 and 2 by ˆ,ˆ1MLML 2ˆp and ML   *1120*212ˆˆˆˆ,,ˆˆ ˆ,,bMLML ML MLML MLvbbbMLyGy pEytyyG ypd, 1,d, 1.bbbbyMy M (67) . Btive density ,1bybN is given by: 4.2.2ayesian Prediction The predicfunction of 1**2*,πdd ,0b byGyp py  tt (68) 00bwhere for 1M and 1m 111***2πyGy00 ,dd.bbp ptt (69) henAlso, w 1M and 1m *2πdd .p p221**00 ,bbyGytt (70) 1) Interval prediction Bayesian prediction interval, for the future observation ,1bybN, can be computed using (69) andwhich can be approximated using MCMC algorithm by the form (70) *,,iibjiijbGypdy (71) e ,,1,2,,iijpi1*ibyt10biGypwherare generated from the pos- terior density function (39) using Gibbs sampler and Me- tropolis-Hastings techniques. A 100%(L,U) o the future observation en by the flowing two nonlinear equations 1 BPI fby is givsolving ol*11*0,d1,2iibjbiLiibjbGyp yG y,diyp (72) *1,d,2,diibjbiUiiGypyyp y1*iG0bjb (73) Numerical methods such as Newton-Raphson are nec-essary to solve the above two nonlinear equations (72) and (73), to obtain L and U for a given . 2) Point prediction a) BPP for the future observation by bafunction can be obtained using  sed on BSEL 1,LbEyt  (74) where ˆbMLy is the ML prediction for the future obser- vany ich caˆbBS bMyy bwhtio n be obtained using (67) and byt E 0d.bbbbEytyyt y (75) 2) BPP for the future observation by baseloss function can be obtained using d on BLINX ˆ1ln 1bMLbBLayye Eea,bay t (76) where ˆbMLy is the ML prediction for the future obser- vation by which can be obtained using (67) and bayEe t can be obtained using 0d.bbay bbEe teyy (77) ayt5. Simulation Procedure In this subsection we will consider the upper record val- ich cues whan be obtained from the GOS by taking 1, 1mk and 1r. In this section, we willnt a com- pute poind interval predictors of future upper record Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI Copyright © 2012 SciRes. OJS 363values in two cases, one sample and two tion as following: The fos are used to obtain ML prediction nd Bayesian prediction (poiininnrsample predic- 5.1. One Sample Prediction llowing stepnterval) ar the rema(point and iinterval) font and g  failure times ,nmkr ,, 1,2ssTTr r 1) Fogiven values of 1,p,s and 2, upper record val rated from the MTR din. rate 1ues ofstributio2) Genedifferent sizes are gene ,iip and i2,1,2,,i from the posterior PDF using MCMC algorithm. merically, we get the recordues. 3) Solving Equations (21), 95% MLPI for unobserved uppenur val4) The MLPP for the future observation st, is com- puted using (22) when p is known and (36) when p and j are . unknown5) The 95% BPI for unobserved upper record are ob- tained by solving Equations (29) and (30) when p is known and (43) and (44) when p and j are un- known. 6) The BPP for the future observation st, is computed based on BSEL function using (31) when p is known and (45) when p and j are unknown. 7) The BPP for the future observation st, is computed based on BLINX loss function using (33) when p knowand (47) when pisn and j are unknown. t pti5.2. Two Sample Prediction The following steps are used to obain MLrediction (point and interval) and Bayesian predicon (point and interval) for future upper record value ,1,2.bsYb  1) For given values of 1,p and 2, upper record vafrolues of different sizes are generated m the MTR distribution. 2) Generate 1,iip and 2,1,2,,ii fromthe posterior PDF using MC algthm. CM ori p is known and (66) 3) Solving equations (53) whenwhen p and j are unknown we get the 95% MLPI nd (67) whn p anfor unobserved upper record values. 4) The MLPP for the future observation by, is com- puted using (54) when p is known aed j are unknown. 5) The 95% BPI for unobserved upper record are ob- tained by solving Equations (59) and (60) when p is er 915, θ = 3.19504, record values *T when (p = 0.4, θ = 1.24Table 1. Point and 95% interval predictors for the future uppΩ = 0.5). s1 2Point predictions (r, s) BLINEX a = (0.01, 2, 3) BSEL ML (3, r + 1) 1.30244 1.26763 1.1.22518 257 1.3027 (3, r + 2) 1.50949 461 1.441.4257(5, r + 154467 1.54467 1.5030+ 1) 1.46631 1.4558 1.49993 1.40629 (7, r + 2) 1.68695 1.62164 1.61.68742 1.51416 3 1.50998 1.36665 3 1.5449 1.46006 ) 1.(5, r + 2) 1.73144 1.67009 1.65177 1.73189 1.57457 (7, r 1.4997 0205 Interval predictions Bayes ML (r, s) L U length L U th leng(3, r + 1) 1.07653 2.16685 11.07261.59494 302 .09032 4 0.522(3, r + 2) 1.14454 2.61449 11.03031.80551 515 11.34031.76244 102 (5, r + 2) 1.41102 2.76539 1.7 1.30879 1.94403 0.635244 (7 ) 1.30032 2.31374 1.0192 1.69241 0.397485 (7, r + 2) 1.37071 1.34801 1.2662 1. 0.601199 .46995 6 0.77(5, r + 1) 1.34499 2.34997 .00498 4 0.4223543, r + 1342 1.2942.71872 8674 T. A. ABUSHAL, A. M. AL-ZAYDI364 Table 2. Point and 95% interedictors fohe future up values when (p = 0.391789, θ1 = 0.307317, θ2 = 3.33166, 5). t predictiorval pr tper record*sTΩ = 0.Poinns (r, s) BLINE 2, 3) EL MX a = (0.01,BSL (3 7 2.2174, r + 1) 2.322 2.23435 2.21354 2.32287 (3, 2) 2.68054 2.52154 2.48465 2.68226 2.4958 (5, 1) 2.81243 2.74901 2.73223 2.813 2.74014 2.66724 2.62719 2.61879 2.6678 2.62507 (7, r + 2) 2.8745 2.79894 2.72.87574 2.79966 r + r + (5, r + 2) 3.13233 3.01438 2.98352 3.13346 3.00112 (7, r + 1) 8385 Interval predictions Bayes ML (r, s) L U lL U gth ength len(3, r + 1) 1.91839 421.91487 2.94288 01 .03255 .11416 1.028(3, r + 2) 1.80949 53.1.82973.35066 09 412.46333.42614 751 522.53063.82785 24 (7, r + 1) 2.44491 3.71097 1.26605 2.44436 3.08759 0.643231 (7, r + 2) 2.39459 4.67043 2.27746 3.37074 0.97328 .02409 2146 7 1.52(5, r + 1) 2.46589 .17173 .70584 9 0.962(5, r + 2) 2.54923 .01563 .46641 1 1.297584 2.39 Table 3. Point and 95% interval predictors foe future upprd values Y b = 1, 2 when ( 0.4, θ1 = 1.22 = 3.1950 5). int predictior ther reco*b, p =4915, θ4, Ω = 0.Pons (r, b) B0.01, 2, BSELINEX a = (3) L ML (3.1) 0.669076 0084 0.552831 0.6690.58789 0.55108 (3.2) 1.06708 0272 0.901294 1.0689 1745 0.554524 0.6680.(5.2) 1.05059 0.926826 0.888678 1.05157 0.879144 (7.2) 0.999761 0.874356 0.838862 1.00077 0.790802 0.9414 0.905675 (5.1) 0.667370.58033 557414 (7.1) 0.637403 0.552511 0.526817 0.638057 0.506104 Interval predictions BayL es M(r, b) L Ulength (CP L ) ) Ulength (CP(3.1) 0.2.031.91158 (96.0.0850494 166 1. 123947 553 16) 1.4738661 (97.70)(3.2) 0.2.702.32429 (94.0.267226 068 1. 0.1.971.85479 (95.0.0926795 062 1. 0.2.582.20882 (94.0.289275 782 1. (7.1) 0.122419 1.93781 1.81539 (96.01) 0.0854495 1.20807 1.12263 (95.03) 0.376222 2.51787 2.14165284 1.55638 1.2901 (95.43) 37762 191 97) 1.963957 (98.54)(5.1) 122991 778 90) 1.3626794 (96.50)(5.2) 377125 594 75) 1.7546854 (97.63)(7.2) (94.49) 0.266Copyright © 2012 SciRes. OJS T. A. ABUSHAL, A. M. AL-ZAYDI Copyright © 2012 SciRes. OJS 365Table 4. Point and 95% interval predictors future upper record values *bY, b = 1, 2 when (.391789, θ1 = 0.307317, θ2 = 3.33166, Ω = 0.5). predifor thep = 0Pointctions (r, b) a =(0.BSEL BLINEX 01, 2, 3) ML (3.1) 869 696151 63902 0.878460.8760.0.2 0.827878 (3.2) 559 22759 09298 1.5681) 0.1 707462 647328 0.889363 ) 375 25773 12298 1.5657(7.2) 1.44473 1.15656 1.04976 1.44702 1.26946 1.561.1.1 1.47625 (5.188790.0.0.828237 (5.21.561.1.9 1.46954 (7.1) 0.813281 0.645203 0.598027 0.814804 0.695196 Interval predictions Bayes ML (rL U length (CP) L U ) , b) length (CP(30.10032.95917 8588 (97.15357 1 5.56) .1) 72 2.) 0.09742.48912.39167 (9(30.32003.97264 65263 (97.32)313335 5 36) (50.09812.80057 5 (96.91446 4 5.42) (50.31903.64706 32801 (97.66)297965 2 16) (7.1) 0.0971018 2.83087 2.73377 (96.94) 0.0826299 2.22201 2.13938 (94.13) (7.2) 0.304841 3.74134 3.4365 (343 2.87667 2.61324 (93.33) .2) 05 3. 0.3.18482.87151 (95..1) 198 2.7024) 0.09152.42922.33769 (9.2) 49 3. 0.3.08572.78776 (95.97.94) 0.26 known and (72) and73) wh p and (enj are un-known. 6) TPP for theeris coputed b onLusi pknow 4) he B future obs1)vatio Point and 95% interval pr for futr- ns are obned using a onsample anle es b MTR ion. e lized record v is evm all tabat, the lengthe nd BPI as thle size increase. r fiple size lengthI PI inincreasiedictorse-ure obsed two-samptaischem ased on adistributOur results arspecia to upperalues. 2) ItMLPI aident frodecreaseles the samps of th3) Foxed samr the s of the MLPand Bcrease by ng s ovating (on by, m- ased61) when is BSE when function n and (7p and j arunk7) TPP for thervatis coputed bon BLon usihen is know6) e nown. ion y, he Be future obsbng (63) wm- ased INXwhe loss functi p n and (7n p andj arn. 8) Ge 10, 000 sameach of size 6 fromTR dtion, then calcue covercentagw mn,e unknownerateistribuples late thN = rage pe a e r b. The pee coveraves be number of observed values. CES ps, “A Concep4) rcentagge improy use of a largM(CP) of bY. The computational (our) results ere coputed by using Mathematica 7.0. When p is know the prior parameters chosen as 12 1 22.3, 2.7,0.5, 1.3 which yield the generated values of 11.24915and 23.19504. While, in the case of four parameters are unknown the prior parameters1212 1 2,,,,, bbcc dd cho- sen as 1.2,2.3, 2, 2,0.3,3 which yield the generat vaREFEREN U. Kamt of Generalized Order Statistics,” Journal of Statistical Planning and Inference, Vol. 48, No. 1, 1995, pp. 1-23. doi:10.1016/0378-3758(94)00147-N  M. Ahsanullah, “Generalized Order Statistics from Two rm Distribution,” Communications in Sta- nd Methods, Vol. 25, No. 10, 1996, pp. .1080/03610929608831840edParameter Unifotistics—Theory alues of 0.391789p, 120.307317, 3.33166. 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