Best Linear Unbiased Estimators of the Three Parameter Gamma Distribution using doubly Type-II censoring

Transcription

1 Best Lea Ubased Estmatos of the hee Paamete Gamma Dstbuto usg doubly ype-ii cesog Amal S. Hassa Salwa Abd El-Aty Abstact Recetly ode statstcs ad the momets have assumed cosdeable teest may applcatos volvg data elatg to lfe testg. I ths atcle, the sgle ad double momets of ode statstcs fom the thee paamete gamma dstbuto ae studed. he best lea ubased estmatos (BLUE s) of the locato ad scale paametes based o the momets of ode statstcs fo doubly type II cesoed samples ae obtaed. I patcula, the BLUE s based o type II ght cesoed ad complete samples ae obtaed as specal cases. Futhemoe, the BLUE s ae used to costuct the cofdece tevals fo the locato ad scale paametes. I addto, the samplg dstbuto of the pvotal quattes ae obtaed fo dffeet sample szes ad cesog types usg Peaso's system techque. Mote Calo smulato s pefomed to obta these estmatos ad the samplg dstbuto umecally usg Mathcad statstcal package. Key wods: Best Lea Ubased Estmato; Ode Statstcs; Momets; Gamma Dstbuto; Doubly ype- II Cesoed Samples; Appoxmate Cofdece Itevals; Pvotal Quattes; Samplg Dstbuto ;Mote Calo Smulato.. Itoducto Gamma dstbuto s the most popula model fo aalyzg lfetme data, so t has wde applcatos the aea of lfe testg ad elablty. It has qute a bt of flexblty fo aalyzg ay postve eal data set. She ad Ba (98) have metoed that the gamma dstbuto has eceved cosdeable atteto the aea of weathe aalyss. he gamma dstbuto has a log hstoy ad t has seveal desable popetes. Johso et al. (994) have peseted some popetes of the thee-paamete gamma dstbuto. It s a geealzato of the ch-squae ad expoetal dstbutos.. Assocate Pofesso of Statstcs, Depatmet of Mathematcal Statstcs, Isttute of Statstcal Studes& Reseach, Cao Uvesty, Oma, Gza, Egypt.. Assstat Pofesso of Statstcs, Depatmet of Statstcs, Faculty of Commece, Al-Azha Uvesty, Gls' Bach-Cao.

2 by he pobablty desty fucto of the thee-paamete gamma dstbuto s gve ( x ) x f ( x;,, ) exp( ( )),,, ( ) x (.) whee, ad ae the shape, locato ad scale paametes, espectvely, [see Johso et al (994)]. he coespodg dstbuto fucto s gve by x ( ) F( x;,, ), x,,, ( ) (.) whee x ( ) ( z ) e dz. x z he stadad fom of the thee paamete gamma pobablty desty fucto (.) ad dstbuto fucto (.) ae obtaed by settg ad equatos (.) ad (.) as follows: f ( x; ) ( x) exp( x), x,, (.) ( ) ad ( ) F ( x; ) x. (.4) ( ) I may pactcal applcatos, such as lfe testg, t s qute commo ot to obseve complete data but oly obseve some foms of cesoed data. hs may be based o cost o tme cosdeatos. he doubly type II cesoed samples ase f the fst ad the last few falue obsevatos ae ukow. I patcula, f the last few falue obsevatos of a gve sample ae ukow, the cesoed sample s efeed to as type II ght cesoed samples. Gupta (96) has obtaed the fst fou momets of the ode statstcs fom gamma dstbuto fo sample szes () ad shape paamete () 5. Pescott (974) has computed the vaaces ad covaace of the gamma ode statstcs fo () ad ad 5. Mehota ad Nada (974) have obtaed appoxmate maxmum lkelhood estmatos fo the odeed sample fom omal ad gamma dstbutos. I addto, they

3 obtaed the BLUE s ad made compaso betwee the two methods to study the effceces betwee them. Fo a ovevew of the applcatos fo BLUE s, Mahmoud et al. () have deved the exact expessos fo the sgle momets of ode statstcs fom the vese Webull dstbuto based o type II ght cesoed sample. he vaace ad covaace matx of odeed sample ae also calculated. hey used these momets to obta the BLUE s of the locato ad scale paametes of the vese Webull dstbuto. Futhemoe, they appled the esults to daw feece of ths dstbuto. Mahmoud et al. (5) have deved the exact explct expessos fo the sgle, double, tple, ad quaduple momets of ode statstcs fom the geealzed Paeto dstbuto. Also, they obtaed the BLUE s of the paametes of ths model. Appoxmate cofdece tevals of the paametes fom the geealzed Paeto dstbuto ae obtaed usg Edgewoth appoxmato ad compag them wth those based o Mote Calo smulato. Saeevkuma ad homas (5) have obtaed the BLUE s of the scale ad locato paametes fo the omal ad double expoetal dstbutos. hs atcle focuses o the momets of ode statstcs fom the thee paamete gamma dstbuto. hese momets wll be used to estmate the locato ad scale paametes based o doubly ad ght type II cesoed samples. Mote Calo smulato s pefomed to obta the pot ad teval estmates of the paametes usg Mathcad statstcal package. hs atcle s ogazed as follows, the sgle ad double momets of ode statstcs fom the thee paamete gamma dstbuto ae dscussed Secto (). I Secto (), BLUE's of the paametes ae calculated usg the momets of ode statstcs. I Secto (4), the cofdece tevals of the paametes ae costucted though the pecetage pots of some pvotal quattes; addto the samplg dstbuto of these pvotal quattes s obtaed. A umecal study s caed out Secto (5). ables fo the umecal study ae dsplayed at the appedx.. Momets of Ode Statstcs Suppose that : :... : ae the ode statstcs coespodg to depedet adom vaables each havg stadad gamma dstbuto (.), ad the

4 pobablty desty fucto of the th ode statstcs : s gve by: f ( x ) (, - ) ( ) :,..., x x e ( x ( )) ad x ( ( )) :,, (.) whee (.,.) stads fo beta fucto. he kth momets of the th ode statstcs deoted by k k k k : ( ) ( ( )) k :, s gve by : E g ( k), (.) (, - ) whee, k- - x - g ( k) x e ( ( )) dx,,,... x Let : ad s: be ay two ode statstcs fom the set :... : coespodg to the depedet adom vaables havg the : stadad gamma dstbuto (.), the the ot desty of : ad s: f ( x, y) - x y x y e (,s - ) (s, - s ) s l s s ( ) ( ( )) ( s l l ls ( ) ( y ( )) ( ( )) l x y ( )) s s gve by x y, (.) he ot momet of : ad s:, s deoted by, s: by s k, k k k k k x y s, s: E :, Ys : x y e ( (,s - ) (s, - s ) x s s s l l ls ( ) ( ( )) ( ( )) y l l x y dydx k, k s gve ) (.4) I vew of equatos (.) ad (.4), t s dffcult to deve exact foms fo the sgle ad ot momets fo the thee paamete gamma dstbuto. So, umecal techque s appled to obta these momets. I addto, the vaace ad covaace matx of ode statstcs ae computed.. Best Lea Ubased Estmato hs Secto dscusses the BLUE s of the scale ad locato paametes of the thee paamete gamma pobablty desty fucto (.), based o doubly type II cesoed 4

5 5 samples. he BLUE's of the paametes ca be detemed usg sgle ad double momets of ode statstcs peseted Secto () Suppose that... : : : s be the avalable doubly type II cesoed samples fom the poposed desty fucto (.) ad let s x,...,, ), ( z (.) be the coespodg ode statstcs fo the stadad gamma dstbuto. Let us deote Z E by, Z va by, ad Z ov, Z c by, ; futhe let s ),..., (, s ),..., (, s ) (,,..., ad, s, ), (, Accodg to Balaksha ad Cohe (99), the BLUE s of the locato paamete ad, the scale paamete ae gve by s x a, (.) ad s x b. (.) I addto, the vaace ad covaace of ad ae gve by ) Va( V, (.4), ) ( V Va (.5)

6 ad Cov(, ) V. (.6) Fo moe detals, efe to Lloyd (95), Davd (98), Balaksha ad Cohe (99), ad Aold et al. (99). Usg equatos (.) ad (.) the coeffcets of BLUE's based o doubly ad ght type II cesoed samples ae calculated. he coeffcets of BLUE's based o doubly type-ii cesoed samples ae dsplayed ables () ad (), whle ables () ad (4) epeset the coeffcets of BLUE's based o type-ii ght cesoed samples. o check the accuacy of the coeffcets ables () to (4), the followg codtos must be vefed s s a ad b. (.7) Accodg to equatos (.4), (.5) ad (.6), the vaace, covaace ad mea squae eos (MSEs) of the estmatos based o doubly ad ght type II cesoed samples ae calculated ad dsplayed ables (7) ad (8). 4. Appoxmate Cofdece Itevals Accodg to Balaksha ad Cohe (99) the cofdece tevals fo the paametes fo the poposed dstbuto wee gve though the pvotal quattes R, V R ad V R (4.) V whee, ad ae the BLUE's of ad wth vaaces V ad V, espectvely as show equatos (.4),(.5). R ca be used to daw feece fo whe s kow, whle R ca be used to daw feece fo whe s ukow. Also, R ca be used to daw feece fo. he cofdece tevals fo ae costucted whe s kow though the fomula P V ( R ) V ( ) ) (4.) ( R whe s ukow, the cofdece tevals fo s gve by P V ( R ) V ( ) ) (4.) ( R whee V s gve equato (.4). 6

7 ad the cofdece tevals fo s costucted though the fomula P (4.4) V ( R ) V ( R ) whee V s gve equato (.5). Sce the samplg dstbutos fo the pvotal quattes R, R ad R ae ukow, so the samplg dstbuto of these pvotal quattes ae deved, usg Peaso's system appoach. he Peaso's system appoach s based o computg the followg cteo fo fxg the dstbuto famly 4 6 K (4.5) 4 whee the two momet atos ad 4 deote the skewess ad kutoss measues, espectvely ad s the th cetal momet. So fo dffeet values of K, thee exst dffeet types of dstbutos [see Eldeto ad Johso (969)]. he samplg dstbuto of the pvotal quattes R, R ad R fo dffeet sample szes usg Peaso's system techque ae llustated able (9). It s clea fom able (9) that the samplg dstbutos of the pvotal quattes R, R ad R follow Peaso's type I, VI, III o IV dstbutos fo some sample szes. It s foud that most of the samplg dstbutos of the pvotal quattes R ad R ae close to the omal dstbuto fo dffeet cesoed samples whe the sample szes =, Numecal Illustato It s obvous that, the sgle ad double momets of ode statstcs fom the thee paamete gamma dstbuto has o exact foms. So the sgle ad double tegato volved the computatos of sgle ad double momets wee pefomed usg Mathcad statstcal package. he method of BLUE's s appled to estmate the paametes of ths model. he accuacy of the estmates of the paametes s studed though the popetes of the estmates. he pefomaces of the obtaed estmatos ae vestgated tems of the mea squae eos (MSEs). he pecetage pots of the estmates based o the pvotal quattes R, R ad R ae used to costuct cofdece tevals wth cofdece 7

8 level at.99. I addto, the samplg dstbuto of the pvotal quattes s also detemed. he smulato pocedues ae descbed though the followg steps: Step (): Let : :... s: be the avalable doubly type II cesoed samples of szes 5(5) 5 fom the thee paamete gamma dstbuto wth shape paamete. Step (): Compute the sgle ad double momets fom the doubly type II cesoed samples wth, ad s,. Step (): As a specal case, the sgle ad double momets fom the type II ght cesoed wll be obtaed wth ad s,,,4. Step (4): Compute the vaace covaace matx usg steps () ad () ad the compute the vese of vaace covaace matx. Step (5): Geeate odeed samples of sze fom the stadad gamma dstbuto, accodg to Mathcad package "the commat pgamma ( x,) s used to geeate adom samples fom stadad gamma dstbuto". Step (6): Fom equatos (.) ad (.), the estmates of the scale paamete ad, locato paamete ae calculated ude doubly ad ght type II cesoed samples. Step (7): Compute the pvotal quattes R, R ad R usg equato (4.). Step (8): Costuct the cofdece tevals of the paametes at Cofdece level 99 %. Step (9): Repeat the steps fom (5) to (9) fo specfed 5, tmes. Step (): he esultg 5, pvotal quattes R, R ad R wll be used to calculate the fst fou cetal momet fo dffeet sample szes ad cesog types. Step (): Use the cetal momet of the pvotal quattes R, R ad R to compute ad. Step (): Use ad to calculate K usg equato (4.5). Numecal esults ae summazed ables () to (9). ables () to (4) epeset the coeffcets of BLUE s based o doubly ad ght type II cesoed samples. he vaaces, covaaces ad the MSE of the estmatos ae appeaed ables (5) ad (6). ables (7) ad (8) cota the appoxmate cofdece tevals fo the estmatos. 8

9 Fally the samplg dstbutos of R, R ad R ude each level of cesog ad fo dffeet sample szes ae dsplayed able (9). Fom these ables, we have: - It s clea fom ables () to (4) that, fo dffeet sample szes ad cesoed levels, the coeffcets of the BLUE s of the paametes based o doubly ad ght type II cesoed sample vefy the codtos (.7). - Ude doubly ad ght type II cesoed samples, the estmates of the paametes appoach the tue values. he covaaces of the estmates, Cov (, ), decease as the double cesog level ceases ad as the sample sze deceases. he MSE of estmatos deceases as the sample sze ceases, whle ceases as the cesog levels cease [see ables (5) ad (6)]. - Regadg to the cofdece tevals, as the cesog levels cease, the aveage wdth of the cofdece tevals cease. Also, as the sample sze ceases the aveage wdth of the cofdece tevals deceases fo the two metoed cases fo the cesoed samples [see ables (7) ad (8)]. 4- he samplg dstbutos of the pvotal quattes fo dffeet sample szes ad cesoed samples take the followg foms: a- Fo small sample szes (=5,) ad dffeet cesoed samples, the samplg dstbuto of the pvotal quattes R, R ad R follow Peaso's type IV, III ad I espectvely. b- Fo modeate sample szes (=5,) the samplg dstbuto of the pvotal quattes R, R ad R follows Peaso's type VI, IV, III o I fo most dffeet cesoed samples. Whle the case of doubly cesoed samples the samplg dstbuto of the pvotal quattes R has a omal dstbuto fo =. c- Fo =5 the samplg dstbuto of the pvotal quattes R ad R follows omal dstbuto fo most dffeet cesoed samples, whle a few cases has Peaso's type VI, IV, III o I. he samplg dstbuto of the pvotal quattes R follows Peaso's type IV, I fo dffeet cesoed samples. d. It s clea fom able (9) that the the samplg dstbutos of the pvotal quattes R ad R ae close to the omal dstbuto fo dffeet cesoed samples at sample szes =, 5. 9

10 Appedx able(): Coeffcets of the BLUE s fo the Locato Paamete Based o the Doubly ype II Cesoed Samples wth s a Note that ad s stad fo complete samples

11 able(): Coeffcets of the BLUE s fo the Scale Paamete Based o the Doubly ype II Cesoed Samples wth. s b

12 able(): Coeffcets of the BLUE s fo the Locato Paamete Based o the ype II Rght Cesoed Samples wth. s a able(4): Coeffcets of the BLUE s fo the scale Paamete Based o the ype II Rght Cesoed Samples wth s b

13 Cotued able 4 s b able (5): BLUE s of ad, Vaaces, Covaaces ad MSEs Based o the Doubly ype II Cesoed Samples wth s, MSE ( ) va va cov MSE able (6): BLUE s of ad, Vaaces, Covaaces ad MSEs Based o the ype II Rght Cesoed Samples wth s, MSE ( ) va va cov MSE

14 able(7): Cofdece Itevals the case of Doubly ype II cesoed sample fo ad, Based o the Pecetage Pot of R, R ad R wth Cofdece level 99 % wth. s ( R ) R ) R ( ) 5 (-.9,.6) (-.54,.67) (.79,.7) (-.787,.98) (-.495,.4) (.,.987) (-.648,.757) (-.948,.5) (.46,.7) (.9,.84) (-.4,.454) (.4,.79) (-.7,.99) (-.54,.) (.547,.4) (-.676,.89) (-.4,.4) (.9,.69) (-.65,.966) (-.45,.4) (.96,.79) 5 (-.6,.) (-.6,.67) (.47,.) (-.6,.) (-.45,.69) (.5,.667) (-.656,.76) (-.68,.64) (.5,.64) (-.56,.775) (-.98,.7) (.99,.58) (-.56,.787) (-.9,.87) (.,.69) (-.5,.57) (-.,.85) (.4,.785) (-.569,.948) (-.44,.47) (.,.75) (-.54,.59) (.9,.4) (.5,.485) (-.55,.44) (-.98,.7) (.99,.58) 5 (-.447,.45) (-.4,.47) (.5,.64) (-.476,.558) (-.97,.7) (.7,.495) (-.58,.557) (-.49,.77) (.8,.84) able(8): Cofdece Itevals the case of ype II Rght cesoed sample fo ad, Based o the Pecetage Pot of R, R ad R wth Cofdece level 99 % wth. s ( R ) R ) R ( ) 5 (-.4,.87) (-5.7,.9) (.59, 6.8) (-.5,.6) (-.59,.9) (.87,.7) (-.687,.) (-.75,.67) (.5,.9) (-.45,.658) (.7,.55) (.8,.644) 5 (-.69,.669) (-.,.97) (.45,.958) (-.65,.74) (-.78,.96) (.5,.854) (-.49,.9) (-.4,.97) (.6,.67) (-.9,.799) (-.,.54) (.8,.779) (-.44,.55) (-.7,.8) (.7,.844) (-.447,.89) (-.6,.74) (.8,.5) (-.9,.799) (-.76,.88) (.8,.644) 5 (-.7,.88) (-.7,.55) (.44,.6) 4 (-.9,.899) (-.4,.655) (.7,.974) 4

15 able(9): Samplg Dstbuto of R, R ad R fo Complete, Rght ype II ad Doubly Cesoed Samples Accodg to Peaso's type of Dstbutos. ypes of cesoed s Peaso's type of R ) ( Peaso's type of R ) ( Peaso's type of R ( ) 5 Complete IV III I Rght IV III I Complete IV III I Rght IV III I Rght IV I I Doubly IV III I Doubly IV III I Doubly VII III I Complete VI VI I Rght IV VI I Rght VI VI I 5 Rght IV III I Doubly IV VI I Doubly IV III I Doubly IV VI I Complete VI IV I Rght III VI I Rght III III I Rght IV III I Doubly IV VI I Doubly IV VI I Doubly Nomal Dst. VI I Doubly Nomal Dst. VI I Complete Nomal Dst. III I Rght Nomal Dst. Nomal Dst. IV Rght Nomal Dst. Nomal Dst. IV Rght Nomal Dst. Nomal Dst. IV 5 Rght 4 Nomal Dst. Nomal Dst. IV Doubly Nomal Dst. Nomal Dst. IV Doubly IV I I Doubly IV VI I Doubly VI I I 5

16 Refeeces - Aold, B. C., Balaksha, N. ad Nagaaa, H. N. (99), A Fst Couse Ode Statstcs, New Yok, Joh Wley & Sos. - Balaksha, N. ad Cohe, A. C. (99), Ode Statstcs ad Ifeece: Estmato Methods, Sa Dego, Academc Pess. - Davd, H. A. (98), Ode Statstcs, d ed. New Yok, Joh Wley & Sos. 4- Eldeto, W.P. ad Johso, N.L. (969), Systems of Fequecy Cuves, Cambdge Uvesty Pess. 5- Gupta, S. S. (96), Ode statstcs fom the gamma dstbuto, echometcs, Vol., Johso, N.L., Kotz, S., ad Balaksha, N. (994), Cotuous Uvaate Dstbutos, d ed, Vol.. New Yok, Joh Wley& Sos. 7- Lloyd, E. H. (95), Least-squaes estmatos of locato ad scale paametes usg ode statstcs, Bometka, Vol.9, Mahmoud, M. A. W.,Sulta, K. S. ad Ame, S. M. (), Ode statstcs fom veese webull dstbuto ad assocated feece, Computatoal Statstcs & Data Aalyss, Vol. 4, Mahmoud, M. A. W.,Sulta, K. S. ad Moshef, M. E. (5), Ifeece based o ode statstcs fom the geealzed paeto dstbuto ad applcato, Commucatos Statstcs-Smulato ad Computato, Vol.4, Mehota, K. G. ad Nada, P. (974), Ubased estmato of paametes by ode statstcs the case of cesoed samples, Bometka, Vol.6, Pescott, P. (974), Vaaces ad covaaces of ode statstcs fom the gamma dstbuo, Bometka, Vol.6, Saeevkuma, N. K. ad homas, P. Y. (5), Applcato of ode statstcs of depedet odetcally dstbuted adom vaables estmato, Commucatos Statstcs-heoy ad Methods, Vol.4, She, W. K. ad Ba, L. J. (98), A two-sample test of equal gamma dstbuto scale paametes wth ukow commo shape paamete, echometcs, Vol.5,

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