Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. EFFICIENT ESTIMATION OF THE WEIBULL SHAPE PARAMETER BASED ON A MODIFIED PROFILE LIKELIHOOD ZHENLIN YANG a ad MIN XIE b a School of Ecoomcs ad Socal Sceces, Sgapore Maagemet Uverst emal: zlag@smu.edu.sg b Departmet of Idustral ad Sstems Egeerg, Natoal Uverst of Sgapore The maxmum lkelhood estmator of the Webull shape parameter ca be ver based. A estmator based o the modfed profle lkelhood s proposed ad ts propertes are studed. It s show that the ew estmator s almost ubased wth relatve bas beg less tha % most of stuatos, ad t s much more effcet tha the regular MLE. The smaller the sample or the heaver of the cesorg, the more effcet s the ew estmator relatve to the regular MLE. Kewords: Webull dstrbuto, Shape parameter, Bas, Mea squared error, Modfed profle lkelhood, Cesored data.. INTRODUCTION Because of ts flexblt modelg both creasg falure rate ad decreasg falure rate, Webull dstrbuto s ow wdel used relablt studes. As the falure rate tred for Webull dstrbuto s characterzed b the value of ts shape parameter, the estmato of the Webull shape parameter s of partcular terest. Several methods exst the lterature, such as the maxmum lkelhood estmato (MLE) method (Lawless, 98), lear estmator (Lawless, 98), method based o probablt plot ad a modfed verso of t (Drapella ad Koszk, 999), shruke estmator (Pade ad Sgh), etc. Amog the above-metoed methods, the MLE s a ver popular oe due to ts smplct ad effcec (Ross, 994). The Webull probablt plot s usuall used to get some rough estmates that mght serve as startg values for umercal procedures solvg the lkelhood equato. However, the MLE of the Webull shape parameter s kow to be b-
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. ased, ad ca ver based the case of small sample or heav cesorg (Macksack ad Stllma, 996) (e.g, for a complete sample of sze 0, the relatve bas s about %). Ross (994) provded a smple ubasg formula for the case of complete sample, whch was show to work qute well, ad added aother formula (Ross, 996) for the case of Tpe II cesored data. I ths paper, we propose a estmator for the Webull shape parameter based o modfed profle lkelhood of Cox ad Red (987, 989, 99) uder the oto of parameter orthogoalzato. It turs out that the modfcato s extremel smple t volves ol a smple adjustmet o the profle lkelhood or the lkelhood equato. The modfed MLE performs surprsgl well as compared wth the regular MLE. The relatve bas ca be reduced to less tha % most cases. Also, the modfed MLE s much more effcet tha the regular MLE. It s also more effcet tha the ubased MLE of Ross (994, 996). The modfed MLE works the best for the complete or Tpe II cesored data. Ths paper s orgazed as follows. Secto dscusses the use of parameter orthogoalzato for the estmato of the Webull parameters that provdes the basc motvato for ths stud. Secto 3 derves the formulas for the profle lkelhood estmates for the Webull shape parameter. Secto 4 presets some smulato results regardg the performace of the ew estmator relatve to the regular lkelhood estmator ad ts adjusted verso.. PARAMETER ORTHOGONOLIZATION A problem wth the MLE for Webull parameters s that the estmators are hghl correlated. The basc dea behd the parameter orthogoalzato s that f the two parameters are orthogoal, the the MLEs of the two parameters are asmptotcall depedet. Hece, makg ferece o oe parameter s ot affected (at least asmptotcall) b whether the other parameter s estmated or gve. Let Y be a Webull radom varable wth the probablt dest fucto (pdf) f(; α, ) α α exp α, ()
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. where s the shape parameter ad α s the scale parameter. We are ow terested estmatg the presece of the usace parameter α. Suppose that a reparameterzato s made from (, α ) to (, λ ) so that ad λ are orthogoal the sese that the elemet of the expected Fsher formato matrx I λ 0 (Cox ad Red, 987). The orthogoalt codto ca be reexpressed terms of the orgal parameter settg as follows αα α where αα E [ f (, α) α ] ( α) + α 0, E [ f (, α) α ] ( γ ) α, ad, α γ 0.5775 s Euler s costat. Solvg the resulted dfferetal equato oe gets α λ [( γ ) ] exp ad hece the orthogoal usace parameter: λ α [( γ) ] exp. The above result s gve Cox ad Red (987), but there s a tpographcal error for the expresso of α. Ths orthogoal parameter settg was used to gve a modfed profle lkelhood ad hece a estmator of the Webull shape parameter, but smulato results show that t s ot qute satsfactor although t s able to reduce the bas b 50% or more, depedg the true value of the shape parameter. It was oted Cox ad Red (989) that f λ s orthogoal to, so s a smooth fucto of λ. Ths suggests that a further mprovemet s possble. Followg the method of Cox ad Red (989), we show that the optmal orthogoal parameterzato takes the log form,.e. o λ log( λ ) log( α ) + ( γ ). However, smlar to the example 3 cosdered Cox ad Red (989), the proportoalt costat volves, whch s a pheomeo ot beg able to be explaed b the orgal authors. Nevertheless, ths leads us wth a flexble choce of ths costat. Aother mportat pot s that the dervato of the orthogoal parameter s based o the complete sample. It would be of terest to vestgate ths orthogoal parameterzato for 3
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. the case of cesored data. Naturall, ths orthogoal parameterzato should be reserved uder cesorg, but t s dffcult to verf as the case of cesored data oe caot work out exact expressos for certa expectatos. We wll take prmarl the log form, but allow some flexblt the form of the compoet wth regard to dfferet tpes of cesorg. 3. MODIFIED PROFILE LIKELIHOOD The purpose of modfcatos to the profle lkelhood s to approxmate more closel the lkelhood fucto used exact margal or codtoal ferece (Cox ad Red, 99). I terms of the lkelhood equato, the modfed verso should provde us wth estmates that are much less based tha those correspodg to the profle lkelhood. Wth these md, we vestgate the usefuless of the method the cotext of estmatg the Webull shape parameter. Let Y, Y,, Y k be a radom sample from WB(α, ) ad l (, α ) be the log lkelhood fucto. The profle lkelhood for s defed as l p ( ) l [, αˆ( )], where α ˆ( ) s the restrcted MLE of α for a gve. The modfed profle lkelhood [] s as follows where (, λˆ( ) ) λλ l m ( ) p ( ) l log det[ (, λˆ( ))] J, () J s the elemet of the observed formato matrx for (, λ ), evaluated for fxed at the correspodg restrcted MLE λ ˆ( ). λλ 3.. The Case of Complete Data Here, we gve a detaled descrpto of the method for the case of complete sample. The case of cesored wll be dscussed later for dfferet cesorg assumptos. Let Y, Y,, Y be a radom sample from WB(α, ). The log lkelhood fucto s l (, α ) + log ( ) log (3) α α α 4
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. For a gve, the restrcted MLE of α s α ˆ( ) ( ). Substtutg α ˆ( ) to (3) gves the profle log lkelhood for : l p ( ) (log ) + ( ) log log + ( ) log. The usual MLE of s obtaed b maxmzg the profle log lkelhood, or equvaletl b solvg the profle lkelhood equato where the profle score fucto s B takg the parameterzato ad usg the relatoshp, S p ( ) dl p ( ) d 0, log S p ( ) + o λ log( λ) log(α) log. (4) γ +, (5) ˆo J o o[, λ ( )] (, αˆ( ) ) λ λ o J ( α λ ) α αˆ( ) αα, oe ca easl see that ˆo J [, λ ( )] 4. Hece, from Equato (), we have that o o λ λ l m () l p ( ) log ad the modfed profle lkelhood equato becomes, S m ( ) log + log. (6) 5
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. Solvg the modfed profle lkelhood equato S m ( ) 0 gves the modfed MLE for. Ths modfcato s smple, but works surprsgl well as show b the Mote Carlo smulatos ext secto. 3.. The Case of Cesored Data For the case of cesored data, t s reasoable to use the same orthogoal parameter settg. Ol extra work s to derve, uder dfferet cesorshp mechasm, the observed formato umber j (, λ ˆ( )), the ke quatt the modfcato term. λλ Let Y, Y,, Y r ow deote the r smallest observatos a radom sample of from WB(,α ),.e., the data are Tpe II cesored. The observed formato s foud to be of the same form as the case of complete sample f the same parameterzato s used. Ths gves the modfed the modfed profle score fucto for : r S m ( ) r r * * r + log log compared wth the usual profle score fucto where a otatoal coveto, r r r S p ( ) * * r + log r r * w w r r + ( r) wr, s used., (8) log, (9) I the stuato of Tpe I cesorg, defe T,,,,, as the actual Webull lfetme, C the cesorg tme, ad Y m(t, C ). Thus, the data (Y, Y,, Y ) ow represet the Tpe I cesored data wth r of them the real lfetmes ad r of them the cesored lfetmes. We assume that r. I ths case, we emplo the orthogoal parameterzato o λ log( λ ) log( α ) + ( γ ). The observed formato s ˆo jλλ [, λ ( )] ad we have from () that l m ( ) l p ( ) log. The profle ad modfed profle scores are 6
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. ad r S p ( ) r + log D log S m ( ) r r + log log D respectvel, where D s the set of ucesored observatos ad r s the total umber of them. (0) () 4. MONTE CARLO SIMULATION I ths secto, some smulato results are preseted to compare modfed MLE wth the tradtoal oe. The bas ad MSE of the two estmators are smulated. The reported results clude: the relatve bas, E( ˆ )/, ad the relatve effcec of ˆ M to ˆ, REF MSE( ˆ )/MSE( ˆ ). We frst cosder the case of complete or Tpe II cesored data. Several sample szes () ad degrees of cesorshp (r) are cosdered. Te dfferet values are used for each combato of ad r. The α value s fxed at 00 as the estmato of for both methods s depedet of α. Smulato results are summarzed Table. The results Table dcate that the ew estmator deoted as MMLE s almost ubased wth relatve bas beg less tha % most of stuatos. It s also much more effcet tha the regular MLE. The smaller the sample or the heaver of the cesorg, the more effcet s the ew estmator relatve to the regular MLE. Ross (994) proposed a smple bas-reducto factor ( ) ( 0.68) for the case of complete data, ad added aother formula { + r[.37 ( r.9) ]} M for Tpe II cesored data (Ross, 996). It was show b smulato that these smple factors work well ad ca reduce the relatve bas to less tha 0.3%. However, these two formulas do ot match whe r, ad there s o smple formula avalable for other tpe of cesored data. It s eas to see that the cases of complete ad Tpe II cesored data, our modfed MLE s more effcet tha the bas-reduced MLE of Ross. For examples, for r 0, the relatve effcec of the bas-reduced MLE to the regular MLE s.36, compared wth.64 for the modfed MLE, for r 0, the umbers are, respectvel,.5 ad.30, ad for 30 ad r 0, the umbers are.67 ad.90, respectvel. 7
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. Table. Relatve Bas (%) ad Relatve Effcec of MLE ad Modfed MLE Complete or Tpe II Cesored Sample Relatve Bas Relatve Bas Relatve Bas Relatve Bas MLE MMLE REF MLE MMLE REF MLE MMLE REF MLE MMLE REF (0, r0) (0, r0) (0, r5) (0, r0) 0.5 7..8.64.57 0.5.8.30 0.70.48 7.75 0.94.30 0.8 7.6.83.64.5 0.48.83.90 0.34.47 7.6 0.80.9.0 7.33.0.64.66 0.59.84.40 0.79.48 7.57 0.76.30.5 6.90.6.64.43 0.38.84.44 0.83.48 7.5 0.7.9.0 6.80.5.63.64.40.86.95 0.39.47 7.76 0.93.3 3.0 6.90.6.6.50 0.45.83.0 0.45.47 7.6 0.8.30 4.0 6.94.64.64.47 0.4.85.36 0.74.48 7.50 0.7.9 5.0 6.9.66.6..03.84.8 0.67.48 7.56 0.76.9 8.0 6.5.05.6.77 0.68.83.79 0.4.46 7.83.0.30 0.0 6.40..6.79 0.70.85.00 0.4.47 7.38 0.59.9 (30, r0) (30, r5) (30, r0) (30, r5) 0.5 3.04 0.53.90 3.08 0.06.5 9. 0.33.36 6.63 0.36.6 0.8.6-0..88 3.54 0.46.53 9. 0.3.36 6.93 0.65.7.0.59 0.6.89 3.6 0..5 8.9 0.4.36 6.53 0.7.5.5 3.86..89 3.9 0.6.5 8.87 0.0.35 7.4 0.85.8.0 3.0 0.5.88 3.55 0.48.54 9.7 0.38.36 6.94 0.66.7 3.0 3.06 0.54.9.8-0.7.5 9.6 0.38.36 6.65 0.39.6 4.0.77 0.3.9 3.76 0.66.55 8.88 0.3.36 6.8 0.54.7 5.0.63 0.9.89 3.47 0.40.53 8.73-0.0.35 6.45 0.0.6 8.0 3.8 0.7.9 3.76 0.66.54 9.8 0.40.36 6.65 0.38.6 0.0 3.65.03.9 3.45 0.39.54 8.85 0.09.35 6.76 0.49.6 For the Tpe I cesored case, we take the smple stuato that the cesorg tme s costat across the observatos. Ths s the case that all the testg uts are put o test at the same tme, ad the test termates at tme C. The smulato s ru at several degrees of cesorshp represeted b p 0, the proporto of cesorg. Fewer values are cosdered relatve to the complete or Tpe II cesored case sce the behavor of the estmators s qute stable wth respect to the value. The α value s aga fxed at 00 sce the estmato of s depedet of the α value. The smulato results are summarzed Table. The results show that the modfed MLE aga performs much better tha the regular MLE terms of basess ad effcec, although ot as good as the cases of complete or Tpe II cesored data. The bas of the MLE creases wth p 0, the proporto of cesorg, but the modfed MLE does ot. The relatve effcec of the modfed MLE over the regular MLE creases wth p 0 as well. 8
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. Table. Relatve Bas (%) ad Relatve Effcec of MLE ad Modfed MLE Tpe I Cesored Data. Relatve Bas Relatve Bas p 0 MLE MMLE REF p 0 MLE MMLE REF ( 0) ( 30) 0.0 0.5 6.5.44.3 0.0 0.5 3.85.0.08 0.0.0 6.6.0.3 0.0.0 3.79.4.08 0.0.0 6.36.9.4 0.0.0 3.66.0.08 0.5 0.5 6.49.0.5 0.5 0.5 4.4 0.58.0 0.5.0 6.84.34.6 0.5.0 4.3 0.76.0 0.5.0 7.00.5.6 0.5.0 4.8 0.63.0 0.50 0.5 0.87 0.8.3 0.50 0.5 6.98 0.68.9 0.50.0 0.37 0.35.30 0.50.0 6.93 0.57.9 0.50.0 0.88 0.79.3 0.50.0 7.9 0.95.9 0.75 * * * * 0.65 0.5 0.4 0.3.33 0.75 * * * * 0.65.0 0.67 0.3.35 0.75 * * * * 0.65.0 0.56 0.6.30 ( 00) ( 50) 0.5 0.5.3 0.8.03 0.0 0.5.34 0.78.05 0.5.0.0-0.03.03 0.0.0.36 0.79.05 0.5.0.8 0.4.03 0.0.0.38 0.8.05 0.50 0.5.09 0.3.06 0.5 0.5.4 0.3.06 0.50.0.99 0..05 0.5.0.43 0.34.06 0.50.0.85 0.09.05 0.5.0.76 0.66.07 0.75 0.5 4.3 0.. 0.50 0.5 3.87 0.3. 0.75.0 4. 0.09. 0.50.0 3.60-0.0.0 0.75.0 4.07 0.07. 0.50.0 4.3 0.58. 0.90 0.5.56 0..4 0.75 0.5 9.00 0.7.8 0.90.0.8 0.43.4 0.75.0 8.89 0..9 0.90.0.58 0.8.36 0.75.0 9.45 0.73.5 Note that all the smulatos are carred out usg Fortra 90, where a IMSL subroute UVMID s used for solvg the lkelhood equatos. For the purpose of real data aalss, t ma be more coveet to use Mathematca or Maple to do the job. The Fortra code ad the Mathematca code are avalable from the fst author upo request. 5. CONCLUSIONS The tradtoal MLEs for the Webull parameters are hghl based. Ths s especall so for the shape parameter, whch s a ver mportat parameter relablt decso makg ad plag such as bur- ad replacemet tme determato. Ths problem s bascall caused b the fact that the estmators are hghl correlated. Through parameter orthogoalzato, a ferece procedure s developed ths paper. Specfcall, for the Webull shape parameter, a modfed profle lkelhood uder the oto of parameter orthogoalzato s studed. The modfcato volves ol a smple ad- 9
Appeared : Joural of Statstcal Computato ad Smulato, 003, 73, 5-3. justmet to the lkelhood equato. Compared wth the tradtoal MLE, the modfed MLE performs surprsgl well. The modfed MLE ot ol reduces the bas sgfcatl, but s also more effcet tha both the regular MLE ad the ubased MLE of Ross (994, 996). Fall, t should be metoed that a good estmator for the shape parameter s crucal obtag a good estmator for the scale parameter as the latter s ofte a fucto of the former ad a estmate of the scale parameter ca be computed whe the shape parameter s estmated. Also, whe the shape parameter s estmated, the dstrbuto ca be trasformed to expoetal wth a smple power trasformato ad statstcal fereces ca be carred out easl (Xe, Yag ad Gaudo, 000). REFERENCES D.R. Cox ad N. Red, Parameter orthogoalt ad approxmate codtoal ferece (wth dscusso), J. R. Statst. Soc. B, vol 49, 987, pp -30. D.R. Cox ad N. Red, O the stablt of maxmum-lkelhood estmators of orthogoal parameters, Caada J. Statst., vol 7, 989, pp 9-33. D.R. Cox ad N. Red, A ote o the dfferece betwee profle ad modfed profle lkelhood, Bometrka, vol 79, 99, pp 408-4. A. Drapella ad S. Koszk, A alteratve rule for placemet of emprcal pots o Webull probablt paper, Qualt ad Relab. Eg. It., vol 5, 999, pp 57-59. J.F. Lawless, Statstcal Models ad Methods for Lfetme Data, 98; Joh Wle & Sos, New York. M.S. Macksack ad R.H. Stllma, A cautoar tale about Webull aalss, IEEE Tras. Relablt, vol 45, 996, pp 44-48. M. Pade ad U.S. Sgh, Shruke estmators of Webull shape parameter from Tpe II cesored samples, IEEE Tras. Relablt, vol 4, 993, pp 8-86. R. Ross, Formulas to descrbe the bas ad stadard devato of the ML-estmated Webull shape parameter, IEEE Tras. Delectrcs ad Electrcal Isulato, vol, 994, pp 47-53. R. Ross, Bas ad stadard devato due to Webull parameter estmato for small data sets, IEEE Tras. Delectrcs ad Electrcal Isulato, vol 3, 996, pp 8-4. M. Xe, Z. Yag ad O. Gaudo, More o the ms-specfcato of the shape parameter wth Webull-to-expoetal trasformato, Qualt ad Relab. Eg. It., vol. 6, 000, pp 8-90. 0