Shrinkage Testimator in Gamma Type-II Censored Data under LINEX Loss Function

Transcription

1 Open Jounal of Statistis, 03, 3, Published Online August 03 ( Shinkage Testimato in Gamma Type-II Censoed Data unde LINEX Loss Funtion Ali Shadokh, Hassan Pazia Depatment of statistis, Payame Nou Univesity of Tehan, , Ian Reeived Septembe, 0; evised Januay, 03; aepted Januay 8, 03 Copyight 03 Ali Shadokh, Hassan Pazia. This is an open aess atile distibuted unde the Ceative Commons Attibution Liense, whih pemits unestited use, distibution, and epodution in any medium, povided the oiginal wok is popely ited. ABSTRACT Pakash and Singh pesented the shinkage testimatos unde the invaiant vesion of LINEX loss funtion fo the sale paamete of an exponential distibution in pesene Type-II ensoed data. In this pape, we extend this appoah to gamma distibution, as Pakash and Singh s pape is a speial ase of this pape. In fat, some shinkage testimatos fo the sale paamete of a gamma distibution, when Type-II ensoed data ae available, have been suggested unde the LINEX loss funtion assuming the shape paamete is to be known. The ompaisons of the poposed testimatos have been made with impoved estimato. All these estimatos ae ompaed empiially using Monte Calo simulation. Keywods: Gamma Distibution; Shinkage Estimato and Fato; Asymmeti Loss Funtion; Level of Signifiane; Testimation; Monte-Calo Simulation. Intodution In life-testing eseah, the most widely used life distibution is the Gamma with pobability density funtion fo any andom vaiable x; x f x;, e ; x0,, 0. (.) Let x, x,, xn be the andom samples of size n taken fom the Gamma distibution. The paamete and ae alled the shape and sale paamete, espetively. It is uial to have in-depth study of the (Classi and Bayes) estimate of the sale paamete of Gamma distibution beause, in seveal ases, the distibution of the minimal suffiient statistis is Gamma (see Pasian and Kimani []). Pazia and Shadokh [] deived Bayes estimatos of the sale paamete of gamma distibution on the two asymmeti loss funtion LINEX and Peautionay by using seveal pio distibutions and then ompaed the effiieny of all estimates. In the pesent pape, onentation is on the gamma distibution. Feguson [3], Zellne and Geisel [4], Aithison and Dunsmoe [5], Vaian [6], and Bege [7] indiated to insuffiient to symmeti loss funtion and just Vaian [6] suggested asymmeti linea loss funtion. This loss funtion was widely used by seveal authos; among of them wee Basu and Ebahimi [8], Pandey [9], Soliman [0], and Pakash and Singh []. Following Basu and x Ebahimi [8], the invaiant fom of the LINEX loss funtion (ILL) fo any paamete is defined as ˆ Le ; 0,, (.) whee is the shape paamete and ˆ is any estimate of the paamete. The LINEX loss funtion is onvex and the shape of this loss funtion is detemined by the value of. The negative (positive) value of gives moe weight to oveestimation (undeestimation) and its magnitude eflets the degee of asymmety. It is seen that, fo =, the funtion is quite asymmeti with oveestimation being ostlie than undeestimation. If < 0, it ises almost exponentially when the estimation eo ˆ 0 and almost linealy when ˆ 0. Fo small values of, the LINEX loss funtion is almost symmeti and not fa fom squaed eo loss funtion. Pandey [9], Pasian and Fasipou [], Singh, Gupta, and Upadhyay [3], Misa and Meulen [4], Ahmadi, Doostpaast, and Pasian [5], Xiao, Takada, and Shi [6], Singh, Pakash, and Singh [7] and othes have used the LINEX loss funtion in the vaious estimation and pedition poblems. In life-testing, fatigue failues and othe kinds of destutive test situations, the obsevations usually oued in an odeed manne suh a way that the weakest items Copyight 03 SiRes.

2 46 A. SHADROKH, H. PAZIRA failed fist and then the seond one and so on. Let us suppose that n items ae put on life test and teminate the expeiment when (< n) items have failed. If X, X,, X denote the fist obsevations having a ommon density funtion as given in (.) then the joint pobability density funtion is given by whee f x, x,, x n! x e i n! i i i T (.3) T x n x (.4) T is a omplete suffiient statisti of θ and distibuted as gamma distibution with paametes λ, θ. The maximum likelihood estimato (MLE) of θ is given by ˆ T ~ (.5) and an easily show that ˆθ is the minimum vaiane unbiased estimato (MVUE) of θ. Roa and Sivastava [8] onsideed a lass fo the total test time as T Y d, and found the value of the onstant d e d (say) whih minimizes the isk of Y unde the ILL. The minimum isk estimato is T Y d with the minimum isk unde ILL Risk Y e, (.6) fo, see Pakash and Singh []. In the pesent pape, some shinkage testimatos fo the sale paamete of a gamma distibution, when Type-II ensoed data ae available, have been suggested unde the ILL loss funtion assuming the shape paamete is to be known.. Shinkage Testimatos and thei Popeties Following Thompson [9], the shinkage estimato fo the paamete is given by T 0 0 ˆ Y k,0k. (.) The value of the shinkage fato k nea to the zeo implies stong belief in the guess value 0 and nea to one implies a stong belief in the sample values. Seveal eseahes have studied the pefomane of the shinkage estimatos and found that the shinkage estimato pefoms bette with espet to any usual estimato when the guess value 0 is lose to the paamete. This suggests that we may test the hypothesis H 0 : 0 against H: 0. A test statisti T ~ 0 is available fo testing the hypothesis H 0. The loss fo estimato Y ˆ unde the ILL is defined as ˆ LY e, whee and k T. The isk of the poposed shinkage estimato Y ˆ unde the ILL is given by 0 ˆ Risk Y e e k k k (.) The value of k kmin (say), whih minimizes the isk Risk Y ˆ is thus obtained by solving the given Equation k k k e e. (.3) The value of k min depends upon the unknown paamete. Hene, an estimate ˆk of k min is obtained by eplaing the paamete to its minimum vaiane unbiased estimato. Based on this, the poposed shinkage testimato fo the sale paamete is defined as ˆ T ˆ 0 ˆ T d k kd ItT t, (.4) whee I A denotes the indiato of A, t m and t m. Hee 0 0 m and m ae the values of the lowe and uppe 00 % Copyight 03 SiRes.

3 A. SHADROKH, H. PAZIRA 47 points of the hi-squae distibution with degees of feedom. The isk unde the ILL fo the shinkage testimato ˆ is given by Risk ˆ f e Gw, w,e e Gw, w, f0 G w w f G w w G w w f whee,,,,,, 0 exp w, w m m, f wd 0 ˆ,, f k w, (.5) v w G u, v, y yw e dw u and y may be a funtion of w. Fo, see Pakash and Singh []. Waika, Shuumann, and Raghunathan [0] has suggested an idea of seleting the shinkage fato whih is the funtion of the test statisti i.e., unde H0: 0 T m m 0. T 0ksay m m m 0 Theefoe, the poposed shinkage testimato based on k is given by ˆ T T d k0 kd ItT t. (.6) The isk unde the ILL fo the shinkage testimato ˆ is given by Risk ˆ f e Gw, w,e e Gw, w, f0 G w w f G w w G w w f whee,,,,,, 0 exp w w f m m m. Fo, see Pakash and Singh []. When H0: 0 is aepted, m m m., (.7) If one is inteested in taking smalle values of the shinkage fato, he an take m. The poposed shinkage testimato is ˆ T T 3 d k 0 k d I whee t T t T k ; m m 0 (.8) it may be possible that the value of shinkage fato is negative so positive is taken. Adke, Waika, and Shuumann [] and Pandey, Malik, and Sivastava [] have onsideed this type of shinkage fato. The isk of the shinkage testimato ˆ 3 is given by Risk ˆ 3 f e Gw, w,e e Gw, w, f0 G w w f G w w G w w f whee,,,,,, 0 exp f w w m m., (.9) Fo, see Pakash and Singh []. The minimum value of onstant d, d obtained fo the lass Y dt, lies between zeo and one. Hene, it may be a hoie fo the shinkage fato. Thus, the poposed shinkage testimato may be onsideed as ˆ T 4 d d0i tt t (.0) The isk of the poposed shinkage testimato ˆ 4 unde ILL is given by whee ˆ f e G w, w,e 4 Risk e G w, w, f d G w, w,, (.) 0 exp d w f3 m m. Fo, see Pakash and Singh []. 3. Numeial Illustation The elative effiieny fo ˆi ; i,, 4, with espet to the minimum isk impoved estimato unde the ILL is defined as Copyight 03 SiRes.

4 48 A. SHADROKH, H. PAZIRA ˆ Risk Y RE i, Y ; i,,4 Risk ˆ i The expession fo the elative effiieny RE ˆ i, Y ; i,, 4, is the funtion of,,, and. Fo the Table. RE ˆ, Y seleted values of 6,8,0 ;,0.5,,.5 ; ; 0.0,0.05,0. and 0.5,.5, the elative effiienies have been alulated and pesented in Tables -8. Only positive values of ae onsideed beause oveestimation in mean life is moe seious when 0.5. = 6 = 8 = Copyight 03 SiRes.

5 A. SHADROKH, H. PAZIRA 49 Table. RE ˆ, Y when 0.5. = 6 = 8 = Copyight 03 SiRes.

6 50 A. SHADROKH, H. PAZIRA Table 3. RE ˆ 3, Y when 0.5. = 6 = 8 = Copyight 03 SiRes.

7 A. SHADROKH, H. PAZIRA 5 Table 4. RE ˆ 4, Y when 0.5. = 6 = 8 = Copyight 03 SiRes.

8 5 A. SHADROKH, H. PAZIRA Table 5. RE ˆ, Y when.5. = 6 = 8 = Copyight 03 SiRes.

9 A. SHADROKH, H. PAZIRA 53 Table 6. RE ˆ, Y when.5. = 6 = 8 = Copyight 03 SiRes.

10 54 A. SHADROKH, H. PAZIRA Table 7. RE ˆ 3, Y when.5. = 6 = 8 = Copyight 03 SiRes.

11 A. SHADROKH, H. PAZIRA 55 Table 8. RE ˆ 4, Y when.5. = 6 = 8 = Copyight 03 SiRes.

12 56 A. SHADROKH, H. PAZIRA than the undeestimation. 3.. When 0.5 Fom these tables it is obseved that the shinkage testimatos ˆ 4 pefom bette than the impoved estimato Y fo all onsideed values of,,, and α. The testimatos ˆ, ˆ and ˆ 3 pefom bette than Y when 0.5. The testimatos ˆ 4 attain maximum effiieny at the point δ = 0.4 and othes nea to the point δ =. Fo fixed and level of signifiane α, as the unensoed sample size ineases, the elative effiieny deeases in all onsideed values of δ fo all the testimatos. Fo fixed and α, when ineases the elative effiieny ineases in all onsideed values of δ fo all testimatos. It has been seen that as the level of signifiane α ineases the elative effiieny ineases in all onsideed values of δ fo all testimatos. 3.. When.5 Fom these tables it is obseved that the shinkage testimatos ˆ 4 pefom bette than the impoved estimato Y fo all onsideed values of,,, and α. The testimatos ˆ, ˆ and ˆ 3 pefom bette than Y when 0.5. The testimatos ˆ 4 attain maximum effiieny at the point 0.4 and othes nea to the point.6. Fo fixed and level of signifiane α, as the unensoed sample size ineases, the elative effiieny deeases in the egion fo the testimato ˆ, and in the egion fo the testimatos ˆ and ˆ 3, and also fo testimato ˆ 4 it deeases fo all onsideed values of δ. Fo fixed and α, when ineases the elative effiieny ineases in all onsideed values of δ fo all testimatos. It has been seen that as the level of signifiane α ineases the elative effiieny ineases in and also in when,0.5, and deeases fo when.5 and 8 fo testimatos ˆ, ˆ and ˆ 3, and fo testimato ˆ 4 it ineases fo all onsideed values of δ. 4. Reommendations In this study, some shinkage testimatos ( ˆ, ˆ, ˆ 3 and ˆ 4 ) fo the sale paamete of a gamma distibution, when type-ii ensoed data ae available, suggested that unde the ILL loss funtion assuming the shape paamete was to be known. The ompaisons of the poposed testimatos made with impoved estimato Y. The eommendations have been pesented, based on the elative effiieny fo all the shinkage testimato. Fom the pevious obsevations, the shinkage testimatos ˆ 4 pefom bette than the impoved estimato Y fo all onsideed values of,,, and α, and the testimatos ˆ, ˆ and ˆ 3 pefom bette than Y when 0.5. Sine the shinkage testimatos ˆ 4 always pefom bette than othe shinkage testimatos if the gain in effiieny does not matte, theefoe we stongly suggest using the shinkage testimatos ˆ 4 fo the sale paamete of a gamma distibution, when Type-II ensoed data ae available, suggested unde the ILL loss funtion. 5. Aknowledgements The authos would like to thank the efeee and the edito fo a aeful eading of the pape and fo valuable omments whih impoved the pesentation of the pape. The seond autho gatefully appeiates his wife, D. Saemeh Dehghan, who kindly helps the eseahe in the study. REFERENCES [] A. Pasian and S. N. U. A. Kimani, Estimation unde LINEX loss funtion, In: A. Ulla, Ed., Handbook of Applied Eonometis and Statistial Infeene, 65 Dekke, New Yok, 00, pp [] H. Pazia and A. Shadokh, Compaison of LINEX and Peautionay Bayes Estimatos on the Gamma Distibution Using Censoed Data, Jounal of Statistis and Management Systems, Vol. 4, No. 3, 0, pp doi:0.080/ [3] T. S. Feguson, Mathematial Statistis: A Deision Theoeti Appoah, Aademi Pess, New Yok, 967. [4] A. Zellne and M.S. Geisel, Sensitivity of Contol to Unetainty and Fom of the Citeion Funtion, In the futue of statistis, Ed Donald G. Watts, Aademi Pess, New Yok, 968, pp [5] J. Aithison and I. R. Dunsmoe, Statistial Pedition Analysis, Cambidge Univesity Pess, Cambidge, 975. doi:0.07/cbo [6] H. R. Vaian, A Bayesian Appoah to Real Estate Assessment, In: L. J. Savage, S. E. Feinbeg and A. Zellne, Eds., Studies in Bayesian Eonometis and Statistis: In Hono of L. J. Savage, Noth-Holland Pub. Co., Amstedam, 975, pp [7] J. O. Bege, Statistial Deision Theoy-Foundation Conepts and Methods, Spinge-Velag, New Yok, 980. doi:0.007/ [8] A. P. Basu and N. Ebahimi, Bayesian Appoah to Life Testing and Reliability Estimation Using Asymmeti Loss Funtion, Jounal of Statistial Planning and Infeenes, Vol. 9, No. -, 99, pp. -3. doi:0.06/ (9)908-c [9] B. N. Pandey, Testimato of the Sale Paamete of the Exponential Distibution Using LINEX Loss Funtion, Communiation in Statistis-Theoy and Methods, Vol. Copyight 03 SiRes.

13 A. SHADROKH, H. PAZIRA 57 6, No. 6, 997, pp doi:0.06/006-74(88) [0] A. A. Soliman, Compaison of Linex and Quadati- Bayes Estimatos Foe the Rayleigh Distibution, Communiation in Statistis-Theoy and Methods, Vol. 9, No., 000, pp doi:0.080/ [] G. Pakash and D. C. Singh, Shinkage Estimation in Exponential Type-II Censoed Data unde LINEX Loss, Jounal of the Koean Statistial Soiety, Vol. 37, No., 008, pp doi:0.06/j.jkss [] A. Pasian and N. S. Fasipou, Estimation of the Mean of the Seleted Population unde Asymmeti Loss Funtion, Metika, Vol. 50, No., 999, pp [3] U. Singh, P. K. Gupta and S. K. Upadhyay, Estimation of Exponentiated Weibull Shape Paametes unde LINEX Loss Funtion, Communiation in Statistis-Simulation, Vol. 3, No. 4, 00, pp doi:0.08/sac [4] N. Misa and E. V. D. Meulen, On Estimating the Mean of the Seleted Nomal Population unde the LINEX Loss Funtion, Metika, Vol. 6, No. 9, 003, pp doi:0.080/ [5] J. Ahmadi, M. Doostpaast and A. Pasian, Estimation and Pedition in a Two-Paamete Exponential Distibution Based on K-Reod Values unde LINEX Loss Funtion, Communiations in Statistis-Theoy and Methods, Vol. 34, No. 4, 005, pp doi:0.08/sta [6] Y. Xiao, Y. Takada and N. Shi, Minimax Confidene Bound of the Nomal Mean unde an Asymmeti Loss Funtion, Annals of Statistial Mathematis, Vol. 57, No., 005, pp doi:0.007/bf [7] D. C. Singh, G. Pakash and P. Singh, Shinkage Testimato fo the Shape Paamete of Paeto Distibution Using LINEX Loss Funtion, Communiation in Statistis-Theoy and Methods, Vol. 36, No. 4, 007, pp doi:0.080/ [8] A. K. Rao and R. S. Sivastava, Bayesian Estimation of the Sale Paamete of Gamma Distibution unde Linex Loss Funtion with Censoing, In: B. N. Pandey, Ed., Statistial Tehniques in Life-Testing, Reliability, Sampling Theoy and Quality Contol, Naosa Pub House, Dayaganj, 00, pp [9] J. R. Thompson, Some Shunken Tehniques fo Estimateing the Mean, Jounal of the Ameian Statistial Assoiation, Vol. 63, No. 3, 968, pp. 3-. doi:0.307/8383 [0] V. B. Waika, F. J. Shuumann and T. E. Raghunathan, On a Two Stage Shunken Testimato of the Mean of a Nomal Distibution, Communiations in Statistis-Theoy and Methods, Vol. 3, No. 5, 984, pp doi:0.080/ [] S. R. Adke, V. B. Waika and F. J. Shuumann, A Two Stage Shinkage Testimato fo the Mean of an Exponential Distibution, Communiation in Statistis-Theoy and Methods, Vol. 6, No. 6, 987, pp doi:0.080/ [] B. N., Pandey, H. J. Malik and R. Sivastava, Shinkage Testimato fo the Vaiane of a Nomal Distibution at Single and Double Stages, Mioeleton Reliability, Vol. 8, No. 6, 988, pp doi:0.06/006-74(88) Copyight 03 SiRes.