BAYESIAN ESTIMATION OF GUMBEL TYPE-II DISTRIBUTION

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1 Data Scece Joural, Volume, 0 August 03 BAYESIAN ESTIMATION OF GUMBEL TYPE-II DISTRIBUTION Kamra Abbas,*, Jayu Fu, Yca Tag School of Face ad Statstcs, East Cha Normal Uversty, Shagha 004, Cha Emal-addresses:* amuaj@gmal.com, zsh0@63.com, yctag@stat.ecu.edu.c Departmet of Statstcs, Uversty of Azad Jammu ad Kashmr, Muzaffarabad, Pasta ABSTRACT I ths paper we cosder the Bayesa estmators for the uow parameters of Gumbel type-ii dstrbuto. The Bayesa estmators caot be obtaed closed forms. Approxmate Bayesa estmators are computed usg the dea of Ldley s approxmato uder dfferet loss fuctos. The approxmate Bayes estmates obtaed uder the assumpto of o-formatve prors are compared wth ther maxmum lelhood couterparts usg Mote Carlo smulato. A real data set s aalyzed for llustratve purpose. Keywords: Bayesa estmator, Maxmum lelhood estmator, Ldley s approxmato, Mote Carlo smulato, Gumbel type-ii dstrbuto INTRODUCTION The Gumbel type-ii dstrbuto was troduced by Germa mathematca Eml Gumbel (89-9) 958, ad s useful predctg the chace of meteorologcal pheomea, such as aual flood flows, earthquaes, ad other atural dsasters. It has also bee foud to be satsfactory descrbg the lfe expectacy of compoets. The radom varable s sad to follow a Gumbel type-ii dstrbuto wth parameters ad, where the cumulatve dstrbuto fucto (CDF) s gve by F x, exp x, x 0,, 0, () The correspodg probablty desty fucto (PDF) of () s ( ) f x, x exp x, x 0,, 0. () Recetly, may authors have cotrbuted to statstcal methodology ad characterzato of Gumbel type-ii dstrbuto. For example, Kotz ad Nadarajah (000) dscussed some propertes of Gumbel dstrbuto. Feroze ad Aslam (0) cosdered Bayesa aalyss of Gumbel type-ii dstrbuto uder doubly cesored samples usg dfferet loss fuctos. Cors et al. (00) dscussed the maxmum lelhood (ML) algorthms ad Cramer-Rao (CR) bouds for the locato ad scale parameters of the Gumbel dstrbuto. Al Mousa et al. (00) studed the Bayesa estmato order to aalyze both the parameters of Gumbel dstrbuto based o record values. Smlarly, Malowsa ad Szyal (008) obtaed Bayesa estmators for two parameters of a Gumbel dstrbuto based o th lower record values. Nadarajah ad Kotz (004) troduced beta Gumbel (BG) dstrbuto whch provdes closed-form expressos for the momets, asymptotc dstrbuto of extreme order statstcs as well as dscussed the maxmum lelhood estmato procedure. Mladovc ad Tsoos (009) studed the sestvty of Bayesa relablty estmates for modfed Gumbel falure models uder fve dfferet parametrc prors usg squared error loss fucto. Al-Badha ad Sclar (987) ad Hossa ad Howlader (996) studed dfferet estmato methods case of complete samples. However, Bayesa estmatos uder dfferet loss fuctos are ot frequetly dscussed. Bayesa estmators uder dfferet loss fuctos volve tegral expressos, whch are ot aalytcally solvable. Therefore, Ldley s approxmato techque s sutable for solvg such problems. The ma objectve of ths study s to develop the Bayesa estmators uder dfferet loss fuctos ad compare them wth maxmum lelhood estmators (MLEs) terms of bas ad mea squared error (MSE) of the estmate. The rest of the paper s orgazed as follows. I Secto, the MLEs ad observed Fsher formato matrx for 33

2 Data Scece Joural, Volume, 0 August 03 parameters are derved. Bayesa estmato uder LINE (lear expoetal) loss fucto ad geeral etropy loss fucto are dscussed Secto 3. Smulato study s preseted Secto 4, ad oe real data set s aalyzed Secto 5. Fally, cocluso s gve Secto 6. MAIMUM LIKELIHOOD ESTIMATION (,,..., ) be a radom sample of sze from the Gumbel type-ii dstrbuto (). The lelhood Let fucto of (, ) s The the log-lelhood fucto ca be wrtte as ad ( ) (, ) exp. L l L l l ( ) l( ), l L l( ) l( ), l L. The maxmum lelhood estmates of ad, say ML of the equatos ad ad ML respectvely, ca be obtaed as the solutos, l( ) l( ) ML. ML Ths may be solvg usg a terato scheme. We use the Laplace approxmato to compute MLEs. Further, the observed Fsher formato matrx s obtaed by tag the secod ad mxed partal dervatves of l L wth respect to ad. We have (3) (4) where I l L l L, l L l L l L l( ), 34

3 Data Scece Joural, Volume, 0 August 03 3 BAYESIAN ESTIMATION l L, l L l L l( ). I Bayesa estmato, we cosder two types of loss fuctos. The frst oe s LINE loss fucto, whch s asymmetrc. The LINE loss fucto was troduced by Vara (975), ad several authors, such as Basu ad Ebrahm (99), Rojo (987), ad Nassar ad Essa (004), have used ths loss fucto dfferet estmato problems. Ths fucto rses approxmately expoetally o oe sde of zero ad approxmately learly o the other sde. The LINE loss fucto ca be expressed as L( ) e, 0., (5) where ( ), ad s a estmate of. The sg ad magtude of the shape parameter represets the drecto ad degree of symmetry, respectvely. Moreover, f 0, the overestmato s more serous compared to the uder estmato ad vce-versa. For close to zero, the LINE loss s approxmately squared error loss ad therefore almost symmetrc. The posteror expectato of the LINE loss fucto (5) s E [ L( )] E [ e ] ( E ( )), (6) e where E () deotes the posteror expectato wth respect to the posteror desty of. The Bayes estmator of, deoted by BL uder LINE loss fucto, s the value, whch mmzes (6). It s BL l E e, (7) provded that the expectato E [ e ] exsts ad s fte. The problem of choosg the value of the parameter s dscussed Calabra ad Pulc (996). The secod type of loss fucto s the geeralzato of the etropy loss, whch s dscussed by Dey ad Lu (99) ad Dey (987). The geeral etropy loss s defed as where ( L BE, ) log, s a estmate of. The Bayes estmator relatve to the geeral etropy loss s provded that E( ) E, (9) BE exsts ad s fte. For =, the Bayes estmator (9) cocdes wth the Bayes estmator uder the weghted squared error loss fucto, ad for = -, the Bayes estmator (9) cocdes wth the Bayes estmator uder the squared error loss fucto. Further, the Bayesa estmators uder LINE loss fucto ad geeral etropy loss fucto are provded Appedx. 4 SIMULATION STUDY To compare the performace of theoretcal results, the samples are geerated from Gumbel type-ii dstrbuto usg the verse trasformato techque by cosderg dfferet values of parameters. Sample sze s vared to observe the effect of small ad large samples o the estmators. For each sample sze, we compute maxmum lelhood estmates of ad ad the Bayesa estmates uder LINE loss ad etropy loss were computed usg Laplace s approxmato ad Ldley s approxmato respectvely. (8) 35

4 Data Scece Joural, Volume, 0 August 03 For Bayesa estmators, we cosder that ad each have depedet Gamma ( a, b ) ad Gamma ( a, b ) prors. Further, the Bayesa estmators of ad are also obtaed usg geeral uform prors. We use dfferet values of loss fucto parameter =± ad o-formatve prors of both ad,.e., a = b = a = b = 0 ad a a b b. The behavor samplg of approxmate Bayesa estmators s vestgated ad compared wth the MLEs terms of ther MSEs. The results are preseted Tables -4. From the results of smulato study, coclusos are draw regardg the behavor of the estmators, whch are summarzed below.. As expected, t s observed that the performaces of both Bayesa ad maxmum lelhood estmators become better whe sample sze creased. Also, t s observed that for large sample szes, the Bayesa estmates ad maxmum lelhood estmates become closer terms of MSEs.. Whe = ad a a b b 0, the MSEs of Bayesa estmators uder LINE loss fucto ad geeral etropy loss fucto usg Gamma prors ad geeral uform prors are lower tha the MSEs of maxmum lelhood estmators. Therefore the Bayesa estmators are more stable tha maxmum lelhood estmators. 3. For = - ad a a b b, the Bayesa estmators uder geeral etropy loss fucto ad LINE loss fucto perform better tha MLEs obtaed by usg Gamma prors ad geeral uform prors terms of ther MSEs. 4. Fgure dcates that MSEs decreased as creases for all methods of estmato studed. It s to be oted that, whe sample sze s small, the ML method teds to have larger MSE tha the Bayesa method, ad oe would prefer Bayesa estmators. Clearly, for small sample szes, the Bayesa estmators should be recommeded for Gumbel type-ii dstrbuto. From Tables -4, we ca see that each scearo, the Bayesa estmators uder assumpto of geeral etropy loss fucto ad LINE loss fucto outperform the maxmum lelhood estmators sce MSEs are sgfcatly smaller. It s worth otg that BLU: Bayesa estmator uder LINE loss fucto usg geeral uform pror. BEU: Bayesa estmator uder geeral etropy loss fucto usg geeral uform pror. BLG: Bayesa estmator uder LINE loss fucto usg Gamma pror. BEG: Bayesa estmator uder geeral etropy loss fucto usg Gamma pror. Table. Average estmates ad correspodg MSEs (wth parethess) for α whe ( = ad a =a =b =b = 0). Estmator α (0.03).0765(0.0478).678(0.086).50(0.889) 0.580(0.03) 0.53(0.0).0698(0.0439).0508(0.04).666(0.039).5660(0.0884).50(0.84).0650(0.474) 0.568(0.0094) 0.559(0.009).0554(0.046).0370(0.0403).6085(0.064).5586(0.093).403(0.887).075(0.594) (0.0068).0470(0.07).576(0.0630).0996(0.9) 0.595(0.0063) 0.50(0.006).049(0.055).0305(0.045).5753(0.06).5430(0.0548).00(0.00).0445(0.0954) 0.50(0.0056) 0.57(0.0054).0330(0.049).00(0.04).569(0.067).537(0.0557).034(0.09).0474(0.00) 36

5 Data Scece Joural, Volume, 0 August (0.0037).08(0.047).546(0.0347).0566(0.0573) 0.50(0.0034) 0.503(0.003).058(0.04).068(0.038).54(0.0340).53(0.030).0569(0.0564).046(0.057) (0.0033) (0.003).099(0.039).07(0.036).5369(0.034).583(0.033).0579(0.0570).056(0.053) (0.00).074(0.0086).560(0.088).038(0.0345) (0.000) (0.000).059(0.0084).04(0.008).556(0.086).54(0.079).0383(0.034).084(0.03) (0.000) 0.503(0.000).0(0.0083).0077(0.008).59(0.086).55(0.080).0387(0.0340).088(0.037) (0.006).034(0.0065).509(0.044).039(0.05) 0.504(0.006) 0.500(0.006).03(0.0064).0087(0.0063).506(0.04).54(0.038).040(0.049).0083(0.040) 0.503(0.005) 0.50(0.005).009(0.0063).0057(0.0063).584(0.043).5093(0.038).04(0.050).0085(0.043) Table. Average estmates ad correspodg MSEs (wth parethess) for β whe ( = ad a =a = b =b = 0). Estmator β (0.049).0380(0.075).6059(0.90).958(0.443) 0.508(0.036) (0.0).0378(0.0735).003(0.0635).6058(0.803).534(0.43).758(0.3763).0398(0.89) (0.035) (0.033).09(0.074).07(0.0663).5966(0.900).534(0.349).950(0.4369).0598(0.349) (0.053).059(0.0437).5663(0.05).9(0.09) (0.048) (0.043).034(0.043).0035(0.039).5660(0.030).544(0.0876).3(0.055).035(0.666) 0.503(0.048) 0.500(0.045).054(0.043).040(0.040).5584(0.04).5055(0.099).7(0.35).036(0.856) 50 βml (0.0090).046(0.038).5373(0.0535).066(0.065) (0.0088).09(0.036).5370(0.059).069(0.033) 37

6 Data Scece Joural, Volume, 0 August (0.0086).006(0.03).5075(0.048).09(0.0909) 0.508(0.0088) 0.505(0.0086).008(0.035).0070(0.06).539(0.0533).50(0.0493).0659(0.044).046(0.0959) (0.0057).007(0.053).504(0.038).0448(0.0634) 0.507(0.0056) 0.50(0.0056).035(0.053).006(0.047).503(0.036).50(0.099).0430(0.06).07(0.057) 0.500(0.0056) (0.0055).0066(0.05).000(0.048).569(0.035).5006(0.0303).044(0.064).08(0.059) (0.0044).008(0.09).50(0.05).0308(0.047) (0.0044) (0.0043).004(0.08).007(0.05).59(0.050).5005(0.037).095(0.0464).0047(0.0436) (0.0043) 0.500(0.004).0049(0.08).000(0.06).50(0.050).5000(0.040).030(0.0466).0053(0.0447) Table 3. Average estmates ad correspodg MSEs (wth parethess) for α whe ( = - ad a =a = b =b =) Estmator α (0.00).0805(0.054).60(0.07).08(0.934) (0.09) (0.009).077(0.05).0705(0.0478).590(0.005).5597(0.0698).74(0.93).730(0.868) (0.08) 0.599(0.0086).0639(0.0467).0608(0.039).5495(0.0845).5(0.060).0954(0.458).0740(0.394) (0.0070).047(0.079).5667(0.0594).090(0.0959) 0.545(0.0069) 0.55(0.0059).04(0.078).0460(0.06).55(0.0585).54(0.0485).0697(0.0945).0630(0.0944) 0.540(0.0070) 0.586(0.0056).0396(0.06).0374(0.039).564(0.058).558(0.0440).036(0.083).034(0.0737) (0.0037).086(0.046).545(0.030).08(0.0607) 0.547(0.0036) 0.535(0.0034).048(0.043).005(0.04).537(0.035).590(0.087).000(0.0596).0066(0.0470) 0.543(0.0036) 0.540(0.0033).08(0.040).039(0.034).579(0.096).543(0.069).03(0.0547).030(0.0473) 38

7 Data Scece Joural, Volume, 0 August (0.00).085(0.0084).53(0.08).0064(0.0346) (0.00) 0.507(0.000).060(0.0084).055(0.008).575(0.08).56(0.07).0053(0.034).0047(0.030) (0.00) 0.506(0.009).00(0.008).05(0.0080).5086(0.074).5073(0.065).003(0.035).008(0.097) (0.006).038(0.0066).500(0.05).0006(0.065) (0.006) 0.508(0.006).08(0.0065).00(0.0065).500(0.050).5048(0.044).000(0.063).000(0.039) 0.503(0.006) 0.505(0.005).0086(0.0064).0008(0.0063).5000(0.043).5007(0.040).000(0.053).0000(0.035) Table 4. Average estmates ad correspodg MSEs (wth parethess) for β whe ( = - ad a =a =b =b =). Estmator β (0.046).036(0.0737).6004(0.9).960(0.783) (0.039) (0.037).064(0.069).05(0.0590).5706(0.89).55(0.53).705(0.763).508(0.699) (0.038) 0.505(0.03).065(0.067).047(0.050).5404(0.49).5357(0.39).650(0.669).578(0.66) (0.056).054(0.0443).5669(0.07).0(0.309) (0.054) 0.508(0.05).04(0.04).0(0.039).5457(0.099).585(0.0684).(0.05).0935(0.66) (0.049) (0.04).030(0.0394).097(0.0353).584(0.0849).50(0.0603).36(0.305).5(0.56) (0.009).035(0.050).5398(0.0545).0733(0.5) (0.0090) (0.0089).0059(0.043).006(0.033).569(0.0530).58(0.0448).0600(0.4).059(0.030) (0.0088) (0.0085).09(0.034).0(0.00).5084(0.0487).507(0.043).040(0.0968).0349(0.0850) 39

8 MSE Data Scece Joural, Volume, 0 August Β-ML (0.0058).0075(0.047).57(0.036).0399(0.060) 0.500(0.0058) 0.500(0.0058).008(0.045).000(0.043).536(0.030).59(0.084).0309(0.0600).007(0.0479) (0.0057) 0.500(0.0056).0050(0.04).003(0.036).5049(0.096).5030(0.07).008(0.055).000(0.0460) (0.0045).007(0.08).556(0.045).00(0.0458) (0.0044) (0.0044).000(0.06).000(0.05).5009(0.04).5005(0.05).005(0.0457).003(0.0388) (0.0043) (0.0043).0003(0.030).000(0.0).500(0.033).500(0.08).000(0.049).0003(0.0377) ML BLU BLG BEU BEG Sample sze () (a) 40

9 MSE MSE Data Scece Joural, Volume, 0 August ML BLU BLG BEU BEG Sample sze () (b) ML BLU BLG BEU BEG Sample sze () (c) 4

10 MSE Data Scece Joural, Volume, 0 August ML BLU BLG BEU BEG Fgure. Plot of sample sze ad MSEs of α ad β usg dfferet methods of estmato such as = ±, a =b =a =b =0 ad a =a =b =b =. I abscssa (a) MSEs of α (whe =), (b) MSEs of α (whe =-), (c) MSEs of β (whe =), (d) MSEs of β (whe =-). 5 DATA ANALYSIS Sample sze () (d) I ths secto we cosder the real data set obtaed from Nchols ad Padgett (006), whch represets the breag stress of carbo fbres ( Gba). The data cosst of 00 observatos ad are preseted Table 5. Table 5. Breag stress of carbo fbres ( Gba). 3.7,.74,.73,.5, 3.6, 3., 3.7,.87,.47, 3., 4.4,.4, 3.9, 3.,.69, 3.8, 3.09,.87, 3.5, 4.9, 3.75,.43,.95,.97, 3.39,.96,.53,.67,.93, 3., 3.39,.8, 4., 3.33,.55, 3.3, 3.3,.85,.56, 3.56, 3.5,.35,.55,.59,.38,.8,.77,.7,.83,.9,.4, 3.68,.97,.36, 0.98,.76, 4.9, 3.68,.84,.59,3.9,.57, 0.8, 5.56,.73,.59,,.,.,.7,.7,.7, 5.08,.48,.8, 3.5,.7,.69,.5, 4.38,.84, 0.39, 3.68,.48, 0.85,.6,.79, 4.7,.03,.8,.57,.08,.03,.6,.,.89,.88,.8,.05, 3.65 Table 6. Pot estmates ad stadard devatos (SD) of α ad β. Estmator α SD β SD ML BLU BLG BEU BEG The pot estmates of α ad β ad ther stadard devatos (SD) are summarzed Table 6. It s observed that the Bayesa estmates uder geeral etropy loss fucto ad LINE loss fucto are close to the ML estmates. Whe we compare the ML estmators wth Bayesa estmators usg Ldley s approxmato terms of ther stadard devatos, the approxmate Bayesa estmators perform better tha the MLEs. 4

11 Data Scece Joural, Volume, 0 August 03 6 CONCLUSION I ths paper we cosder classcal ad Bayesa estmators uder the assumpto of LINE loss ad geeral etropy loss fuctos. Nether Bayesa or maxmum lelhood estmators ca be obtaed closed forms. Ldley s approxmato s used to obta the Bayesa estmates, ad t s cocluded that the approxmato wors very well eve for small sample szes though the computato of Ldley s techque based o the maxmum lelhood estmators. We compare the performace of dfferet methods by Mote Carlo smulatos. Smulatos showed that the Bayesa estmators uder geeral etropy loss fucto ad LINE loss fucto perform better tha the maxmum lelhood estmators. However, t s observed that for large sample szes the Bayesa ad maxmum lelhood estmates become closer terms of ther MSEs. 7 ACKNOWLEDGEMENT The authors would le to tha the aoymous revewers for ther costructve commets ad suggestos whch led to a large mprovemet the earler verso of ths mauscrpt. 8 REFERENCES Al-Badha, F. A. & Sclar, C. D. (987) Comparso of methods of estmato of parameters of the Webull dstrbuto. Commucatos Statstcs Smulato ad Computato 6, pp Al Mousa, M. A. M., Jahee, Z. F., & Ahmad, A. A. (00) Bayesa estmato, predcto ad characterzato for the Gumbel model based o records. Statstcs 36, pp Basu, A. P. & Ebrahm, N. (99) Bayesa approach to lfe testg ad relablty estmato usg asymmetrc loss fucto. Joural of Statstcal Plag ad Iferece, pp Calabra, R. & Pulc, G. (996) Pot estmato uder asymmetrc loss fuctos for left-trucated expoetal samples. Commucatos Statstcs - Theory ad Methods 5(3), pp Cors, G., G, F., & Gerco, M. V. (00) Cramer-Rao bouds ad estmato of the parameters of the Gumbel dstrbuto. IEEE 9, pp -3. Dey, D. K., Gosh, M., & Srvasa, C. (987) Smultaeous estmato of parameters uder etropy loss. Joural of Statstcal Plag ad Iferece, pp Dey, D. K. & Lao Lu P.S. (99) O comparso of estmators a geeralzed lfe model. Mcroelectrocs Relablty 3, pp 07-. Gumbel, E. J. (958) Statstcs of Extremes. New Yor: Columba Uversty Press. Hossa, A. & Howlader, H. A. (996) Uweghted least squares estmato of Webull parameters. Joural of Statstcal Computato ad Smulato 54, pp Kotz, S. & Nadarajah, S. (000) Extreme Value Dstrbutos: Theory ad Applcatos. Imperal College Press. Ldley, D. V. (980) Approxmate Bayesa method. Trabajos de Estadstca 3, pp Malowsa, I. & Szyal, D. (008) O characterzato of certa dstrbutos of th lower (upper) record values. Appled Mathematcs ad Computato 0(), pp Mladovc, B. & Tsoos, P. C. (009) Sestvty of the Bayesa relablty estmates for the modfed Gumbel falure model. Iteratoal Joural of Relablty, Qualty ad Safety Egeerg (IJRQSE) 6(40), pp

12 Data Scece Joural, Volume, 0 August 03 Nadarajah, S. & Kotz, S. (004) The beta Gumbel dstrbuto. Mathematcal Problems Egeerg 4, pp Nassar, M. M. & Essa, F. H. (004) Bayesa estmato for the expoetated Webull model. Commucatos Statstcs - Theory ad Methods 33(0), pp Navd, F. & Aslam, M. (0) Bayesa aalyss of Gumbel type-ii dstrbuto uder doubly cesored samples usg dfferet loss fuctos. Caspa Joural of Appled Sceces Research (0), pp -0. Nchols, M. D. & Padgett, W. J (006) A bootstrap cotrol chart for Webull percetles. Qualty ad Relablty Egeerg Iteratoal, pp 4-5. Rojo, J. (987) O the admssblty of wth respect to the LINE loss fucto. Commucatos Statstcs - Theory ad Methods 6(), pp Vara, H. R. (975) A Bayesa Approach to Real Estate Assessmet. Amsterdam: North Hollad, pp APPENDI For Bayesa estmato, we eed pror dstrbuto of ad. Assumg that ad each have depedet Gamma ( a, b ) ad Gamma ( a, b ) prors respectvely for a, b, a, b 0,.e., ( ) a b e ad ( ) a b e. Based o the prors, the jot posteror desty of ad ca be wrtte as L( data, ) ( ) ( ) f, x L( data, ) ( ) ( ) dd (0) 0 0 Therefore, the Bayesa estmator of ay fucto of ad, say g( ; ), uder the LINE loss fucto s g(, ) E [ g(, )], data g(, ) L( data, ) ( ) ( ) dd L( data, ) ( ) ( ) dd It s ot possble for () to have a closed form. Therefore, we adopt Ldley s approxmato (980) procedure to approxmate the rato of the two tegrals such as (), whch ca be evaluated as g g(, ) l s l A l A l B l B p C p C, where j j () j j l(, ) j,, 0,,,3, 3, l j j l (, ) l (, ) g(, ) g(, ) p, p, l, l, g(, ) g(, ) g(, ) g(, ) l, l, l, l, A ( l s l s ) s, B 3l s s l ( s s s ), C l s l s,, j,, j j j j j j jj j j j j () 44

13 Data Scece Joural, Volume, 0 August 03 where l() s the log-lelhood fucto of the observed data, sjs the (, j) th elemet of the verse of Fsher s formato matrx. Therefore, the approxmate Bayesa estmators of ad uder LINE loss fucto are l e e s (l ) 3 e s 3 BLG e s s s s e 3 (l ) 3 a a e s b b e s, (3) 3 3 BLG l e e s (l ) e ss e s 3 ss s e (l ) a a b e s b e s. The Bayesa estmators of ad uder geeral etropy loss fucto are (4) ( ) 3 ( ) BEG ( ) s (l ) 3 s s s s s ( ) ( ) 3 3 (l ) ( ) ( ) a a b s b s, (5) ( ) 3 ( ) BEG ( ) s (l ) 3 ( ) ( ) 3 s s s s s s (l ) ( ) a ( ) a, b s b s (6) 45

14 Data Scece Joural, Volume, 0 August 03 v u w s, s, s s, uv w uv w uv w where r w (l ), v r u (l ),, ad ad equatos ()-(6) are the maxmum lelhood estmators of ad from (3) ad (4). Smlarly, the approxmate Bayesa estmators of ad uder LINE loss fucto ad geeral etropy loss fucto usg geeral uform prors.e. ( ) a b ad ( ) ca be obtaed. 3 4 (Artcle hstory: Receved 9 Aprl 03, Accepted 5 July 03, Avalable ole 4 August 03) 46