BAYESIAN AND NON BAYESIAN ESTIMATION OF ERLANG DISTRIBUTION UNDER PROGRESSIVE CENSORING

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1 3_8.pdf BAYESIAN AND NON BAYESIAN ESTIMATION OF ERLANG DISTRIBUTION UNDER PROGRESSIVE CENSORING R.A. Bakoban Departent of Statstcs, Scences Faculty for Grls, Kng Abdulazz Unversty, Saud Araba E-al: ABSTRACT Based on progressvely Type-II censored saples, the axu lkelhood and Bayes estators for the scale paraeter, relablty and cuulatve hazard functons are derved. The Bayes estators are studed under syetrc (squared error) loss functon and asyetrc (LINEX and general entropy) loss functons. Tow technques are used for coputng the Bayes estates; standard Bayes and portance saplng ethods. The perforance of the estates are copared by usng the ean square error and the relatve absolute bas through Monte Carlo sulaton study. Keywords: Erlang dstrbuton; Iportance saplng technque; Monte Carlo Sulaton; Progressve Type-II censorng; Syetrc and asyetrc loss functons. Matheatcs Subject Classfcaton: 6F0; 6F5; 6N0; 6N05.. INTRODUCTION The progressve Type-II censorng schee s played a vtal role n lfe-testng. Ths type of censorng allows the experenter to reove tes fro the experent before ts end, thus resultng n a savng n cost as well as experental te. Also, ths type of censorng has been attractng n any felds of applcaton, for exaple, scence, engneerng and edcne (see, Balakrshnan and Aggarwala (000)). The progressve Type-II censorng can be descrbed as follows: n unts are placed on a lfe-testng experent and only ( n) unts are copletely observed untl falure. The censorng occurs progressvely n stages. These stages offer falure tes of the copletely observed unts. At the te of the frst falure (the frst stage), r of the n survvng unts are randoly wthdrawn fro the experent. At the te of the second falure (the th second stage), r of the n r survvng unts are wthdrawn and so on. Fnally, at the te of the falure (the th stage), all the reanng r n r r... r survvng unts are wthdrawn. We wll refer ( r, r,..., r ). to ths as progressve Type-II rght censorng wth schee The Erlang varate s the su of a nuber of exponental varates. It was developed as the dstrbuton of watng te and essage length n telephone traffc. If the duratons of ndvdual calls are exponentally dstrbuted, the duraton of a successon of calls has an Erlang dstrbuton. The Erlang varate s a gaa varate wth an nteger shape paraeter. The probablty densty functon (pdf) of an Erlang varate s gven by x / b ( x / b) e f ( x ;, b), x 0, b 0, 0, b ( )! Where b and are the scale and the shape paraeters, respectvely. Such that s an nteger nuber. For ore detals about ths dstrbuton see, Evans et al. (000). Several authors nterested n estaton of the scale paraeter of gaa dstrbuton wth known shape paraeter, aong the Ghosh and Sngh (970) and Berger (980). Also, Johnson et al. (994) gave a good revew about ths dstrbuton n the lterature. Soe authors nterested n gaa dstrbuton wth an nteger shape paraeter for exaple, Constantne et al. (986) studed the estaton of P( Y X ) n the gaa case wth known nteger-valued shape paraeter. Fang (00) presented the study of the hyper-erlang dstrbuton odel and ts applcatons n wreless networks and oble coputng systes. Also, the oents of order statstcs fro nondentcally dstrbuted Erlang varables coputed by Abdelkader (003). Wllot and Ln (0) presented a revew of analytcal and coputatonal propertes of the xed Erlang dstrbuton n the context of rsk analyss. () 54

2 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton In ths paper, we consder the Erlang dstrbuton wth shape paraeter ( 3) n (), whch has pdf and cuulatve dstrbuton functon (cdf), respectvely, as x x / b f ( x ; b) e, x 0, b 0, () 3 b x / b e x F ( x ; b), x 0, b 0. (3) b Also, the relablty and cuulatve hazard functons of ths dstrbuton are gven, respectvely, by x / b e x R ( x ; b), x 0, b 0, (4) b H ( x ; b) Log[ R( x ; b)], x 0, b 0. (5) We wll denote the Erlang dstrbuton n () by Er(b). Also, for splcty, we wll denote the scale paraeter, relablty and cuulatve hazard functons by, b, R and H, respectvely. Based on progressve Type-II censored saples, we consder the estaton of the scale paraeter, relablty and cuulatve hazard functons fro Er(b). In Secton, the axu lkelhood (ML) estaton for b, R and H are derved. Also, confdence nterval (CI) based on the asyptotc dstrbuton of the ML of b are obtaned n ths secton. Bayesan estaton under squared error (SE), LINEX and general entropy (GE) loss functons are dscussed n Secton 3 by usng tow technques for coputng the Bayes estates. The perforance of the estates are copared by usng the ean square error (MSE) and the relatve absolute bas (RABas) through Monte Carlo sulaton study based on dfferent censorng schees are nvestgated n Secton 4. Fnally, concludng rearks are presented n Secton 5.. MAXIMUM LIKELIHOOD ESTIMATION Suppose that x ( x, x,..., x ) s a progressve Type-II censored saple fro a lfe test on n tes whose lfetes have an Erlang dstrbuton, Er(b), wth pdf gven n (), and r, r,..., r denote the correspondng nubers of unts reoved (wthdrawn) fro the test. The lkelhood functon based on the progressve Type-II censored saple (Balakrshnan and Aggarwala (000)) s gven by where r l ( b; x ) A f ( x ; b)[ F ( x ; b )], (6) A n ( n r )( n r r )...( n ( r )), f ( x ) and Fx ( ) are gven, respectvely, by () and (3). Substtutng () and (3), nto (6), then the lkelhood functon for Er(b) s ( r ) 3 b ( r ) x x l ( b; x ) A b x e. b And the natural logarth of the lkelhood functon s gven by 3 ( ; ) ( ) ( ) ( ) ( ) L b x Log A Log b Log x b r x (7) x r Log /. (8) b To derve the ML estaton of the unknown paraeter b, say b ˆ ML, we dfferentate (8) wth respect to b and then solve the followng non-lnear equaton nuercally by usng Newton-Raphson ethod r 55

3 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton x x 3 b ( r ) x r x b. b b (9) The ML of the relablty functon, R, and the cuulatve hazard functon, H, are gven by replacng ˆML n (4) and (5), respectvely. The observed asyptotc varance of ML estaton for the paraeter b s gven by droppng the expectaton operator fro the eleent of the nverse of the Fsher nforaton atrx as follows ˆ l Var ( bml ) b where 3 { b [3 b [ x r x ( ( ) [ b ( x b x )( ) x b ])]}, x b. (0) The asyptotc noralty of the ML estator can be used to copute the approxate confdence nterval for the paraeter b. Thus, ( )00% confdence nterval for b becoes bˆ z Var ( bˆ ), ML / ML where z / s a standard noral percentle. 3. BAYES ESTIMATION In ths secton we studed Bayes estators under three loss functons. One s syetrc (squared error) loss functon and the others are asyetrc (LINEX and general entropy) loss functons. The squared error (quadratc) loss functon assocates equal portance to the losses due to overestaton and underestaton of equal agntude. However, n real applcatons, the estaton of the paraeters or functon as relablty functon an overestaton s ore serous than the underestate; thus, the use of a syetrcal loss functon s napproprate. (see, Canfeld (970) and Basu and Ebrah (99)). In ths case, an asyetrc loss functons ust be consdered. The LINEX loss functon rses approxately exponentally on one sde of zero and approxately lnearly on the other sde. Ths functon was ntroduced by Varan (975) and several authors nterested n, aong the Solan (000, 00) and Bakoban (00). The general entropy (GE) loss s also asyetrc loss functon whch s used n several papers, for exaple, Dey et al. (987), Dey and Lu (99) and Solan (005, 006). In our estaton of the paraeters, we copute the estators by usng two technques, standard Bayes and portance saplng ethods. Iportance saplng ethod s used for nuercally approxatng ntegrals. Also, t s vewed as a varance reducton technque. Many authors nterested n ths ethod aong the, Kundu and Pradhan (009) and Kundu and Howlader (00) and Klakattaw et al. (0). Furtherore, Yaun and Druzdzel (006) and Tokdar and Kass (00) gave an algorth for the portance saplng ethod. In Bayes study, we assue that the paraeter b has an nverted gaa ( IG ( c, )) pror dstrbuton wth pdf / b c e c p( b) / b, b 0, c, 0, () c where c s the shape paraeter, the scale paraeter and the varate /b s a gaa varate wth the sae shape and scale paraeters. Cobnng (7) and (), we obtan the posteror densty of b as where 3 ( ) (, ) ( ) b c x b x k b e b, b () () x ( x, b) Log ( x ) r Log, b (3) 56

4 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton ( ) x ( r), (4) 3 b c ( ) ( x, b ) k b e db. (5) 0 The Bayes estators of a functon of b, say ( b), wll be derved under syetrc and asyetrc loss functons n the followng subsectons. For coputng the Bayes estators, we use the standard Bayes technque and the portance saplng technque. The portance saplng technque s desgned to copute the Bayes estates. The posteror densty functon () can be wrtten as: 3c [ ( )] 3c b ( ) ( x, b ) ( b x ) b e, (3 c) IG ( b;3 c, ( )) g ( b x ), (6) where g ( b x ) exp[ ( x, b)]. The rght-hand sde of (6), say ( bx ), and ( bx) approxate Bayes estators can be coputed usng ( bx ) portance saplng technque. N (7) N dffer only by the proportonalty constant. An as a posteror densty functon based on the 3. Syetrc Bayes estates The quadratc loss for Bayes estate of a paraeter, say ( b), s the posteror ean assung that exsts, say BS ( b ), whch defne as 3 b c ( ) ( x, b ) (8) ( b BS ) k ( b ) b e db. 0 Equaton (8) provded the standard Bayes estate under the quadratc loss functon. Accordng to the portance saplng technque, the approxate Bayes estator under the quadratc loss functon, say SPBS ( b ), can be coputed by the followng Algorth: Step. Generate b ~ IG ( b;3 c, ( )). Step. Repeat Step to obtan b, b,..., bn. Step 3. Copute the value N ( b ) g ( b x ) SPBS ( b), N g ( b x ) where g ( b x ) exp[ ( x, b )]. (9) (0) 3. Asyetrc Bayes estates The LINEX loss functon ay be expressed as ( ) e a a, a 0, () where ˆ. The sgn and agntude of the shape paraeter a reflects the drecton and degree of asyetry, respectvely. (If a 0, the overestaton s ore serous than underestaton, and vce-versa). For a closed to zero, the LINEX loss s approxately squared error loss and therefore alost syetrc. The posteror expectaton of the LINEX loss functon Equaton () s E [ ( ˆ )] exp( a ˆ ) E [exp( a )] a( ˆ E ( )), () 57

5 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton where E (.) denotng posteror expectaton wth respect to the posteror densty of. By a result of Zellner (986), the (unque) Bayes estator of, denoted by ˆ under the LINEX loss s the value ˆ whch nzes (), s gven by ˆ log{ E[exp( a)]}, (3) a provded that the expectaton E [exp( a )] exsts and s fnte [ Calabra and Pulcn (996)]. Next, under the assupton that the nal loss occurs at u u, the general entropy for u u( b) (see, Solan (005)) s q (, ) u u. u u q Log (4) u u When q 0, a postve error ( u u) causes ore serous consequences than a negatve error. The Bayes estate u of u under GE loss (4) s / q q u Eu ( u ), (5) q provded that E ( u ) exsts and s fnte. u 3.. Bayes estates under LINEX loss functon Accordng to (3), the standard Bayes estators of ( b) where ( x, b), ( ) and under the LINEX loss functon, say ( b ), s gven by 3 [ ( ) ( ) (, )] ( ) a b b c x b b Log k b e db, a (6) 0 k are gven n (3), (4) and (5), respectvely. Accordng to the portance saplng technque, the approxate Bayes estator under LINEX loss functon, say SP ( b ), can be coputed by applyng the steps and n the Algorth that gven n subsecton (3.), then the thrd step can be conducted fro (3) as Step 3. Copute the value where g ( b x ) s defned n (0). N a ( b ) ( ) e g b x SP ( b) Log, N a g ( b x ) 3.. Bayes estates under general entropy loss functon The standard Bayes estate ( b ) of ( b ) under GE loss (5) s gven by 3 [ ( ) (, )] ( b ) k q c b x b [ ( b )] b e db, (8) 0 where ( x, b), ( ) and are gven n (3), (4) and (5), respectvely. k Accordng to the portance saplng technque, the approxate Bayes estator under GE loss functon, say can be coputed by applyng the steps and n the Algorth that gven n subsecton (3.), then the SP ( b ), thrd step can be conducted fro (5) as Step 3. Copute the value / q (7) 58

6 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton where g ( b x ) s defned n (0). N / q q [ ( b)] g ( b x ) SP ( b), N g ( b x ) (9) 4. SIMULATION STUDY In ths secton we dscuss the nuercal results of a sulaton study testng the perforance of the Bayes ethods of estaton wth the MLEs. Usng the Algorth presented n Balakrhnan and Sandhu (995), we generate progressvely Type-II censored saple of dfferent szes fro a rando saple of dfferent szes n fro Er(b) as follows Step. Generate ndependent Unfor (0, ) observatons W, W,..., W. Step. Deterne the values of the censored schee r, for,,...,. Step 3. Set Step 4. Set V Step 5. Set E / ( r) U E j W for,,...,. V j j for,,...,. for,,...,. Then U, U,..., U s progressve Type-II censored saple fro the Unfor (0, ) dstrbuton. Step 6. For gven values of the pror paraeters ( c 3, ), generate a rando value for b fro the nverted gaa dstrbuton whose densty functon gven by Equaton (). Step 7. Usng.707 U U U fro Step 5, we can obtan progressve Type-II b obtaned n step 6 and,,..., censored saple x, x,..., x fro Er(b) by solvng the followng equaton nuercally U x / b x e, b Step 8. Copute the estates as the followng: a) Estaton of the scale paraeter b:,,...,. x, x,..., x fro step 7, the MLE of b, say b ˆ, Usng ML were coputed by solvng Equaton (9) nuercally usng Newton-Raphson ethod. The Bayes estates of b, are coputed usng the results that obtaned n Secton 3, by settng ( b) b n the correspondng equatons. Equatons (8), (6) and (8) are used for standard Bayes estates, say b BS, b and b, and Equatons (9), (7) and (9) are used for approxate Bayes estates say bbs, b and b, accordng to the portance saplng technque. b) Estaton of the relablty functon Rt ( ) : Substtutng the ML of b, b ˆ ML, nto (4), we obtan the ML of the relablty functon, say R ˆ ML. The Bayes estates of R, are coputed usng the results that obtaned n Secton 3, by settng ( b) R ( t ) n the correspondng equatons. Equatons (8), (6) and (8) are used for standard Bayes estates, say R BS, R and R, and Equatons (9), (7) and (9) are used for approxate Bayes estates say RBS, R and R, accordng to the portance saplng technque. At t, we have Rt ( ) as a true value. c) Estaton of the cuulatve hazard functon Ht ( ) : 59

7 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton Substtutng the ML of b, b ˆ ML, nto (5), we obtan the ML of the cuulatve hazard functon, say H ˆ ML. The Bayes estates of H, are coputed usng the results that obtaned n Secton 3, by settng ( b) H ( t ) n the correspondng equatons. Equatons (8), (6) and (8) are used for standard Bayes estates, say H BS, H and H, and Equatons (9), (7) and (9) are used for approxate Bayes estates say HBS, H and H, accordng to the portance saplng technque. At t, we have 0 Ht ( ) as a true value. 0 All above steps are repeated 000 tes to evaluate the ean square error (MSE) and the absolute relatve bas (RABas) of the estates, where MSE ( ˆ ) ( ˆ ) E and RABas ˆ ( ˆ ). The coputatonal results are presented n Tables (-8). All results are obtaned by usng Matheatca 7.0. Table provdes dfferent censorng schees for dfferent saple szes n 0,0,30,40,50 and 00. For splcty n notaton, we denoted the censorng schee (0, 0, 0, 0, 0, 0, 3) by (6 *0, 3). The MSE and RABas of the ML estates of b, R and H for dfferent censorng schees are represented n Table. Tables 3, 4 and 5 contans the standard Bayes estates of b, R and H, respectvely. Also, Tables 6, 7 and 8 contans the approxate Bayes estates of b, R and H, respectvely, accordng to the portance saplng technque. Table. Censorng schees (C.S.) r,,,..., for saple sze n and observed saple sze. n S C.S. n S C.S. 0 7 S (6 *0,3) S 7 (, 0,, *0,, 0,, 3 *0,, 3 *0,, *0,, 0 *0, ) 0 9 S (, 8 *0 ) 40 0 S 8 (9 *0, 30) 0 5 S 3 (5, 4 *0 ) 50 0 S 9 (40, 9 *0 ) 0 0 S 4 (9 *0, 0) 50 0 S 0 (5 *, 5 *0 ) 30 5 S 5 (4, 0,, *0, 4, 3 *0,, 0,, 0,, 00 0 S (80, 9 *0 ) 30 0 S 6 (0,, 4 *0,, 3 *0,, *0, 3, *0,, *0, ) S (49 *0, 50) Table. MSEs and RABas (between parentheses) of ML estates for the scale paraeter, R and H. n S b ˆ R H ˆ ML ˆML ML 0 7 S (0.0033) (0.056) (0.655) 0 9 S (0.0096) ( ) 0.00 (0.4533) 0 5 S ( ) (0.0468) (0.7668) 0 0 S (0.003) (0.0064) (0.85) 30 5 S ( ) (0.0057) (0.0458) 30 0 S ( ) (0.009) ( ) 40 0 S (0.005) (0.0068) (0.5) S (0.0055) (0.005) ( ) 50 0 S ( ) (0.0056) (0.03) 50 0 S (0.0033) ( ) (0.0607) 00 0 S (0.0000) (0.0038) (0.0587) S ( ) (0.003) (0.045) 530

8 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton Table 3. MSEs and RABas (between parentheses) of Bayesan estates for the scale paraeter usng standard Bayes ethod. n S b b, BS a=- 0 7 S (0.069) 0 9 S (0.050) 0 5 S (0.0886) 0 0 S (0.0066) 30 5 S (0.03) 30 0 S (0.0003) 40 0 S (0.055) S (0.005) 50 0 S (0.0054) 50 0 S ( ) 00 0 S (0.0075) S (0.0080) (0.09) ( ) (0.085) (0.0083) (0.0089) ( ) (0.0046) (0.0076) (0.0943) (0.046) (0.003) (0.0080) b, a= (0.07) (0.0504) (0.089) (0.0067) (0.03) (0.0003) (0.056) (0.005) ( ) ( ) (0.0076) (0.008) b, a= ( ) (0.0356) ( ) (0.0046) (0.050) (0.049) (0.096) (0.0076) (0.0348) (0.009) 0.0 (0.09) ( ) b, q= (0.0646) (0.046) ( ) (0.07) (0.089) (0.068) (0.0884) ( ) (0.0306) (0.078) (0.063) ( ) b, q= (0.069) (0.050) (0.0886) (0.0066) (0.03) (0.0003) (0.055) (0.005) (0.0054) ( ) (0.0075) (0.0080) b, q= (0.0855) ( ) (0.0077) (0.0569) (0.0395) (0.0344) (0.0466) (0.06) (0.0603) (0.0706) 0.04 ( ) (0.0463) Table 4. MSEs and RABas of Bayesan estates for the relablty functon usng standard Bayes ethod. n S R R, R, R, R, R, R, BS a=- a=0.00 a= q=-3 q=- q=3 0 7 S (0.0088) (0.006) (0.0088) (0.00) (0.0938) (0.0088) (0.056) 0 9 S (0.087) (0.0769) (0.087) (0.0896) (0.0705) (0.087) (0.0097) 0 5 S (0.0658) (0.057) 0.00 (0.0658) (0.0634) (0.040) 0.00 (0.0658) (0.0309) 0 0 S (0.045) (0.037) (0.045) (0.064) (0.086) (0.045) (0.0383) 30 5 S (0.006) (0.0086) (0.006) ( ) (0.0079) (0.006) (0.0039) 30 0 S ( ) (0.0066) ( ) ( ) (0.0064) ( ) (0.0079) 40 0 S (0.054) ( ) (0.055) (0.0053) ( ) (0.054) (0.045) S (0.0055) ( ) (0.0055) ( ) ( ) (0.0055) (0.0057) 50 0 S 9 (0.0000) (0.03) (0.0000) (0.0538) (0.085) (0.0000) 0.00 (0.069) 50 0 S 0 (0.0066) (0.0055) (0.0066) (0.0065) ( ) (0.0066) ( ) 00 0 S ( ) S (0.005) ( ) ( ) ( ) (0.005) (0.0088) (0.0097) ( ) (0.0097) ( ) (0.005) ( ) (0.0035) 53

9 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton Table 5. MSEs and RABas of Bayesan estates for the cuulatve hazard functon usng standard Bayes ethod. n S H H, H, H, H, H, H, BS a=- a=0.00 a= q=-3 q=- q=3 0 7 S ( ) ( ) (0.3965) ( ) (0.6738) ( ) (0.640) 0 9 S (0.3499) ( ) (0.3498) (0.3355) ( ) (0.3499) (0.357) 0 5 S ( ) (0.5356) ( ) ( ) (0.8868) ( ) (0.6563) 0 0 S (0.344) (0.607) (0.343) (0.96) (0.43) (0.344) ( ) 30 5 S (0.898) (0.603) (0.898) (0.6946) (0.7446) (0.898) (0.0630) 30 0 S (0.9) (0.85) (0.8) (0.87) (0.778) (0.9) (0.055) 40 0 S (0.438) (0.900) (0.438) (0.855) (0.3043) (0.438) ( ) S 8 ( ) ( ) ( ) (0.0843) (0.6007) ( ) (0.046) 50 0 S (0.859) (0.479) (0.8589) (0.773) (0.43) (0.859) (0.0735) 50 0 S (0.7) (0.038) (0.7) (0.47) (0.807) (0.7) (0.0509) 00 0 S (0.3093) S ( ) (0.308) (0.057) (0.309) ( ) (0.549) (0.053) (0.6) (0.088) (0.3093) ( ) ( ) (0.007) Table 6. MSEs and RABas (between parentheses) of Bayesan estates for the scale paraeter usng portance saplng ethod. n S b b, b, b, b, b, b, BS a=- a=0.00 a= q=-3 q=- q=3 0 7 S (0.04) (0.06) (0.044) (0.0343) (0.07) (0.04) (0.087) 0 9 S (0.0459) ( ) (0.046) ( ) (0.068) (0.0459) ( ) 0 5 S (0.0037) ( ) (0.003) (0.0648) (0.056) (0.0037) ( ) 0 0 S ( ) (0.0044) ( ) (0.0066) (0.098) ( ) (0.0304) 30 5 S (0.0038) (0.080) (0.0039) ( ) (0.0447) (0.0038) (0.0309) 30 0 S (0.007) 0.09 ( ) (0.007) (0.0408) (0.057) (0.007) (0.034) 40 0 S (0.708) (0.43) 0.6 (0.708) (0.87) (0.435) 0.6 (0.708) 0.64 (0.33) S (0.0046) (0.00) (0.0046) (0.0069) (0.0064) (0.0046) (0.007) 50 0 S (0.3806) (0.0836) (0.3806) (0.0508) ( ) (0.3806) (0.037) 50 0 S (0.06) (0.0745) 0.09 (0.06) (0.0598) ( ) 0.09 (0.06) ( ) 00 0 S (0.0747) S (0.354) ( ) (0.304) ( ) (0.354) ( ) (0.337) (0.0963) (0.39) (0.0747) (0.354) ( ) (0.38) 53

10 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton Table 7. MSEs and RABas (between parentheses) of Bayesan estates for the relablty functon usng portance saplng ethod. n S R R, R, R, R, R, R, BS a=- a=0.00 a= q=-3 q=- q=3 0 7 S (0.009) (0.0005) ( ) (0.0005) (0.097) (0.009) (0.033) 0 9 S (0.08) (0.0766) (0.08) (0.090) (0.070) (0.08) (0.00) 0 5 S (0.058) (0.0049) 0.00 (0.058) (0.076) (0.0964) 0.00 (0.058) (0.046) 0 0 S (0.00) (0.08) (0.00) (0.0009) (0.0097) (0.00) (0.0089) 30 5 S ( ) ( ) ( ) (0.0076) ( ) ( ) (0.0088) 30 0 S (0.006) ( ) (0.006) (0.0073) (0.0068) (0.006) ( ) 40 0 S (0.087) (0.096) (0.087) (0.09) (0.0963) (0.087) (0.095) S ( ) (0.0050) ( ) ( ) ( ) ( ) ( ) 50 0 S (0.09) (0.00) (0.09) ( ) ( ) (0.09) ( ) 50 0 S ( ) (0.0049) ( ) (0.0037) ( ) ( ) ( ) 00 0 S ( ) S (0.044) (0.0047) (0.0395) ( ) (0.044) (0.0096) (0.048) ( ) (0.0395) ( ) (0.044) (0.0074) (0.047) Table 8. MSEs and RABas (between parentheses) of Bayesan estates for the cuulatve hazard functon usng portance saplng ethod. n S H H, H, H, H, H, H, BS a=- a=0.00 a= q=-3 q=- q=3 0 7 S (0.397) (0.4076) (0.3979) ( ) (0.6669) (0.397) (0.69) 0 9 S (0.3487) (0.3546) (0.3486) ( ) ( ) (0.3487) (0.989) 0 5 S (0.43) (0.464) (0.4309) (0.3973) ( ) (0.43) (0.708) 0 0 S (0.94) (0.049) (0.93) (0.799) (0.377) (0.94) (0.0783) 30 5 S (0.593) (0.307) (0.593) (0.3576) (0.038) (0.593) ( ) 30 0 S (0.57) (0.364) (0.56) (0.73) (0.043) (0.57) (0.057) 40 0 S (0.38) ( ) (0.38) ( ) (0.3677) (0.38) (0.376) S 8 (0.0957) (0.0967) (0.0957) (0.0836) (0.5734) (0.0957) (0.0466) 50 0 S (0.5083) ( ) (0.5083) ( ) (0.0557) (0.5083) (0.78) 50 0 S ( ) (0.0857) ( ) (0.066) (0.0606) ( ) (0.3374) 00 0 S (0.0699) S (0.4994) (0.075) (0.468) (0.0699) (0.4994) (0.049) (0.4768) (0.04) (0.39) (0.0699) (0.4994) (0.464) (0.545) 533

11 Bakoban Bayesan & Non Bayesan Estaton of Erlang Dstrbuton 5. CONCLUSIONS In ths paper we have presented Erlang dstrbuton wth shape paraeter 3. Based on progressve Type-II censored saples drawn fro Er(b), we have coputed the ML and Bayes estates of b, relablty, R, and cuulatve hazard, H, functons. Under SE, LINEX and GE loss functons Bayesan estates are coputed by usng tow technques; standard Bayes and portance saplng. The perforance of the estates are conducted by usng the MSE and RABas through Monte Carlo sulaton study based on dfferent censorng schees. Fro the results n Tables (-8), we observe the followng:. All of the obtaned results can be specalzed to both the coplete saple case by takng ( n, r 0,,,3,..., ) and the Type-II rght censored saple for ( r 0,,,3,...,, r n ).. For fxed n, the MSEs of the estates are decreasng as the observed saple proporton / n s ncreasng. 3. The MSEs and RABases of the Bayes estates under LINEX loss functon when the LINEX constant s close to zero, ( a 0.00 ), are very slar to ther correspondng MSEs and RABases under squared error loss functon. 4. The MSEs and RABases of the Bayes estates under GE loss functon when q, are very slar to ther correspondng MSEs and RABases under squared error loss functon. 5. For S, S,..., S, the results show that the standard Bayes and portance saplng technques are gven 6 resultng estates very close to each other. 6. The Bayesan estates of b are slar to each other based on MSE. On the other hand, based on RABas, BS estates have saller values than the others, also, n soe cases the standard Bayes ethod for S, S,..., S, 8, a have the sallest values. But ML estates of b perfors the best based on RABas for the two technques. 7. By coparng the Bayesan estates, estates, q 3, of R have the nu MSEs and RABases. Furtherore, the ML estates have the nu MSEs and RABases by coparng all estates. 8. estates, q 3, of H have the nu RABases for ost cases. Fro the prevous dscusson, we conclude that the portance saplng technque perfors as well as standard Bayes technque. The estates perfors better than the Bayesan estates for estatng R and H. Based on RABas, we recoend the ML estator for estatng b. Also, we recoend ML estators for estatng R. But we prefer to use estators to estate H. REFERENCES [] Abdelkader, Y. H. (003). Coputng the oents of order statstcs fro nondentcally dstrbuted Erlang varables. Statstcal Papers, 45, [] Bakoban, R. A. (00). A study on xture of exponental and exponentated gaa dstrbutons. Advances and Applcatons n Statstcal Scences, (), 0-7. [3] Balakrshnan, N. and Aggarwala, R. (000). Progressve Censorng: Theory, Methods and Applcatons. Boston: Brkhauser. [4] Balakrhnan, N. and Sandhu, R. A. (995). A sple sulaton algorth for generatng progressve Type-II censored saples. The Aercan Statstcan, 49(), [5] Basu, A. P. and Ebrah, N. (99). Bayesan approach to lfe testng and relablty estaton usng asyetrc loss functon, Journal of Statstcal Plannng and Inference, 9, -3. [6] Berger, J. (980). Iprovng on nadssble estators n contnuous exponental fales wth applcatons to sultaneous estaton of gaa scale paraeters. Annals of Statstcs, 8, [7] Calabra, R. and Pulcn, G. (996). Pont estaton under asyetrc loss functons for lft-truncated exponental saples. Councatons n Statstcs Theory and Methods, 5(3), [8] Canfeld, R. V. (970). A Bayesan approach to relablty estaton usng a loss functon. IEEE Transactons on Relablty, R-9, 3-6. [9] Constantne, K., Karson, M. and Tse, S. K. (986). Estaton of P( Y X ) n the gaa case. Councatons n Statstcs-Sulaton and Coputaton, 5, [0] Dey, D. K., Ghosh, M. and Srnvasan, C. (987). Sultaneous estaton of paraeters under entropy loss. Journal of Statstcal Plannng and Inference, 5, [] Dey, D. K. and Lu, P. L. (99). On coparson of estators n a generalzed lfe odel. Mcroelectroncs Relablty, 33,

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