Bayesian Estimation of Shift Point in Shape Parameter of Inverse Gaussian Distribution Under Different Loss Functions

Transcription

1 Joural of Otzato Idustral Egeerg 8 (05) - Bayesa Estato of Shft Pot Shae Paraeter of Iverse Gaussa Dstrbuto Uder Dfferet Loss Fuctos Mohaad Saber Fallah Nezhad a,, Batul Rast b a Assocate Professor, Deartet of Idustral Egeerg, Yazd Uversty, Yazd, Ira b M.Sc, Deartet of Idustral Egeerg, Yazd Uversty, Yazd, Ira Receved 5 March, 04 Revsed 0 February, 05 Acceted 8 February, 05 Abstract I ths aer, a Bayesa aroach s roosed for shft ot detecto a verse Gaussa dstrbuto. I ths study, the ea araeter of verse Gaussa dstrbuto s assued to be costat ad shft ots shae araeter s cosdered. Frst the osteror dstrbuto of shae araeter s obtaed. The the Bayes estators are derved uder a class of rors ad usg varous loss fuctos. We assued ufor, Jeffreys, eoetal, gaa ad ch square dstrbutos as ror dstrbutos. The squared error loss fucto (SELF), etroy loss fucto (ELF), le loss fucto (LLF) ad recautoary loss fucto (PLF), are used as loss fuctos. We attet to fd out the best estator for shft ot uder varous rors ad loss fuctos. The roosed Bayesa aroach ca be adated to ay slar roble for shft ot detecto. Sulato studes were doe to vestgate the erforace of dfferet loss fuctos. The results of sulato study deote that the Jeffrey ror dstrbuto uder PLF has the ost accurate estato of shft ot for sale sze of 0, ad the gaa ror dstrbuto uder SELF has the ost accurate estato of shft ot for sale sze of 50. Key words: Bayes estators, shft ot, verse Gaussa dstrbuto, loss fucto.. Itroducto Whe the data set of dvdual observatos s avalable, a cotrol chart ca be used to detect a shft the araeters. A lkelhood rato test aroach s used to detere chages araeters. The lkelhood rato statstc s lotted for all ossble shft ot values, ad a arorate Uer Cotrol Lt (UCL) s chose. The locato of the au test statstc value corresods to the au lkelhood locato of oe shft ot (Sullva & Woodall, 000). Aother aroach for shft ot detecto s the Bayesa ferece. Srvastava (0) estated shft ot whch occurs ay sequeces of deedet observatos Posso odel about statstcal rocess cotrol ad couted the Bayes estators uder Asyetrc Loss Fuctos (ALF), Squared Error Loss Fuctos (SELF), Le Loss Fucto (LLF), Precautoary Loss Fucto (PLF) ad Geeral Etroy Loss Fucto (GELF). He foud that the asyetrc loss fucto was ore arorate. Soe of the loss fuctos lke le loss fucto (LLF) suggested ad studed by Vara (975) ad Basu ad Ebrah (99), geeral etroy loss fuctos (GELF) roosed by Calabra ad Pulc (996) ad recautoary loss fucto (PLE) studed by Norstro (996), are eloyed ths artcle. I ths aer we estated the shft ot shae araeter of verse Gaussa dstrbuto. The Bayesa aroach s eloyed for chage ot estato. Fallahezhad, Rast, ad Abooe (04) successfully aled Bayesa ethod for chage ot estato a sequece of eoetal dstrbuto. Kadlar ad Karasoy (007) reseted aother Bayes ethod for chage ot estato. Marcos D Agelo (0) roosed a fault detecto ethod ducto ache usg a fuzzy- Bayesa chage ot estato aroach. Other aroaches of chage ot estato are vestgated by Keraatour, Nak, Khedat, ad Soleyaa (0) ad Khedat ad Nak (03). The deftos of varous loss fuctos are gve Secto. We reset Lkelhood, ror ad osteror dstrbutos for, uder the assuto of dfferet rors Secto 3. The Bayes estato of chage ot usg dfferet Prors uder dfferet loss fuctos s obtaed Secto 4. The results of sulato studes are reseted Secto 5. A suary of results coes Secto 6. Fally we coclude the aer Secto 7. Corresodg author Eal address: Fallahezhad@Yazd.ac.r

2 Mohaad Saber Fallah Nezhad et al./ Bayesa Estato of Shft Pot.... Loss Fuctos A loss fucto reresets losses curred whe we estate the araeter by (Naz Sdhu & Asla, 03). A uber of asyetrc loss fuctos are roosed for dfferet alcatos. We use the followg loss fuctos for the chage ot detecto... The squared error loss fucto (SELF) I decso theory the loss crtero s secfed order to obta a good estator. The slest for of the loss fucto s the squared error loss fucto (SELF) that assgs equal agtudes to both ostve ad egatve errors. The squared error loss fucto (SELF) s oe of the ost wdely used loss fuctos decso theory. The wde alcato of ths syetrc loss fucto s due to ts atheatcal coveece. However, as ths loss fucto s syetrc, t fals to cosder the dffereces betwee overestato ad uderestato of ay araeter, thus ths loss fucto ay be arorate ost of the estato robles. Soetes overestato leads to ay serous cosequeces. The troducto of the asyetrc le loss fucto (LLF) by Vara (975) has led to a creased dscusso agast the arorateess of the SELF. The syetrc square-error loss s wdely eloyed ferece ad t s defed as follows (Schroeder, 0). L (, ) ( ) () The SELF s also ofte used because t does ot lead to etesve uercal coutato (Rao & Padey, 009). Padey ad Roa (009) estated the shae araeter of a geeralzed Pareto dstrbuto uder asyetrc loss fucto. The Bayes estator of uder SELF s: Self E ( ).. The geeral etroy loss fucto (GELF) The Etroy Loss Fucto has bee aled estato of araeters but ts stadard Mea Square Error of estato s large (Schroeder, 0). The etroy loss was frst troduced Jaes ad Ste (96) for estato of the ultoal varace-covarace atr (Parsa & Neatollah, 996). Pulc ad Calabra (996) suggested the geeral etroy loss fucto (GELF) for estato of araeters whch ca be defed as: L (, ) {( ) l( ) }, 0 (3) GELF Ths loss fucto has a u at ot. Ths loss s a geeral for of the etroy loss fucto that has bee used by several authors choosg the shae araeter. Ths geeral verso allows dfferet shaes of loss fucto 0 for eale whe a overestato causes worse cosequeces tha a () uderestato (Parsa & Neatollah, 996). The Bayes estator of uder the geeral etroy loss s: GELF ( ) [ E ( )] (4).3. The Le loss fucto (LLF) The le loss fucto s sutable for the estato of the locato araeter but ot for the estato of the scale araeter ad other araetrc fuctos (Dey, 0). A uber of asyetrc loss fuctos have bee roosed for alcatos but oe of the ost oular oe s the le (lear-eoetal) loss fucto (LLF). Ths loss fucto whch was troduced by Vara (975), ad several others cludg Basu ad Ebrah (99), Rojo (987), Sulta, Ellah, ad Sola (006) ad Sola (00), have aled ths ethod dfferet estatos ad redcto robles (Dey, 0). Ths loss fucto s cove ad ts shae s detered by the value of ts shae araeter. The ostve (egatve) values of the shae araeter, gves ore weght to uderestato (overestato) (Prakash, 03). Srvastava ad Taa (00), Xu ad Sh (004), Prakash ad Sgh (006), Sgh, Prakash, ad Sgh (007), Prakash ad Sgh (008) ad others have recetly dscussed the estato rocedures uder LLF (Prakash, 03). Uder the assuto whch the al loss occurs at, the Le loss fucto for LLe (, ) ca be eressed as L Le (, ) {e( c ( )) c ( ) }, c 0 (5) Uder the Le loss fucto, the Bayes estator s defed as (Prakash, 03): Le l{ E (e( c ))} (6) c Basu ad Ebrah (99) cosdered the le (leareoetal) loss fucto ad studed Bayesa estato uder ths asyetrc loss fucto for a eoetal lfete dstrbuto. Ths loss fucto s sutable for stuatos where overestato of s worse tha ts uderestato (Padey& Rao, 009). Here c reresets the shae araeter of the loss fucto. The behavor of the LINEX loss fucto chages wth the choce ofc..4. The recautoary loss fucto (PLF) Norstro (996) troduced a alteratve asyetrc recautoary loss fucto, ad also reseted a geeral class of recautoary loss fuctos as a secal case (Norstro, 996). These loss fuctos revet uderestato. These estators are very useful whe uderestato ay lead to the worst cosequeces. Precautoary loss fucto s defed as follows (Padey & Rao, 009):

3 Joural of Otzato Idustral Egeerg 8 (05) - ( ) L (, ) (7) The Bayes estator of uder the recautoary loss fucto s: PLF E ( ) [ ( )] (8) It should be also oted that estato wth PLF s equal to the Bayes estator uder the geeral etroy loss fucto (GELF) whe the GELF (shae) araeter s equal to Lkelhood Fucto, Posteror Dstrbutos for uder the Assuto of Dfferet Prors, Let,,...,,( 3) be a sequece of observato that coes fro a verse Gaussa dstrbuto wth robablty desty fucto defed as follows: f e 3 (9) Where 0 s the ea ad 0 s the shae araeter. As teds to fty, the verse Gaussa dstrbuto becoes ore slar to a oral (Gaussa) dstrbuto. Let be the chage ot shae araeter of the verse Gaussa dstrbuto that leads to the evet that two sequeces,,..., ad,,..., have dfferet shae araeters. The robablty desty fucto of the sequece,,..., s obtaed as follows: f e 3 (0) where 0, 0, 0. Also the robablty desty fucto of the sequece,,..., s as follows: f e 3 0, 0, 0 () The lkelhood fuctos of the sequeces are obtaed as follows: L, 3 e,,..., () L, 3 e,..., (3) We ote that the eas ad are kow (ad ossbly dfferet). The osteror dstrbutos uder the assuto of ufor, Jeffreys, eoetal, gaa ad ch-square rors are reseted the et sectos. 3.. The ufor ror The ufor ror s assued to be: (4) (5) The osteror dstrbuto uder the assuto of the ufor ror s: e 0 (6) e 0 (7) 3.. The Jeffreys ror The Jeffreys ror s defed as: I (8) I (9) The osteror dstrbuto uder the Jeffreys ror s: 3

4 Mohaad Saber Fallah Nezhad et al./ Bayesa Estato of Shft Pot... e 0 (0) e 0 () 3.3. The eoetal ror The eoetal ror s assued to be: e( k ), 0, k 0 () e( k ), 0, k 0 (3) Where k, k are cosdered as the araeters of the eoetal dstrbuto. The osteror dstrbuto uder the assuto of eoetal ror s: e k (4) 0 e k (5) 0 a e b (8) 0 a e b 0 (9) 3.5. The ch-square ror The geeral ch-square ror s assued to be: h e 0 h 0 (30) h e 0 h 0 (3) Where h, h are take as the araeters of ch-square dstrbuto. The osteror dstrbuto uder the chsquare ror s: h e 0 (3) h e 0 (33) 3.4. The gaa ror The gaa ror s assued to be: a b e 0 a, b 0 (6) a b e, 0, a, b 0 (7) Where a, b, a, b ad are araeters of gaa dstrbuto. The osteror dstrbuto uder the assuto of the gaa ror s: 4. Estato of chage ot wth Bayesa Iferece 4.. Ufor ror We take the argal ror dstrbuto of dscrete ufor over the set {,, 3 & ( -)}. Therefore, the Jot robablty dstrbuto fucto of,, for ufor ror s obtaed as equato (34). The argal dstrbuto fucto of s deoted by P ( ) that s obtaed as equato (35). Therefore, the equato (36) s obtaed: f (,, ) L(,, ) L, L, ( ) ( ) e( ) e( ) / P ( ) f (,, ) d d (35) (34) 4

5 Joural of Otzato Idustral Egeerg 8 (05) - P ( ) ( ) ( ). ( ) ( ) ( ) ( ) (36) k k ( ) ( ) (. ) k k k ( k ) ( ) ( ) ( ) k Srvastava (0) estated the shft ot uder dfferet loss fuctos sequeces of Posso dstrbuto. The Bayes estate of uder SELF s gve as: E P ( ) ( ) (37) Ad the Bayes estato of uder PLF s gve as: ( P ( )) (38) Also the Bayes estato of uder GELF s gve as: P ( ( )) (39) Where s the shae araeter that s assued to be equal oe. The Bayes estato of uder LLF s gve as c L( e P ( )) (40) c Where c s the shae araeter that s assued to be equal oe. 4.. Jeffrey ror The Jot robablty dstrbuto fucto of, for Jeffrey ror s gve as: f (,, ) ( ) e( ) ( ) e( ), (4) The argal dstrbuto fucto of shft ot s obtaed as: P ( ) f (,, ) d d (4) Therefore equato (43) s as followg. Now the estato of the shft ot uder dfferet loss fuctos ca be obtaed Eoetal ror The Jot robablty dstrbuto fucto of, for eoetal ror s gve as:, P ( ) ( ) ( ). ( ) ( ) ( ) ( ) k k ( ) ( ) (. ) ( ) ( ) k k k k ( ) ( ) k ( ) ( ) (44) f (,, ) e( ( k )) e( ( k )) The argal dstrbuto fucto of shft ot s: P ( ) f (,, ) d d (45) (43) 5

6 Mohaad Saber Fallah Nezhad et al./ Bayesa Estato of Shft Pot... Therefore, P ( ) ( ) ( ). ( ) ( ) ( ) ( ) k k (46) k k ( ) ( ) (. ) k k k ( k ) ( ) ( k ) ( k ) k Now the estato of the shft ot uder dfferet loss fuctos ca be obtaed Gaa ror The Jot robablty dstrbuto fucto of,, for gaa ror s gve as: ( ) ( ) f (,, ) e( ( b )) e( ( b )) a a The argal dstrbuto fucto of shft ot s: P ( ) f (,, ) d d Thus, P ( ) (47) (48) ( a ) ( a). ( ) ( ) a a ( b ) ( b ) (49) k k ( a ) ( a) (. ) k k k k ( ) a ( ) a ( b ) ( b ) k Now the estato of the shft ot uder dfferet loss fuctos ca be obtaed Ch-square ror The Jot robablty dstrbuto fucto of,, for gaa ror s gve as: h h ( ) ( ) f (,, ) e( ( )) e( ( )) The argal dstrbuto fucto of shft ot s: P ( ) f (,, ) d d Hece, (50) (5) 6

7 Joural of Otzato Idustral Egeerg 8 (05) - P ( ) h h. ( ) ( ) ( ) ( ) h h ( ) ( ) (5) k h k h ( ) ( ) (. ) k k h k h k ( ) ( ) ( ) ( ) k Now the estato of the shft ot uder dfferet loss fuctos ca be obtaed. 5. Sulato Study I ths secto, we reset sulato studes of chage ot estato uder dfferet rors ad dfferet loss fuctos. To fd out the otu estator, we test varous values for araeters of ror dstrbuto ad also dfferet sale sze. Data geerato for sulato s erfored 'Matlab' software. A sestvty aalyss s erfored o the araeters of rors. The sulato results for 00 rus are reseted the followg Tables. The ubers arethess are MSE of estatos. As t ca be see the followg Tables, the stadard MSE MSE devato of estatos s equal to 0 ad ts value s suffcetly sall thus gatherg data fro 00 rus s suffcet. 5.. Ufor ror dstrbuto The results for ufor ror dstrbuto are reseted Table. As t ca be see the results, the erforace of dfferet loss fuctos s ot slar to each other, where PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of Jeffrey ror dstrbuto The results for Jeffrey ror dstrbuto are reseted Table. It s cocluded that PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of Eoetal ror dstrbuto We erfored the sestvty aalyss o araeters of ror eoetal dstrbuto for deterg the best set of the araeters. Table 3 shows the Bayes estato uder eoetal ror wth araeters k =k = 0. usg dfferet loss fuctos. As t ca be see Table3, PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. We chaged the araeters of eoetal ror dstrbuto to vestgate ther effects. Table 5.3. shows Bayes estato uder eoetal ror wth araeters k =k = 0.5 usg dfferet loss fuctos. As t ca be see Table 4, PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table Bayes estato uder ufor ror usg dfferet loss fuctos Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (5.58) (3.6594) 6.73 (39.78).089 (8.9509) (.499) (.433) (3.5563).785 (.73) Table Bayes estato uder Jeffrey ror usg dfferet loss fuctos Sale sze Shft ot Shft ot Estato(λ =5, λ = 0, µ = µ =) (4.5403).0745 (3.3585).376 (8.947).7648 (9.439) (0.495) (0.8908).967 (.0606).694 (.3743) Table 3 Bayes estato uder eoetal ror usg dfferet loss fucto (k=k= 0.) Sale sze Shft ot Shft ot Estato(λ =5, λ = 0, µ = µ =) (4.940) (3.3874).9364 (8.608).044 (8.9760) (0.9333) (.0) (.6877).7757 (.344) 7

8 Mohaad Saber Fallah Nezhad et al./ Bayesa Estato of Shft Pot... I ths case, we creased the araeters of eoetal ror dstrbuto. Table 5 shows Bayes estato uder eoetal ror wth araeters k =k = usg dfferet loss fuctos. It ca be see that PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table 6 shows Bayes estato uder eoetal ror wth araeters k =k = usg dfferet loss fuctos. As t ca be see Table6, PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Wth chagg the araeters of eoetal ror dstrbuto, t s cocluded that the erforace of dfferet ror eoetal dstrbutos are relatvely slar to each other. Table 7 shows the total results for Bayes estato of the chage ot uder eoetal ror wth dfferet araeters usg dfferet loss fuctos. It s observed fro Table 7 that whe k =k =0. the the PLF has the best estato of shft ot sale sze of 0 ad the SELF has the best estato of shft ot sale sze of 50 whe araeters of eoetal ror dstrbuto are k =k = Gaa ror dstrbuto The results for gaa ror dstrbuto are reseted the followg Tables. The values of (a,b )=(a,b )=(0.,0.) are assued as the araeters of gaa ror dstrbuto.as t ca be see Table 5.4., PLF has the best erforace deterg the shft ot for sale sze of 0, SELF has best erforace deterg the shft ot for sale sze of 50. Table 9 shows the Bayes estato for gaa ror dstrbuto wth araeters (a,b )=(a,b )=(0.,0.5). As you ca see, PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table 4 Bayes estato uder eoetal ror usg dfferet loss fuctos (k=k= 0.5) Sale sze Shft ot Shft ot Estato(λ =5, λ = 0, µ = µ =) (3.7844).8696 (.504).8373 (8.3953).0756 (8.937) (9.0507) (0.6490).4079 (0.6445).8353 (.694) Table 5 Bayes estato uder eoetal ror usg dfferet loss fuctos (k=k= ) Sale sze Shft ot Shft ot Estato(λ =5, λ = 0, µ = µ =) (3.30).584 (.7).63 (8.4389).0633 (8.9435) (7.834) (0.883).48 (0.7696).8093 (.938) Table 6 Bayes estato uder eoetal ror usg dfferet loss fuctos (k=k=) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (.666).459 (.944).4956 (8.560).04 (8.968) (6.7887) (0.3479) 0.00 (0.00).84 (.880) Table 7 The loss fuctos wth the best erforaces for eoetal ror No. of Table Sale sze Loss fucto Paraeters of ror dstrbuto 0 50 For sale sze=0 For sale sze= PLF SELF k =k = PLF SELF k =k = PLF SELF k =k = PLF SELF k =k = Table 8 Bayes estato uder gaa ror usg dfferet loss fuctos ((a,b)=(a,b)=(0.,0.)) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (4.366).08 (3.5).04 (8.98).787 (8.9760) (0.030) (0.7884).9850 (.0370).6439 (.344) Table 9 Bayes estato uder gaa ror usg dfferet loss fuctos ((a,b)=(a,b)=(0.,0.5)) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (.9830).30 (.373).675 (8.85).8390 (8.937) (7.9488) (0.487).0679 (0.958).70 (.694) 8

9 Joural of Otzato Idustral Egeerg 8 (05) - Table 0 shows the Bayes estato for gaa ror dstrbuto wth araeters (a,b )=(a,b )=(0.,). As t s show, PLF has the best erforace deterg the shft ot for sale sze 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table shows the Bayes estato for gaa ror dstrbuto wth araeters (a,b )=(a,b )=(0.,). PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table shows the suarzed results for Bayes estato of the chage ots uder gaa ror usg dfferet loss fuctos. It s see that whe (a,b )=(a,b )=(0.,0.) the PLF has the best erforace to estate the chage ots based o the sale sze of 0 ad the SELF s the best estator for shft ot sale sze of 50 whe araeters (a,b )=(a,b )=(0.,) are assued for gaa ror dstrbuto Ch-square ror dstrbuto The results for ch-square ror dstrbuto are reseted the followg Tables. The values of h =h =0. are assued as the araeters of ch-square ror dstrbuto. As t coes Table 3, PLF has the best erforace deterg the shft ot for sale sze of 0 also PLF has the best erforace deterg the shft ot for sale sze of 50. We chaged the araeters of ch-square ror dstrbuto to detere the best estator. Table 4 shows Bayes estato uder ch-square ror wth araeters h =h =0.5 usg dfferet loss fuctos. As you ca see Table4, PLF has the best erforace deterg the shft ot for sale sze of 0 ad the sale sze of 50. Table 0 Bayes estato uder gaa ror usg dfferet loss fuctos ((a,b)=(a,b)=(0.,)) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (.499).464 (.994).450 (8.8650).8455 (8.9435) (6.9787) (0.4).9973 (.0089).6998 (.938) Table Bayes estato uder gaa ror usg dfferet loss fuctos ((a,b)=(a,b)=(0.,)) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (.3488).5068 (.9506).60 (8.8898).8386 (8.968) (5.9906) (0.3783) (0.00).75 (.880) Table The loss fuctos wth the best erforaces for gaa ror No. of Table Sale sze Loss fucto Paraeters of ror dstrbuto 0 50 For sale sze=0 For sale sze= PLF SELF (a,b )=(a,b )=(0.,0.) PLF SELF (a,b )=(a,b )=(0.,0.5) PLF SELF (a,b )=(a,b )=(0., ) PLF SELF (a,b )=(a,b )=(0.,) Table 3 Bayes estato uder ch-square ror usg dfferet loss fuctos (h=h=0.) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (4.997) (.6).678 (8.407).000 (8.9847) (.964).8358 (6.97) () (0.8057) Table 4 Bayes estato uder ch-square ror usg dfferet loss fuctos (h=h=0.5) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (3.8935).3940 (.9534).6090 (8.4398).063 (8.990) (9.695) (7.3439).847 (0.939).086 (0.9744) 9

10 Mohaad Saber Fallah Nezhad et al./ Bayesa Estato of Shft Pot... Table 5 shows Bayes estato uder ch-square ror wth araeters h =h = usg dfferet loss fuctos. As you ca see Table5, PLF has the best erforace deterg the shft ot for sale sze of 0 ad SELF has the best erforace deterg the shft ot for sale sze of 50. Table 6 shows Bayes estato uder ch-square ror wth araeters of h =h = usg dfferet loss fuctos. As t ca be see ths Table, SELF has the best erforace deterg the shft ot for sale sze of 0 ad 50. Table 7 shows the suarzed results for Bayes estato of the chage ot uder ch-square ror usg dfferet loss fuctos. It ca be cocluded that whe h =h = the SELF coes u wth the best estato for shft ot sale sze of 0 ad the PLF has best erforace to estate the shft ot sale sze of 50 whe araeter h =h =0. are assued for the ch-square ror dstrbuto. 6. Results The suarzed results of sulato studes for all ror dstrbutos ad loss fuctos SELF, PLF, GELF ad LLF have bee show Table8, based o whch, the followg coclusos are ade:. For sale sze of 0, the Jeffrey ror dstrbuto uder PLF s the ost accurate shft ot estator.. For sale sze of 50, the gaa ror dstrbuto uder SELF s the ost accurate shft ot estator whe the araeters of ror dstrbuto are (a,b )=(a,b )=(0.,). 3. Also t s show that the loss fucto GELF ad LLF do ot estate the shft ot accurately therefore they are ot useful to be aled the roble of chage ot detecto for shae araeter of the Iverse Gaussa dstrbuto. Table 5 Bayes estato uder ch-square ror usg dfferet loss fuctos (h=h=) Sale sze Shft ot Shft ot Estato (λ =5, λ = 0, µ = µ =) (3.93).0543 (.497).495 (8.689).9500 (9.0575) (8.06) (0.388).40 (0.8753).759 (.50) Table 6 Bayes estato uder ch-square ror usg dfferet loss fuctos (h=h=) Sale Shft sze ot Shft ot Estato (λ =5, λ = 0, µ = µ =) 0.3 (3.7655) 3.9 (3.4885).039 (9.039).765 (9.46) (0.7037) (0.557) (0.00).98 (.785) Table 7 The loss fuctos wth the best erforaces for the ch-square ror No. of tables Sale sze Loss fucto Paraeters of ror dstrbuto 0 50 For sale sze=0 For sale sze= PLF PLF h =h = PLF PLF h =h = PLF SELF h =h = SELF SELF h =h = Table8 The ror dstrbuto fuctos wth the best erforaces Sale sze Loss fucto Paraeters of ror dstrbuto Pror For sale For sale dstrbuto 0 50 For sale sze=0 For sale sze=50 sze=0 sze=50 ufor PLF SELF - - Jeffrey PLF SELF - - eoetal PLF SELF k =k =0. k =k = gaa PLF SELF (a,b )=(a,b )=(0.,0.) (a,b )=(a,b )=(0.,) Ch-square SELF PLF h =h = h =h =0. 6. Dscusso I the Bayes estato of a teger valued araeter, such as the shft ot, the loss fucto ust be defed oly for teger values both for the ossble ot estator ad for the ukow value of the araeter:,,,..., -. Ideed, as wrtte the book of Martz ad Waller (98) the case of Bayesa ot estato, the acto sace A cossts of the ossble ot estates of the araeter ad thus s a subset of the araeter sace. As a cosequece, the osteror 0

11 Joural of Otzato Idustral Egeerg 8 (05) - eected loss E L (, ) s ot a cotuous fucto ad s ot dfferetable. For eale, the SELF relatve to s o loger a real valued fucto, but s gve by: L (, ) ( - ),,,,..., -. The, the Bayes estator of uder the SELF s o loger the osteror ea. Geerally, the Bayes estator of uder ay error loss fucto should be obtaed by uercally zg the corresodg osteror eected loss whch s, ts tur, defed oly for teger values of the ossble ot estator. As a cosequece, all the Bayes estates Equatos (4..4)-(4..7) see to be ot accetable. A sle way to obta the Bayes estate of uder a gve (teger-valued) loss fucto s to treat tally as a cotuous (realvalued) araeter ad to obta the (real) value of, say, that zes the cotuous eected loss. The, the Bayes estate of the shft ot, s gve by the teger art of or of, deedg o whch of the rovdes the saller osteror eected loss. Also t s see that the rage of vestgato ths aer s very lted ad the araeters of the loss fuctos are ot aalyzed. It s ecessary to eto that selecto of otal araeters for loss fuctos deeds o the alcato, ad otalty geeral case s eagless. We have show that araeter of ror dstrbuto affects the fal results ad also a ethod for deterg the otal ror alog wth otal loss fucto has bee roosed. Moreover, coarg the statstcal erforaces of the Bayes estators of a quatty of terest uder dfferet loss fuctos has o sese. Ideed, the loss fucto should ot be chose o the bass of ts statstcal erforaces, but o the bass of costs cosderatos, that s, o the bass of the effect of over-estatg or uder-estatg the quatty of terest. For eale, f over-estato roduces ore serous cosequeces tha uderestato, the SELF should ot be chose, whereas the Geeral Etroy loss fucto wth ostve shae araeter s a sutable choce. Lkewse, the ror dstrbuto should be selected deedg o the ror forato avalable to the aalyst, ad ot o the bass of ts statstcal erforaces. 7. Cocluso I ths aer we estated the shft ot sequeces of verse Gaussa dstrbuto wth Bayesa aalyss uder varous loss fuctos. Preseted Bayesa ferece s based o the dfferet ror dstrbutos cludg ufor, Jeffrey, eoetal, gaa ad chsquare uder varous loss fuctos cludg SELF, ELF, LLF, ad PLF. Frst the osteror dstrbuto of the shae araeter was obtaed usg Bayesa Iferece. The the Bayes estators were derved alog wth ea stadard error of estatos. Sulato studes were erfored to vestgate the erforace of dfferet loss fuctos. We tred to fd out the best estator for shft ot whe dfferet sale szes were avalable. The results showed that the sale sze was ortat suerorty of each loss fucto. The results of sulato study deoted that the Jeffrey ror dstrbuto uder PLF was the ost accurate chage ot estator for sale sze of 0, ad the gaa ror dstrbuto uder SELF was the ost accurate chage ot estator for sale sze of 50. Refereces [] Basu, A.P., Ebrah, N. (99). Bayesa Aroach to Lfe Testg ad Relablty Estato Usg Asyetrc Loss Fucto. J. Stats. Pla. Iferece, Vol. 9, -3. [] Calabra, R, Pulc, G. (996). Pot Estato Uder Asyetrc Loss Fuctos for Left Trucated Eoetal Sales. Coucato Statstcs-Theory &Methods, Vol. 5(3), [3] Dey, S. (0). Bayesa Estato of the Paraeter ad Relablty Fucto of a Iverse Raylegh Dstrbuto. Malaysa Joural of Matheatcal Sceces, Vol. 6(), 3-4. [4] Fallahezhad MS, Rast B, Abooe MH, Irovg the Perforace of Bayesa Estato Methods Chage ot Detectos ad coarso wth MLE aroach, Iteratoal Joural of Egeerg, Islac Reublc of Ira, IJE TRANSACTIONS C: Asects Vol. 7, No. 6,. 9-93, Jue 04, [5] Kadlar, C., Karasoy, D. S. A ew Bayes estate of the chage ot the hazard fucto, Coutatoal Statstcs & Data Aalyss, Vol. 5, (007), [6] Keraatour M., Nak S.T.A., Khedat M. Soleyaa M.E., Motorg ad Chage Pot Estato of AR() Autocorrelated Polyoal Profles, Iteratoal Joural of Egeerg, Islac Reublc of Ira, Vol. 5, No. 6, (0), [7] Khedat M., Nak S.T.A. Idetfyg the chage te of ultvarate boal rocesses for ste chages ad drfts, Joural of Idustral Egeerg Iteratoal, Vol. 9, No. 3 (03), -. [8] Marcos D Agelo, R. M. Icet fault detecto ducto ache stator-wdg usg a fuzzy-bayesa chage ot estato aroach, Aled Soft Coutg, Vol., (0), [9] Martz H. & Waller, R. (98). Bayesa Relablty Aalyss, Kreger Publshg Coay,. 98. [0] Naz sdhu, T., & Asla,. (03). Bayesa Estato o The roortoal verse Webull dstrbuto uder dfferet loss fuctos. Advaces Agrculture, Sceces ad Egeerg, Vol. 3 (), [] Naz Sdhu, T., Feroze, N., & Asla, M. (03). Bayesa Aalyss of the Kuarasway Dstrbuto uder Falure Cesorg Salg Schee. Iteratoal Joural of Advaced Scece ad Techology, Vol. 5, [] Norstro, J. (996). The use of recautoary loss fuctos rsk aalyss. IEEE Tras. Relab 45 (3),

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