Estimation of Parameters of Misclassified Size Biased Borel Distribution

Transcription

1 Joural of Moder Appled Statstal Methods Volume 5 Issue Artle Estmato of Parameters of Mslassfed Sze Based Borel Dstrbuto Bhatda S. Trved H L Isttute of Commere, Ahmedabad Uversty, Guarat, Ida, bhatda.trved@ahdu.edu. M. N. Patel Departmet of Statsts, Guarat Uversty, Ahmedabad, mpatel.stat@gmal.om Follow ths ad addtoal wors at: Part of the Appled Statsts Commos, Soal ad Behavoral Sees Commos, ad the Statstal Theory Commos Reommeded Ctato Trved, Bhatda S. ad Patel, M. N. (6) "Estmato of Parameters of Mslassfed Sze Based Borel Dstrbuto," Joural of Moder Appled Statstal Methods: Vol. 5 : Iss., Artle 9. DOI:.37/masm/4783 Avalable at: Ths Regular Artle s brought to you for free ad ope aess by the Ope Aess Jourals at DgtalCommos@WayeState. It has bee aepted for luso Joural of Moder Appled Statstal Methods by a authorzed edtor of DgtalCommos@WayeState.

2 Joural of Moder Appled Statstal Methods November 6, Vol. 5, No., do:.37/masm/4783 Copyrght 6 JMASM, I. ISSN Estmato of Parameters of Mslassfed Sze Based Borel Dstrbuto B. S. Trved Ahmedabad Uversty Navragpura, Ahmedabad, Ida M. N. Patel Guarat Uversty Ahmedabad, Ida A mslassfed sze-based Borel Dstrbuto (MSBBD), where some of the observatos orrespodg to = + are wrogly reported as = wth probablty α, s defed. Varous estmato methods le the method of mamum lelhood (ML), method of momets, ad the Bayes estmato for the parameters of the MSBB dstrbuto are used. The performae of the estmators are studed usg smulated bas ad smulated rs. Smulato studes are arred out for dfferet values of the parameters ad sample sze. Keywords: Borel dstrbuto, mslassfato, sze based, method of momets, mamum lelhood, Bayes estmato Itroduto The Borel dstrbuto s a dsrete probablty dstrbuto, arsg otets ludg brahg proesses ad queueg theory. If the umber of offsprg that a orgasm has s Posso-dstrbuted, ad f the average umber of offsprg of eah orgasm s o bgger tha, the the desedats of eah dvdual wll ultmately beome ett. The umber of desedats that a dvdual ultmately has that stuato s a radom varable dstrbuted aordg to a Borel dstrbuto. Borel (94) defed a oe parameter Borel dstrbuto as P X p ; e ;,,,3, () B. S. Trved s a Assoate Professor the H. L. Isttute of Commere. Emal her at: bhatda.trved@ahdu.edu.. M. N. Patel s the Departmet of Statsts. Emal them at: mpatel.stat@gmal.om. 475

3 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION Ths dstrbuto desrbes a dstrbuto of the umber of ustomers served before a queue vashes uder odto of a sgle queue wth radom arrval tmes (at ostat rate) of ustomers ad a ostat tme ouped servg eah ustomer. Gupta (974) defed the Modfed Power Seres Dstrbuto (MPSD) wth probablty futo gve by g P X a, T () f where a() >, T s a subset of the set of o-egatve tegers, g(θ) ad f(θ) are postve, fte, ad dfferetable, ad θ s the parameter. Hassa ad Ahmad (9) showed the Borel dstrbuto s a partular ase of modfed power seres dstrbuto (MPSD) wth a, g e, f =e (3) (). The Borel-Taer dstrbuto geeralzes the Borel dstrbuto. Let be a postve teger. If,,, are depedet ad eah has Borel dstrbuto wth parameter θ, the ther sum w = s sad to have the Borel-Taer dstrbuto wth parameters θ ad. Ths gves the dstrbuto of the total umber of dvduals a Posso-Galto-Watso proess startg wth dvduals the frst geerato, or of the tme tae for a M/D/ queue to empty startg wth obs the queue. The ase = s smply the Borel dstrbuto above. Here, the M/D/ queue represets the queue legth a system havg a sgle server, where arrvals are determed by a Posso proess ad ob serve tmes are fed (determst). A eteso of ths model wth more tha oe server s the M/D/ queue. Sze-Based Borel Dstrbuto Sze-based dstrbutos are a speal ase of the more geeral form ow as weghted dstrbutos. Weghted dstrbutos have umerous applatos forestry ad eology. Sze-based dstrbutos were frst trodued by Fsher (934) to model asertamet bas; weghted dstrbutos were later formalzed a ufyg 476

4 TRIVEDI & PATEL theory by Rao (965). Suh dstrbutos arse aturally prate whe observatos from a sample are reorded wth uequal probablty, suh as from probablty proportoal to sze (PPS) desgs. I short, f the radom varable X has dstrbuto f (; θ), wth uow parameter θ, the the orrespodg weghted dstrbuto s of the form w w f ; f ; (4) E w where w() s a o-egatve weght futo suh that E{w()} ests. The sze-based Borel dstrbuto s also derved from the sze-based MPSD as t s a partular ase of the MPSD. A sze-based MPSD s obtaed by tag the weght of MPSD () as, gve by P X b b g f g * f (5) where b = a() ad f * (θ) = μ(θ)f(θ). Now, by tag b a,, g e, f e, (6) a sze-based Borel dstrbuto s obtaed wth p.m.f. gve by P X e,,,3, (7) Mslassfed Sze-Based Borel Dstrbuto A depedet varable whh s a dsrete respose auses the estmated oeffets to be osstet a probt or logt model whe mslassfato s preset. By 477

5 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION 'mslassfato' we mea that the respose s reported or reorded the wrog ategory; for eample, a varable s reorded as a oe whe t should have the value zero. Ths mstae mght easly happe a tervew settg where the respodet msuderstads the questo or the tervewer smply hes the wrog bo. Other data soures where the researher suspets measuremet error, suh as hstoral data, ertaly est as well. It wll be show that, whe a depedet varable s mslassfed a probt or logt settg, the resultg oeffets are based ad osstet. Assume that some of the values ( + ) are erroeously reported as, ad let the probabltes of these observato be α. The the resultg dstrbuto of the sze-based radom varable X s alled the mslassfed sze-based dstrbuto. Trved ad Patel (3) have osdered mslassfed sze-based geeralzed egatve bomal dstrbutos ad parameter estmato. The mslassfed szebased Borel dstrbuto a be obtaed as p P X e e e, e e, e e, S (8) where S s the set of o-egatve tegers eludg tegers ad +, α, < θ <, ad =,, 3,. The mea ad varae of ths dstrbuto are obtaed from the momets of mslassfed sze-based MPSD gve by Hassa ad Ahmad (9) as Mea =μ g e b (9) 478

6 TRIVEDI & PATEL Varae =μ 3 g e g e b b () Method of Mamum Lelhood Estmato Let,,, be the probable values of the radom varable X a radom sample of mslassfed sze-based Borel dstrbuto ad deote the umber of observatos orrespodg to the value the sample (where > ). Thus the lelhood futo L s gve by,, L e e e e e e e P P P P () where N

7 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION l L l e e e e l e e e l, e l l l l l l l l l l l, l l l () Let the dervatve of l L wth respet to α ad θ be zero. The solutos of l l l l ad gves us the ML estmators of α ad θ: e e l l e e e l l (3) (4) 48

8 TRIVEDI & PATEL Equatg l l ad l l to zero, we get e e (5) e (6) I the equato (6), substtutg α from the equato (5), we get a equato osstg oly parameter θ, say g(θ) =. By solvg ths equato for θ usg ay teratve method, we get the soluto, ow as the MLE of θ. Usg ths MLE of θ (5), we get the MLE of α. Asymptot Varae Covarae Matr of ML Estmators The seod order dervatves wth respet to α ad θ of the lelhood futo L are obtaed as below: l L e e e l L e (7) (8) 48

9 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION l L e e e e e e (9) Usg the above equatos, the asymptot varae ovarae matr Σ of MLE s obtaed from the verse of the Fsher formato matr l L l L E E J, l L l L E E () That s v ov, Σ, SE ˆ v, SE ˆ v ˆ () ov, v Method of Momets The mea ad varae of the mslassfed sze-based Borel dstrbuto are Mea =μ g e b () 48

10 TRIVEDI & PATEL Varae μ 3 3 b g e 4 b g e (3) The reurree relato of row momets of the mslassfed sze-based Borel dstrbuto s μ g μ g r μr g r r fμ b μμ r (4) where g(θ), f(θ), μ(θ), ad b are as per (6). By tag dfferet values of r, dfferet row momets are obtaed. Tag r = wll obta the seod row momets of the mslassfed sze-based Borel dstrbuto. g μ μ μ g μ g b fμ (5) Solvg () ad (5) for α ad θ yelds momet estmators of α ad θ. The eplt form aot be obtaed for the momet estmators but, by the method of terato, the soluto for the equatos may be obtaed. Asymptot Varae Covarae Matr of Momet Estmators Deote μ by H (θ, α) ad μ by H (θ, α),.e. H, g e b (6) ad μ g b H, μ g fμ g μ (7) 483

11 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION The, the asymptot varae ovarae matr of momet estmators ad are gve by V A Σ A v ov, ov, v (8) where the matr A s H H a a A H H a a (9) ad vm v m ov m,m ov m,m Σ (3) where μ r μr Vm r, r, μr s μμ r s COV m r,m s, r s, ad m r s the r th sample raw momet of the MSBPL dstrbuto,.e. r mr 484

12 TRIVEDI & PATEL Bayes Estmato The ML method, as well as other lassal approahes, s based oly o the empral formato provded by the data. However, whe there s some tehal owledge o the parameters of the dstrbuto avalable, a Bayes proedure seems to be a attratve feretal method. The Bayes proedure s based o a posteror desty, say π(α, θ ), whh s proportoal to the produt of the lelhood futo L(α, θ ) wth a pror ot desty, say g(α, θ), represetg the uertaty o the parameters values. Assume before the observatos were made owledge about the parameters α ad θ was vague. Cosequetly, the o-formatve vague pror π (α) = g (α) = s applable to a good appromato. The o-formatve prors of α ad θ are Hee, the ot pror of θ ad α s gve by π g (3) π g (3) g, g g g, π π (33) If L s the lelhood futo deed by a otuous parameter Θ = (θ, α) wth pror desty g(θ, α), the the posteror desty for Θ s gve by 485

13 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION 486 L g, π L g, e e e e e e d d d d Θ Θ Θ, e e e e e e e e d, e e d (34)

14 TRIVEDI & PATEL where Usg the result gve by Gradshte ad Ryzh (7, p. 347), e e e π Θ (35), ;,, 3 4 m m m 3 3 Φ ; ; m 3 m 3 m 4 3 where N, N, N (36) From (35), the margal posteror of α wll be 487

15 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION 488,, Φ ; ;, d e e e d,,, Φ ; ; Φ ; ; (37) From (37), the Bayes estmate of α s gve by,,, Φ ; ; Φ ; ˆ ; BS d d

16 TRIVEDI & PATEL 489,,,, Φ ; ; Φ ; ; (38) Smlarly, from (35), the margal posteror of θ wll be,, e Φ ; ;, d e e e d,,, ; ; (39) From (39), the Bayes estmate of θ s gve by

17 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION ˆ d BS, e ; ;,, Φ ; 3;,, Φ ; ;,, d (4) where γ, μ, ad η are as gve (36) above. 49

18 TRIVEDI & PATEL Smulato Study Oe thousad radom samples, eah of sze, were geerated by usg Mote Carlo smulato wth dfferet hoes of sample sze, θ, α, ad value of = from the mslassfed sze-based Borel dstrbuto defed equato (8). Usg these dfferet values of sample sze, θ, ad α, we alulated the smulated rs (SR) ad smulated bas of estmators α ad θ by the method of MLE, method of momets, ad Bayes estmato. The smulated results are show Tables ad. The SR s defed as SR= ˆ Coluso A omparso was made betwee dfferet methods of estmato for the parameters of the mslassfed sze-based Borel dstrbuto. From Table ad, t was foud that the method of mamum lelhood estmator wors better ompared to the momet estmator ad the Bayes estmator o the bass of SR. As sample sze reases, SR of both parameters of all three methods dereases. For fed mslassfato error α, as θ reases, the SR of α ad θ dereases the ase of mamum lelhood estmato, momet estmato method, ad Bayes estmato. For fed values of θ ad sample sze, as α reases, there s ot muh dfferee the SR of α as well as θ. At the same tme, f these values were ompared otet of sample sze, observe that, for a fed value of θ ad as α reases, the SR of α ad θ dereases most of the ases wth the rease sample sze. As sample sze reases, the bas α ad θ dereases the ase of all the three methods. Referees Borel, E. (94). Sur l emplo du theoreme de Beroull pour falter le alul d u fte de oeffets. Applato au probleme de l attete a u guhet. Comptes Redus, Aedem des Sees, Pars, Seres A, 4,

19 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION Fsher, R. A. (934). The effets of methods of asertamets upo the estmato of frequees. Aals of Euges, 6(), 3-5. do:./ tb5. Gradshte, I. S., & Ryzh, I. M. (7). Tables of tegrals, seres ad produts (7th ed.). Oford: Aadem Press. Gupta, R. C. (974). Modfed power seres dstrbuto ad some of ts applatos. Sahyā: The Ida Joural of Statsts, Seres B, 36(3), Avalable from Hassa, A., & Ahmad, P. B. (9). Mslassfato sze-based modfed power seres dstrbuto ad ts applatos. Joural of the Korea Soety for Idustral ad Appled Mathemats South Korea, 3(), Retreved from Rao, C. R. (965). O dsrete dstrbutos arsg out of asertamet. I G. P. Patl (Ed.), Classal ad otagous dsrete dstrbutos (pp. 3-33). Calutta, Ida: Pergamo Press ad Statstal Publshg Soety. Trved, B. S., & Patel, M. N. (3). Estmato mslassfed sze-based geeralzed egatve bomal dstrbuto. Mathemats ad Statsts, (), do:.389/ms.3. 49

20 TRIVEDI & PATEL Apped A Table. Smulated rs of ML, momet, ad Bayes estmators for dfferet values of α, θ, ad sample sze ML Momet Bayes θ α SR(θ) SR(α) SR(θ) SR(α) SR(θ) SR(α)

21 MISCLASSIFIED SIZE-BIASED BOREL DISTRIBUTION Table. Smulated Bas of ML, Momet ad Bayes estmators for dfferet values of α, θ, ad sample sze ML Momet Bayes θ α Bas(θ) Bas(α) Bas(θ) Bas(α) Bas(θ) Bas(α)