Shrinkage Estimation of Reliability Function for Some Lifetime Distributions

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1 Ameican Jounal of Computational and Applied Mathematic 4, 4(3): 9-96 DOI:.593/j.ajcam Shinkage Etimation of eliability Function fo Some Lifetime Ditibution anjita Pandey Depatment of Statitic, niveity of Delhi, 7, Delhi Abtact In thi pape, the poblem of hinkage etimation of eliability function i conideed fom a type II cenoed ample, auming availability of pio etimate of eliability function. We develop fou empiical hinkage etimato fo the eliability function petaining to a cla of lifetime ditibution compiing of five ditinct failue time ditibution commonly ued fo modelling life-data in the induty. The popoed etimato ae compaed Maximum Likelihood Etimato (MLE) in tem of the mean quaed eo (MSE). All the popoed empiical hinkage etimato ae hown to be pefeable to MLE a the hunken etimato emain moe efficient in compaion. A the pio value of the eliability function appoache cloe to the tue value of the eliability function, the efficiency of the popoed etimato impove, fo all the tuncation numbe. Keywod Family of Lifetime Ditibution, Type II Cenoing, Shinkage Etimation, eliability Function. Intoduction Shinkage etimato wee intoduced in liteatue by Thompon (968). Infeential etimation i often undetaken in the backdop of pio knowledge in the fom of exiting data, which a eeache may utilize in ode to get a moe pecie and accuate etimate. In many ituation, the expeimente ha ome pio infomation egading paamete in the fom of a point gue value. To utilize thi gue value the hinkage etimato fo mean, hape and cale paamete of vaiou lifetime ditibution have been dicued by a numbe of autho unde Bayeian and non-bayeian et up. See, fo intance, Pandey (983), Caella and Lehmann (988), Pakah and Singh (6, 8, 9), Singh et al. (). Howeve, hinkage etimation of eliability function fo any of the known lifetime ditibution except exponential ditibution (Te and To (996) and Chiou (993)) ha not been addeed o fa, in the autho' knowledge. The advantage of incopoating pio infomation in the eliability function though the hinkage etimato ha been tudied by Baklizi (3). The cuent wok i an effot to coin unified hinkage etimato of eliability function fo five lifetime ditibution commonly ued to model lifetime data by biologit, phyicit, enginee and tatitician fo living and non-living entitie. Chatuvedi and Singh (6) have deived a family of lifetime ditibution which compie of * Coeponding autho: anjitapandey@gmail.com (anjita Pandey) Publihed online at Copyight 4 Scientific & Academic Publihing. All ight eeved thee five impotant lifetime ditibution, namely, exponential, Weibull, Bu, Paeto and ayleigh ditibution. They have tudied behavio of the hazad-ate fo diffeent membe of thi family. The peent pape popoe fou diffeent veion of hinkage etimato fo the eliability function of a family of lifetime ditibution when the data i type II cenoed and a pio etimate of the eliability i given. The peent wok i an effot to povide a ingle efficient eliability etimato fo the five mot commonly ued lifetime ditibution in Induty, unde diffeent cenoing cenaio, fo Type II cenoed data. The etimato uggeted in the peent pape ae baed on MLE, MVE and the P-value. Thee etimato pefom bette than the moe popula Maximum Likelihood Etimate (MLE) ued in ditibution theoy, ee fo intance, Zack and Even (996). The Shinkage etimato ae developed in ection. Numeical computation ae caied out to invetigate the behavio of thee etimato unde ection 3. eult ae dicued in the final ection.. Shinkage Etimation andom vaiable X follow the family of lifetime ditibution paamete and. The pobability denity function of X i given by ' fx ( x;, ) = ( / ) g ( x)exp( g ( x) / ) () whee ( x,, ) > Let g(x), x >a be eal- valued, diffeentiable, tictly inceaing function of x (inceaing to infinity) g(a)=

2 Ameican Jounal of Computational and Applied Mathematic 4, 4(3): and g (x) tand fo the deivative of g(x).we aume that the invee function g (x) of g(x) exit. The following ae pecial cae: (i) Fo g(x) = x, =, we obtain exponential ditibution. (ii) Fo g(x) = x, we obtain Weibull ditibution. (iii) Fo g(x) = log(+x b ), b> and =,we obtain Bu ditibution. (iv) Fo g(x) = log(x/a) and =, we obtain Paeto ditibution. (v) Fo g(x) = x, =, we obtain ayleigh ditibution. The quantity we want to etimate i (t) = P(X>t) =-F(t) =exp[-g (t)/ ] We conide a andom ample of ize n fom thi ditibution uch that the data i cenoed when a pedetemined numbe of failue occu: Type II cenoing. In type II cenoing teting i teminated at the obeved time of the th failue. The cenoing time i a andom vaiable while i fixed. Joint pdf of,,, ) i given by i= ( x( ) x() x( ) x x x h(,,, ; ) () () ( ) = nn ( ) ( n + )( ) g ( x() i ) { g'( x() i ) g ( x() i )}exp( ) g ( x( ) ) n {exp( )} Auming to be a known quantity, mle of i = i= whee () i ( ) = g ( x ) + ( n) g ( x ) Baklizi (3) ha uggeted hinkage etimato of the following fom = α + ( α) whee = exp( ) Such that i etimated by mle of, i the pio gue of the expeimente which may be baed on infomation baed on the pat expeience. Fo a pecified value of, t, we can calculate () (3) α g () t ln = (4) i aumed to lie between and incluive, which may be detemined in the following way : α i choen uch that it minimize mean quae eo of given by α + ( α). Theefoe, α = whee ( ) )[ ) ] ) + α (5) It i known that unde Type II cenoing exactly. Thu, we have =, Va( ) = E We potulate the following notation: ( t) = M MLE of the eliability ( t) = MVE of eliability We now obtain g ( t) M ) = exp( ) + '' ) Va ( M ) = [ ' )] = ( ) = exp E ( ) = exp E ' M = ' = exp ' M E = exp ( E ) E χ () (6)

3 94 anjita Pandey: Shinkage Etimation of eliability Function fo Some Lifetime Ditibution '' '' M = ( ) = exp 4 3 '' M E exp( = = ( ) = exp )exp ( E) E ' M E = exp ( E ) E g t g t E ( M ) = exp + exp( ) () (), S / 4 3 = Va( M ) = exp, 4 3 = S / ) = Va( ) + ( E ) M M M Next uing MVE of eliability function, () t =, < (Chatuvedi & Singh, 6) ing Taylo expanion, ( ) =, e x = + x! + x! + x3 +, < xx < 3! ' ( ) = ( )) + ( )) ( )) + ( )) '' ( )) Theefoe we can now wite a unde, (7) (8) (9) () () () E 3 By definition, ( ) = + ( ) 3 3 ( ) 4 3 Va ( ( )) = 4 ( ) g ( t) 4 (3) (4) = + (5) ) Va( ) ( E ) The fit hinkage etimato which i obtained by ubtituting M, E E M i baed on MLE E M fo a pecified value of and, E, each, in expeion.3 and.5. The econd hinkage etimato i baed on MVE which i obtained by ubtituting, E E fo a pecified value of expeion.3 and.5 The thid hinkage etimato of the following tet T H χ E = v H : and, : =, i aumed unde H. () E, each, in 3 i baed on P-value Let t epeent the obeved value of the tatitic T. The P-value fo thi tet i z = min {P H (T>t), P H (T<- t)} = min {-F(t), F(-t)} (6) whee F(.) i the ditibution function of T. A lage P-value indicate that i cloe to the pio etimate (Te & To, 996). The popoed hinkage etimato i obtained by eplacing -α the P-value of

4 Ameican Jounal of Computational and Applied Mathematic 4, 4(3): thi tet. The Fouth hinkage etimato 4 i baed on the quae oot of the P-value of the above tet uch that the hinkage etimato i obtained by eplacing -α the quae oot of the P-value of thi tet. 3. elative Efficiencie of the Etimato Baed on the following data taken fom Matz and Walle (98, p.395), vaiou hinkage etimato ae calculated and thei pefomance i aeed elative to MLE uing atio of Mean Squae Eo, epeented by E i,, i=,,3,4. Nine failue time fo a heat exchange ued in alkylation unit of a gaoline efiney wee ecoded a unde:.4,.58,.75,.83,.,.8,.7,.5, and.35 yea. A Weibull ditibution (g(x)=x), paamee = 3.5 ha been ued to model thi data. The family of lifetime ditibution that ha been conideed in thi pape contain Weibull ditibution a a pecial cae. nde Type II cenoing, fo the ubequent computation, the following ae potulated:. : the tue value of the eliability, which i taken a.5,.75 and.9.. Θ : taken a.5,.8, and Tuncation numbe =,5,7,9 have been conideed. We, thu, invetigate efficiency of the newly contucted hunken etimato elative to the MLE by potulating the above pemutation of cenaio fo the tue value of eliability and the unknown paamete coeponding to fou diffeent cenoing popotion fom the oiginal ample. 4. Concluion nde type II cenoing, the elative efficiencie of the popoed hinkage etimato (elative to MLE) ae hown in table. The hunken etimato have been found to be bette than the MLE which ae conventionally employed by pactitione fo the pupoe of paamete etimation, epect to elative efficiency. The developed hinkage etimato pefom bette than the uual ML etimato when the gue (o the aumed tue) value of the eliability function i appoximately the tue value of the paamete fo mall ample ize. It i evident that i mot efficient followed by 3, 4 and, in that ode, fo the entie ange of Θ. Highe paamete value give moe efficiency to all the popoed hinkage etimato. It i obeved that lee the cenoing popotion, moe i the efficiency in the contucted etimato, which implie that the popoed etimato emain moe pecie and cloe to the tue paamete value even when a lage ample popotion ha been tuncated. nde the empiical tudy etup, the popoed etimato ae found to be highly efficient even coeponding to a Cenoing Popotion which i a low a.3. Thu, all the fou popoed hinkage etimato ignificantly impove the taget eliability function in compaion the tandad MLE. Table. Pefomance unde type II cenoing uing elative efficiency indice CP Θ E E E3 E4 = CP Θ E E E3 E4 = CP Θ E E E3 E4 = CP Θ E E E3 E4 = CP = Cenoing Popotion = tuncation numbe/actual ample ize

5 96 anjita Pandey: Shinkage Etimation of eliability Function fo Some Lifetime Ditibution EFEENCES [] Baklizi, A., 3, Shinkage Etimation of the Exponential eliability cenoed data. Focu on Applied Statitic, Nova Science Publihe Inc. Ed. Mhd. Ahanullah [] Caella, G. and Lehmann, E., 998, Theoy of Point Etimation, Spinge-Velag. [3] Chatuvedi, A. and Singh K.G., 6, Bayeian Etimation pocedue fo a Family of Lifetime Ditibution unde quaed Eo and Entopy Loe, Meton, 64, [4] Chiou, P., 993,, Communication in Statitic- Theoy and Method, (5), [5] Matz, H. F. and Walle,.A., 98, Bayeian eliability Analyi, John Wiley Son Inc. [6] Pandey, M., 983, Shinkage etimation of the Exponential cale paamete. IEEE Tan. el., 3-5. [7] Pakah, G. and Singh, D.C., 6, Shinkage Tetimato fo the Invee Dipeion of the Invee Gauian Ditibution unde the LINEX Lo Function, Autalian Jounal of Statitic, 35(4), [8] Pakah, G. and Singh, D.C., 8, Shinkage Etimation in ExponentialType II Cenoed Data unde the LINEX Lo Function, Jounal of Koean Statitical Society, 37(), [9] Pakah, G. and Singh, D.C., 9, A Bayeian hinkage appoach in Weibull Type-II cenoed data uing pio point infomation. EVSTAT, 7(), [] Singh, H. P. and Shukla, S. K.,, Etimation in the Two Paamete Weibull Ditibution Pio Infomation, IAPQ Tanaction, 5(), 7-8. [] Thompon, J., 968, Some Shinkage Technique fo etimating the Mean, JASA, 63, 3-. [] Te, S. and To, G., 996, Shinkage Etimation of eliability fo Exponentially Ditibuted Lifetime, Comm. Stat. Simul. Comp, 5(), [3] Zack, S. and Even, M., 996, The Efficiencie in Small Sample of the Maximum Likelihood and Bet nbiaed Etimato of eliability Function, JASA, 6, 33-5.