On Type-II Progressively Hybrid Censoring

Size: px

Start display at page:

Download "On Type-II Progressively Hybrid Censoring"

Lucy Mosley
5 years ago
Views:

oural of Moder Appled Statstcal Methods Volue 8 Issue Artcle 8 --009 O Type-II Progressvely Hybrd Cesorg Debass Kudu Ida Isttute of Techology Kapur Ida kudu@tk.ac.

edu/jas Part of the Appled Statstcs Coos Socal ad Behavoral Sceces Coos ad the Statstcal Theory Coos Recoeded Ctato Kudu Debass; oarder Avjt; ad Krsha Hare (009 "O Type-II Progressvely Hybrd Cesorg"

1 oural of Moder Appled Statstcal Methods Volue 8 Issue Artcle O Type-II Progressvely Hybrd Cesorg Debass Kudu Ida Isttute of Techology Kapur Ida kudu@tk.ac. Avjt oarder Reserve Bak of Ida Muba ajoarder@rb.org. Hare Krsha C.C.S. Uversty Meerut Ida hkrshastats@yahoo.co Follow ths ad addtoal works at: Part of the Appled Statstcs Coos Socal ad Behavoral Sceces Coos ad the Statstcal Theory Coos Recoeded Ctato Kudu Debass; oarder Avjt; ad Krsha Hare (009 "O Type-II Progressvely Hybrd Cesorg" oural of Moder Appled Statstcal Methods: Vol. 8 : Iss. Artcle 8. DOI: 0.37/jas/ Avalable at: Ths Regular Artcle s brought to you for free ad ope access by the Ope Access ourals at DgtalCoos@WayeState. It has bee accepted for cluso oural of Moder Appled Statstcal Methods by a authorzed edtor of DgtalCoos@WayeState.

2 oural of Moder Appled Statstcal Methods Copyrght 009 MASM Ic. Noveber 009 Vol. 8 No /09/$95.00 O Type-II Progressvely Hybrd Cesorg Debass Kudu Avjt oarder Hare Krsha Ida Isttute of Techology Reserve Bak of Ida C.C.S. Uversty Kapur Ida Muba Ida Meerut Ida The progressve Type-II cesorg schee has becoe qute popular. A drawback of a progressve cesorg schee s that the legth of the experet ca be very large f the tes are hghly relable. Recetly Kudu ad oarder (006 troduced the Type-II progressvely hybrd cesored schee ad aalyzed the data assug that the lfetes of the tes are expoetally dstrbuted. Ths artcle presets the aalyss of Type-II progressvely hybrd cesored data whe the lfete dstrbutos of the tes follow Webull dstrbutos. Maxu lkelhood estators ad approxate axu lkelhood estators are developed for estatg the ukow paraeters. Asyptotc cofdece tervals based o axu lkelhood estators ad approxate axu lkelhood estators are proposed. Dfferet ethods are copared usg Mote Carlo sulatos ad oe real data set s aalyzed. Key words: Maxu lkelhood estators; approxate axu lkelhood estators; Type-I cesorg; Type-II cesorg; Mote Carlo sulato. Itroducto The Type-II progressve cesorg schee has becoe very popular. It ca be descrbed as follows: cosder uts a study ad suppose < s fxed before the experet addto other tegers R.. R are also fxed so that R R + =. At the te of the frst falure for exaple Y : : R of the reag uts are radoly reoved. Slarly at the te of the secod falure for exaple Y : : R of the reag uts are radoly reoved ad so o. Fally at the te of the th falure the reag Y Debass Kudu s a Professor of the Departet of Matheatcs ad Statstcs Kapur IIT. E- al: kudu@tk.ac.. Avjt oarder s a Assstat Advser of Departet of Statstcs ad Iforato Maageet RBI. Please ote that the vews ths artcle are A. oarder s persoal vews ad ot those of the RBI. E-al: ajoarder@rb.org.. Hare Krsha s the Head of the Departet of Statstcs CCS Uversty. E- al: hkrshastats@yahoo.co. R uts are reoved. Extesve work has bee coducted o ths partcular schee durg the last te years; see Balakrsha ad Aggarwala (000 ad Balakrsha (007. Ufortuately the ajor proble wth the Type-II progressve cesorg schee s that the te legth of the experet ca be very large. Due to ths proble Kudu ad oarder (006 troduced a ew cesorg schee aed Type-II Progressvely Hybrd Cesorg whch esures that the legth of the experet caot exceed a pre-specfed te pot T. The detaled descrpto ad advatages of the Type-II progressvely hybrd cesorg s preseted Kudu ad oarder (006 (see also Chlds Chadrasekar & Balakrsha 007; both publcatos the authors assued the lfete dstrbutos of the tes to be expoetal. Because the expoetal dstrbuto has ltatos ths artcle cosders the Type-II progressvely hybrd cesored lfete data whe the lfete follows a two-paraeter Webull dstrbuto. Maxu lkelhood estators (MLEs of the ukow paraeters are provded ad t was observed that the MLEs caot be obtaed explct fors. MLEs ca 534

3 KUNDU OARDER & KRISHNA be obtaed by solvg a o-lear equato ad a sple teratve schee s proposed to solve the o-lear equato. Approxate axu lkelhood estators (AMLEs whch have explct expressos are also suggested. It s ot possble to copute the exact dstrbutos of the MLEs so the asyptotc dstrbuto s used to costruct cofdece tervals. Mote Carlo sulatos are used to copare dfferet ethods ad oe data aalyss s perfored for llustratve purposes. Type-II Progressvely Hybrd Cesorg Schee Models If t s assued that the lfete rado varable Y has a Webull dstrbuto wth shape ad scale paraeters ad respectvely the the probablty desty fucto (PDF of Y s y y fy ( y; γ = e ; y> 0 ( where > 0 > 0 are the atural paraeter space. If the rado varable Y has the desty fucto ( the X = l Y has the extree value dstrbuto wth the PDF x e x σ σ fx ( x; σ = e ; < x< ( σ where = l σ =. The desty fucto as descrbed by ( s kow as the desty fucto of a extree value dstrbuto wth locato ad scale paraeters ad σ respectvely. Models ( ad ( are equvalet odels the sese that the procedure developed uder oe odel ca be easly used for the other odel. Although they are equvalet odels ( ca be the easer wth whch to work copared to odel ( because odel ( the two paraeters ad σ appear as locato ad scale paraeters. For = 0 ad σ = odel ( s kow as the stadard extree value dstrbuto ad has the followg PDF z ( z e f z;0 = e ; < z<. (3 Z ( Type-II Progressvely Hybrd Cesorg Schee Data Uder the Type-II progressvely hybrd cesorg schee t s assued that detcal tes are put o a test ad the lfete dstrbutos of the tes are deoted by Y.. Y. The teger < s pre-fxed R.. R are pre-fxed tegers satsfyg R R + = ad T s a pre-fxed te pot. At the te of the frst falure Y : : R of the reag uts are radoly reoved. Slarly at the te of the secod falure Y : : R of the reag uts are reoved ad so o. If the th falure Y occurs before te T the experet stops at te pot Y. If however the -th falure does ot occur before te pot T ad oly falures occur before T (where 0 < the at te T all reag R uts are reoved ad the experet terates. Note that R = ( R R. The two cases are deoted as Case I ad Case II respectvely ad ths s called the cesorg schee as the Type-II progressvely hybrd cesorg schee (Kudu ad oarder 006. I the presece of the Type-II progressvely hybrd cesorg schee oe of the followg s observed Case I: { Y: :... Y }; f Y < T (4 or Case II: { Y: :... Y }; f Y : : < T < Y + : :. (5 For Case II although Y + : : s ot observed but Y : : < T < Y + : : eas that the th falure took place before T ad o falure took place betwee Y : : ad T (.e. Y + : :.. Y are ot observed. The covetoal Type-I progressve cesorg schee eeds the pre-specfcato of R.. R ad also T... T (see Cohe for detals. The choces of T.. T are ot 535

4 ON TYPE-II PROGRESSIVELY HYBRID CENSORING trval. For the covetoal Type-II progressve cesorg schee the experetal te s ubouded. I the proposed cesorg schee the choce of T depeds o how uch axu experetal te the expereter ca afford to cotue ad also the experetal te s bouded. Maxu Lkelhood Estators (MLEs Based o the observed data the lkelhood fucto for Case I s ( + R y = l( = K e (6 ad for Case II the MLE s l( = K e T y y T ( + R R y + f > 0 (7 = e f = 0 (8 where ad K = R + k= ( K = + R k ( k k= both are costat. The logarth of (6 ad (7 ca be wrtte wthout the costat ters as L d = ( d ( l l + ( l y d l (. Here d = W : : d = W( = ( + R y : : (9 ad for Case-I ad Case-II respectvely. It s assued that d > 0 otherwse the MLEs do ot exst. Takg dervatves wth respect to ad of (9 ad equatg the to zero results W( = ( + R y + R T : : L( d = + W ( = 0 (0 + ( L d = + d ly dl V ( + W( l 0. : : = ( Here V( = ( + R y : : l y : : ad ( ( + R y l y + R T lt V = : : : : for Case-I ad Case-II respectvely. Note that ad the MLE of where h ( W ( = = u( ( say d ca be obtaed by solvg ( = h( (3 d = d = l y + u ( W (. A sple teratve schee s proposed to obta the MLE of fro (3. Startg wth a tal guess of for exaple (0 obta ( (0 = h( ad proceed ths way to obta ( + ( = h(. The teratve procedures stops ( + ( whe < whch s soe preassged tolerace lt. Oce the MLE of s obtaed the MLE of ca be obtaed fro (. Sce the MLE s whe they exst are ot copact fors the followg approxate MLE s ad ts explct expressos are proposed. Approxate Maxu Lkelhood Estators (AMLEs Usg the followg otatos x = l y ad S = l T the lkelhood equato of the observed data x for Case-I s 536

5 KUNDU OARDER & KRISHNA R ( ( ( ( ( l σ = RK g z G z σ + k= (4 ad for Case II s ( ( ( ( ( ( ( l σ = R R RK g z G z G V σ + k= (5 where z = x σ V = ( S σ ( x ( e x e x = G( x e e g x = = l ad σ =. Igorg the costat ter the followg loglkelhood results fro (5 s ( L( σ = l l( σ = lσ+ l( g( z :: + R l G( z ::. (6 Fro (6 the followg approxate MLE s of ad σ are obtaed (see Appedx ( c c ˆ σ + d = c where B B 4AC σ = + A (7 c De = c D e = d = D X e = d3 = D X e A = c = c ( d3 + X d( c C = d cd = G ( p = l ( l q p = ( + d DX e B + q = ad D = + R for =. p For Case-II gorg the costat ter the log-lkelhood s obtaed as L( σ = l l( σ = lσ+ l( g( z + R l( G( z + R l G( V. (8 I ths case the approxate MLE s are (see Appedx ( c ' ' ' c σ + d ' ' ' ' B B 4AC = σ = + ' ' c A (9 where ' c = = De + Re ' c = De + Re ' d = = DX e + RSe ' d = = DX e + R S e ' 3 d = = DX e + R Se A ' c ' B ' = c ' d3 ' + X d ' c ' + = ( ( ' ' ' C = d cd. Here ad D are the sae as above for =.. = G ( p = l( l q p = ( p + p + / ad q = p. Results Because the perforace of the dfferet ethods caot be copared theoretcally Mote Carlo sulatos are used to copare the perforaces of the dfferet ethods proposed for dfferet paraeter values ad for dfferet saplg schees. The ter dfferet saplg schees ea dfferet sets of R ' s ad dfferet T values. The perforaces of the MLEs ad AMLEs estators of the ukow paraeters are copared ters of ther bases ad ea squared errors (MSEs for dfferet cesorg schees. The average legths of the asyptotc cofdece tervals ad ther coverage percetages are also copared. All coputatos were perfored usg a Petu IV processor ad a FORTRAN- 77 progra. I all cases the rado devate geerator RAN was used as proposed Press et al. (99. Because s the scale paraeter all cases = have bee take wthout loss of geeralty. For sulato purposes the results are preseted whe T s of the for T. The reaso for choosg T that for s as follows: f ˆ represets the MLE or AMLE of the the dstrbuto of ˆ becoes depedet of the case for =. For that purpose the result s reported oly for = wthout loss of geeralty however these results ca be used for ay other also. 537

6 ON TYPE-II PROGRESSIVELY HYBRID CENSORING Type-II progressvely hybrd cesored data s geerated for a gve set R... R ad T by usg the followg trasforato for expoetal dstrbuto as suggested Balakrsha ad Aggarwala (000. Z = E : : Z = ( R ( E E : : : : Z = ( R... R + ( E E : : (0 It s kow that f E ' s are..d. stadard expoetal the the spacg Z ' s are also..d. stadard expoetal rado varables. Fro (0 t follows that E: : = Z E = Z + Z : : R E = Z Z. R... R + ( Usg ( ad paraeters ad Type-II progressvely hybrd cesored data for the Webull dstrbuto ca be geerated for a gve R.. R Y: :... Y. If Y < T the Case I results ad the correspodg saple s {( Y: : R...( Y R }. If Y > T the Case II results ad s foud such that Y : : < T < Y + : :. The correspodg Type-II hybrd cesored saple s {( Y: : R...( Y R } ad R where R s sae as defed before. Cosder dfferet ad T. Two dfferet saplg schees have bee used aely Schee : R =... = = 0 ad R =. R Schee : R =... = R = ad R = +. Note that Schee s the covetoal Type-II cesorg schee ad Schee s a typcal progressve cesorg schee. I each case the MLEs ad AMLEs are coputed as estates of the ukow paraeters. The 95% asyptotc cofdece tervals are calculated based o MLEs by replacg the MLEs by AMLEs. The process was replcated 000 tes. Average estates MSEs ad average cofdece legths wth coverage percetages were reported Tables -8. Based o Tables -4 (for MLEs ad Tables 5-8 (for AMLEs the followg observatos are ade: As expected for fxed as creases the bases ad the MSEs decrease for both ad however for fxed as creases ths ay ot be true. Ths shows that the effectve saple sze ( plays a portat role whe cosderg the actual saple sze (. It s also observed that the MLEs for schees ad behave qute slarly ters of bases ad MSEs uless both ad are sall. The perforaces ters of bases ad MSEs prove as T creases. Slar results are also observed for AMLEs. Coparg dfferet cofdece tervals ters of average legths ad coverage probabltes t s geerally observed that both the ethods work well eve for sall ad. For both ethods t s observed that the average cofdece legths decrease as creases for fxed or vce versa. For both the MLE ad AMLE ethods schee ad schee behave very slarly although the cofdece tervals for schee ted to be slghtly shorter tha schee. Data Aalyss Kudu ad oarder (006 aalyzed the followg two data sets obtaed fro Lawless (98 usg expoetal dstrbutos. Data Set I ths case = 36 ad f = 0 T = 600 R = R =... = R9 = R0 = 8 the the Type II progressvely hybrd cesored 538

7 KUNDU OARDER & KRISHNA saple s: Fro the above saple data D= = 0 s obtaed whch yelds ad of based o MLEs ad AMLEs are ˆ ( ˆ = = ( = = respectvely. Usg the above estates the 95% asyptotc cofdece terval for ad s obtaed based o MLEs ad AMLEs whch are ( ( ad ( ( respectvely. Data Set Cosder = 0 T = 000 ad R ' s are sae as Data Set. I ths case the progressvely hybrd cesored saple obtaed as: ad D= = 7. The MLE ad AMLEs of ad are ( ˆ = ˆ = ad ( = = respectvely. Fro the above estates the 95% asyptotc cofdece tervals are obtaed for ad based o MLEs ad AMLEs whch are ( ( ad ( ( respectvely. I both cases t s clear that f the tested hypothess s H : = 0 t wll be rejected ths ples that ths case the Webull dstrbuto should be used rather tha expoetal. Cocluso Ths artcle dscussed the Type-II progressvely hybrd cesored data for the two paraeters Webull dstrbuto. It was observed that the axu lkelhood estator of the shape paraeter could be obtaed by usg a teratve procedure. The proposed approxate axu lkelhood estators of the shape ad scale paraeters could be obtaed explct fors. Although the exact cofdece tervals could ot be costructed t was observed that the asyptotc cofdece tervals work reasoably well for MLEs. Although the frequetest approach was used Bayes estates ad credble tervals ca also be obtaed uder sutable prors alog the sae le as Kudu (007. Refereces Arold B. C. & Balakrsha N. (989. Relatos bouds ad approxatos for order statstcs lecture otes statstcs (53. New York: Sprger Verlag. Balakrsha N. (007. Progressve cesorg: A apprsal (wth dscussos. I press. To appear TEST. Balakrsha N. & Aggrwala R. (000. Progressve cesorg: Theory ethods ad applcatos. Bosto MA: Brkhauser. Balakrsha N. & Varada. (99. Approxate MLEs for the locato ad scale paraeters of the extree value dstrbuto wth cesorg. IEEE Trasactos o Relablty Chlds A. Chadrasekhar B. & Balakrsha N. (007. Exact lkelhood ferece for a expoetal paraeter uder progressvely hybrd schees: Statstcal odels ad ethods for boedcal ad techcal systes. F. Vota M. Nkul N. Los & C. Huber-Carod (Eds.. Bosto MA: Brkhauser. Cohe A. C. (963. Progressvely cesored saples lfe testg. Techoetrcs Cohe A. C. (966. Lfe-testg ad early falure. Techoetrcs Davd H. A. (98. Order Statstcs ( d Ed.. New York: oh Wley ad Sos. Kudu D. & oarder A. (006. Aalyss of type II progressvely hybrd cesored data. Coputatoal Statstcs ad Data Aalyss Kudu D. (007. O hybrd cesored Webull dstrbuto. oural of Statstcal Plag ad Iferece

8 ON TYPE-II PROGRESSIVELY HYBRID CENSORING Table : MLE Estate for T = 0.75 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( (94.9.0( (9.3.43( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (93..06( ( ( ( ( ( ( ( ( ( ( (96. 08( ( ( ( ( (93.0 Table : MLE Estate for T =.00 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (9..046( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (94..00( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

9 KUNDU OARDER & KRISHNA Table 3: MLE Estate for T =.50 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (94.7 Table 4: MLE Estate for T =.00 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (94. 54

10 ON TYPE-II PROGRESSIVELY HYBRID CENSORING Table 5: Approxate MLE Estate for T = 0.75 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (9..095( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (93.9 Table 6: Approxate MLE Estate for T =.00 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

11 KUNDU OARDER & KRISHNA Table 7: Approxate MLE Estate for T =.50 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (9..093( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (9.6 Table 8: Approxate MLE Estate for T =.00 N.M. Schee Schee ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (9..093( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

12 ON TYPE-II PROGRESSIVELY HYBRID CENSORING Lawless. F. (98. Statstcal odels ad ethods for lfete data. New York: Wley. Ma N. R. (97. Best lear varat estato for Webull paraeters uder progressve cesorg. Techoetrcs Press W. H. Flaery B. P. Teukolsky S. A. & Vetterlg W. T. (99. Nuercal recpes: The art of scetfc coputg. Cabrdge U.K.: Cabrdge Uversty Press. Thoas D. R. & Wlso W. M. (97. Lear order statstcs estato for the two-paraeter Webull ad extree value dstrbuto fro Type-II progressvely cesored saples. Techoetrcs Appedx For case-i takg dervatves wth respect to ad σ of L( σ as defed (6 results L( σ gz ( g ( z = R 0 σ = Gz = ( = gz ( ( ' L( σ g( z g ( z = Rz : : z: : = 0. σ = G( z = g( z (3 Clearly ( ad (3 do ot have explct aalytcal solutos. Cosder a frst-order Taylor approxato to g ( z / g( z ad gz ( / Gz ( by expadg aroud the actual ea of the stadardzed order statstc Z where = G ( p = l( l q ad p = ( + q = p for =... slar to Balakrsha ad Varada (99 Davd (98 or Arold ad Balakrsha (989. Otherwse the ecessary procedures for obtag =.. were ade avalable by Ma (97 ad Thoas ad Wlso (97. Note that for =... ' ( g( z ( G( z g z:: βz (4 g z + β z (5 :: where g ( g"( g ( = g( g ( g ( = + l q( l( l q g"( g ( β = + = l q g ( g( Usg the approxato (4 ad (5 ( ad (3 results get De De σ DX : : e De ad (7 The above two equatos (6 ad (7 ca be wrtte as where + = 0 (6 De σ + : : DX : : e : : De DX : : e De D X e + X σ + DX e D e + σ = 0 ( c c σ + d c = 0 Aσ Bσ C + + = 0 (8 (9 = = = : : d DX : : e c De c D e d D X e = = 3 : : 3 d = D X e A= c B = c ( d + X d ( c + C = d cd 544

13 KUNDU OARDER & KRISHNA ad D = + R for =... The soluto to the precedg equatos yelds the approxate MLE s are ( c c σ + d = (30 c B + B 4AC σ = (3 A Cosder oly postve root of σ ; these approxate estators are equvalet but ot ubased. Ufortuately t s ot possble to copute the exact bas of ad theoretcally because of tractablty ecoutered fdg the expectato of B 4AC. Appedx For case-ii takg dervatves wth respect to ad σ of L( σ as defed (8 gves (slar to Case-I ' L( σ g( z ( ( g z g V = R + = 0 ( R σ G z g( z : : G ( V (3 L ( σ g( z : : = R z : σ G ( z '( z : : ( z ( V ( V : z : : + RV = : : g : : G g g 0. (33 Here aga cosder the frst-order Taylor approxato to g ( z / g( z ad gz ( / Gz ( by expadg aroud the actual ea of the stadardzed order statstc Z where ' s are defed Appedx. Here gv ( / GV ( s also exploded the Taylor seres aroud the pot where ( ( = G p = l l q p = ( p + p + ad q = p. Note that g ( V βv (34 gv ( where ( ( ( ( ( ( ' " ' g g g = q ' g = + g g β ( ( ( ( (35 ( ( q l l l " ' g g = + = l q ' g g Usg the approxato (4 (5 (34 ad (35 (3 ad (33 gves De + R e De + R e σ D X : : e + R Se De + Re = 0 (36 ad De + Re σ + De + R e D X : D X : : e + R Se gv ( + βv GV ( D e + R e : + σ + e + R Se D X : : + X σ e + R Se De + Re DX : : e + RS e = 0. (37 The above two equatos (36 ad (37 ca be wrtte as ' ' ' ' ( c c σ + d c = 0 (38 A σ + B σ + C = 0 where ' c = De + Re ' = + c D e R e ' : : = + d D X e R Se ' : : = + d D X e R S e ' 3 : : d = D X e + RSe ' ' A c B ' = c ' d3 ' + X d ' c ' + = ( ( ' ' ' C = d cd ad D = + R for =. (39 545

14 ON TYPE-II PROGRESSIVELY HYBRID CENSORING The soluto to the precedg equatos yelds the approxate MLE s are ( c c σ + d = (40 ' ' ' ' c ' B + B 4AC σ = (4 A σ Cosder oly postve root of ; these approxate estators are equvalet but ot ubased. Ufortuately t s ot possble to copute the exact bas of ad theoretcally because of tractablty ecoutered fdg ' the expectato of B 4AC. 546

A Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions

A Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions Appled Matheatcs, 1, 4, 8-88 http://d.do.org/1.4/a.1.448 Publshed Ole Aprl 1 (http://www.scrp.org/joural/a) A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple