Keywords- Weighted distributions, Transmuted distribution, Weibull distribution, Maximum likelihood method.

Transcription

1 Volum 7, Issu 3, Marh 27 ISSN: X Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig Rsarh Papr Availabl oli at: O Siz-Biasd Wightd Trasmutd Wibull Distributio Moa Abdlghafour Mobarak, Zohdy Nofal, Mrvat Mahdy Dpartmt of Statistis, Mathmatis ad Isura, Collg of Commr, Bha Uivrsity, Egypt DOI:.23956/ijarss/V7I3/32 Abstrat- This papr offrs a w wightd distributio alld siz biasd wightd trasmutd wibull distributio, dotd by (SBWTWD). Various usful statistial proprtis of this distributio ar drivd i this papr suh as, th umulativ distributio futio, Rliability futio, hazard rat, rvrsd hazard rat ad th rth momt. Plots for th probability dsity futio at diffrt valus of shap paramtrs ar providd. Th maimum liklihood stimators of th ukow paramtrs of th proposd distributio ar obtaid. O data st has b aalyzd for illustrativ purposs. Kywords- Wightd distributios, Trasmutd distributio, Wibull distributio, Maimum liklihood mthod. I. INTRODUCTION Addig a tra paramtr to a istig family of distributio futios is vry ommo i th statistial distributio thory. Oft itroduig a tra paramtr brigs mor flibility to a lass of distributio futios ad it a b vry usful for data aalysis purposs. Espially th wibull distributio ad its gralizatios i th litratur attrat th most of th rsarhrs du to its wid rag appliatios. Th Wibull distributio iluds th potial ad th Rayligh distributios as sub modls, th usfulss ad appliatios of paramtri distributios iludig Wibull, Rayligh ar s i various aras iludig rliability, rwal thory, ad brahig prosss whih a b s i paprs by may authors suh as i {[6], [7], [25]}. Diffrt gralizatios of th Wibull distributio ar ommo i th litratur as i {[4], [5], [2], [22], [28], [38]} ad aothr gralizatio of th wibull distributio usig th opt of wightd distributios is availabl as i {[6], [8], [9], [24], [3], [34], [36], [37]}. Th us ad appliatio of wightd distributios i rsarh rlatd to rliability, bio-mdii, ology ad svral othr aras ar of trmdous pratial importa i mathmatis, probability ad statistis. Ths distributios aris aturally as a rsult of obsrvatios gratd from a stohasti pross ad rordd with som wight futio. Th opt of ths distributios has b mployd i wid varity appliatios i may filds of ral lif suh as mdii, rliability, ad survival aalysis, aalysis of family data, ology ad forstry. It a b trad to th work of Fishr [4] i otio with his studis o how mthod of asrtaimt a iflu th form of distributio of rordd obsrvatios. Azzalii [] was first to itrodu a shap paramtr to a ormal distributio dpdig o a wight futio whih is alld th skw-ormal distributio. Diffrt works o itroduig shap paramtrs for othr symmtri distributios ar availabl i th litratur, svral proprtis ad thir ifr produrs ar disussd by svral authors s for ampl i {[2], [3]}. O th othr sid, Rtly svral authors itrodud shap paramtrs for osymmtri distributios as b show i {[7], [9], [], [2], [3], [5],[8], [23], [26], [29], [32], [33], [35]}. I this papr w ostrut th siz biasd wightd trasmutd wibull distributio ad th sub-modls whih ar th spial ass of our proposd distributio. Various usful statistial proprtis of this modl ar drivd i th t stios. W also prst a umrial ampl of th proposd distributio osidrig th ral lif data-st for illustrativ purposs. This papr is orgaizd as follows. Stio 2 dfis som basi matrials ad i Stio 3, w provid th drivatio of PDF of th proposd modl ad som partiular ass ar obtaid i Stio 4. Stio 5 disusss th diffrt statistial proprtis of this modl. Estimatio of th ukow paramtrs of th proposd modl by maimum liklihood mthod is arrid out i Stio 6. Th ral data-st has b aalyzd i Stios 7 ad stio 8 givs som brif olusio. II. MATERIALS AND METHODS Wightd distributios opt a b trad from th study of Fishr ad Rao. Lt X b a o-gativ radom variabl with its probability dsity futio (pdf), f, th th pdf of th wightd radom variabl X w is giv by: f w w. f =, < E w X <, > () E w X whr, f is th pdf of th bas distributio ad th wight futio w is a o- gativ futio, that may dpd o th paramtr. Wh th wight futio dpds o th lgth of uits of itrst, w =, th rsultig distributio is alld lgth-biasd whih fids various appliatios i biomdial aras suh as arly dttio 27, IJARCSSE All Rights Rsrvd Pag 37

2 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp of a disas. Rao [27] also usd this distributio i th study of huma familis ad wild-lif populatios. I this as th pdf of a lgth-biasd radom variabl is dfid as: f LB. f =, >, < E X <. E X Mor grally, wh th samplig mhaism slts uits with probability proportioal to som masur of th uit siz, wh w =, >, th th rsultig distributio is alld siz-biasd ad th pdf of a siz-biasd radom variabl is dfid as: f SB =. f E X, < E X <, >. This typ of samplig is a gralizatio of lgth- biasd samplig. I this papr w us this wight futio,w =, osidrig th trasmutd wibull distributio as basli distributio to gt a w wightd distributio. Aordig to th Quadrati Rak Trasmutatio Map (QRTM) approah proposd by Shaw ad Bukly [3] a radom variabl X is said to hav trasmutd probability distributio if its df, F T ad pdf, f T ar giv by: F T = + α F αf 2, α, ad, f T = f + α 2αF, whr, F, f, ar th df, pdf of th bas distributio, rsptivly ad α is th trasmutd, shap paramtr. Th, th df ad th pdf of th trasmutd wibull distributio (TWD) ar giv as follow: F T = λ + α λ, ad f T = λ λ α + 2α λ, (2) whr, λ >, > ar th sal, shap paramtrs rsptivly, th pdf, f, ad th df, F, of th wibull distributio tak th forms as follow: f = λ λ, λ >, >, >, ad F = λ. Th distributio i quatio (2) iluds spially th trasmutd potial ad trasmutd Rayligh distributios as spial ass whr = ad = 2, rsptivly. III. DERIVATION OF THE SIZE BIASED WEIGHTED TRANSMUTED WEIBULL DISTRIBUTION I this stio, w driv th probability dsity futio of siz biasd wightd trasmutd wibull distributio. Th plot of pdf of this distributio at various hois of shap paramtrs valus a also b show i this stio. W a gt th pdf of siz biasd wightd trasmutd wibull distributio as follows: Wh, w =. (3) Substitutig (3) ad (2) i() th w gt: H, E X = f SBWTWD, λ, α,, = λ + + λ Γ + α + α 2 λ λ α + 2α Γ + α + α 2 Th dsity futio (4) a b kow as siz biasd wightd trasmutd wibull distributio, dotd by SBWTWD. Figurs (), [(2-a), (2-b)] ad (3) rprst th possibl shaps of probability dsity futio of th SBWTWD at diffrt valus of shap paramtrs, α ad, rsptivly wh th sal paramtr, λ =... (4) Figur(). Th plot of pdf of SBWTWD for diffrt valus of shap paramtr,. Figur(2-a). Th plot of pdf of SBWTWD for diffrt valus of shap paramtr, α , =α== 2, =α== 3, =α== , α=2==.2, α=2==.5, α=2== 27, IJARCSSE All Rights Rsrvd Pag 38

3 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp Figur(2-b). Th plot of pdf of SBWTWD for diffrt valus of shap paramtrα wh =, =2. Figur(3) Th plot of pdf of SBWTWD for diffrt valus of shap paramtr, , α=2, ==, α=2, ==.2.5, α=2, == , =.5, α=3=, =.5, α=3= 2, =.5, α=3= 3 IV. SOME PARTICULAR CASES OF SBWTWD This stio prsts som sub-modls that ddud from Equatio (4) ar: Cas. Puttig =, th rsultig distributio is lgth biasd wightd trasmutd wibull distributio (LBWTWD)giv as: f, λ, α, = λ + 2 λ λ α + 2α, >, λ >, >, α. Γ α + α 2 Cas2. Puttig =, =, th rsultig distributio is lgth biasd wightd trasmutd potial distributio (LBWTED)giv as: f ; λ, α = 2 2α λ2 λ α + 2α λ, >, λ >, α. Cas3. Puttig α =, th rsultig distributio is sizd biasd wightd wibull distributio (SBWWD)giv as: f ; λ,, = λ + + λ Γ +, >, λ >, >, >. as: Cas4. Puttig α =, th rsultig distributio is sizd biasd wightd wibull distributio (SBWWD)giv f, λ,, = 2λ + + 2λ Γ +, >, λ >, >, >. Cas5. Puttig α =, =, =, th rsultig distributio is lgth biasd wightd potial distributio (LBWED)giv as: f, λ = 2λ 2 2λ, >, λ >. Cas6. Puttig α =, = 2, =, th rsultig distributio is lgth biasd wightd Rayligh distributio (LBWRD)giv as: f, λ = 25 2 λ λ 2, >, λ >. Γ 3 2 Cas7. Puttig α =, =, th rsultig distributio is lgth biasd wightd wibull distributio (LBWWD)giv as: f, λ, = 2λ + 2λ Γ +, >, λ >, >. Cas8. Puttig α =, =, =, th rsultig distributio is lgth biasd wightd potial distributio (LBWWD)giv as: f ; λ = λ 2 λ, >, λ >. Cas9. Puttig =, th rsultig distributio is trasmutd wibull distributio (TWD)giv as: f ; λ, α, = λ λ α + 2α λ, >, λ >, >, α. Cas. Puttig =, α =, th rsultig distributio is wibull distributio (WD)giv as: f ; λ, = λ λ, >, λ >, >. Cas. Puttig =, =, th rsultig distributio is trasmutd potial distributio (TED)giv as: Cas2. Puttig =, = 2, th rsultig distributio is lgth biasd wightd trasmutd Rayligh distributio (LBWTRD)giv as: 27, IJARCSSE All Rights Rsrvd Pag 39

4 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp f, λ, α = 2λ3 2 2 λ 2 α + 2α λ 2, >, λ >, α. Γ 3 2 α + α 2 2 Cas3. Puttig =, = 2, th rsultig distributio is trasmutd Rayligh distributio (TRD)giv as: f ; λ, α = 2λ λ 2 α + 2α λ 2, >, λ >, α. Cas4. Puttig =, = 2, α =, th rsultig distributio is Rayligh distributio (RD)giv as: f ; λ = 2λ λ 2, >, λ >. Cas5. Puttig α =, =, = 2 ad multiplyig by (), this modl givs th ivrs Rayligh distributio (IRD)giv as: f, λ = 2λ 3 λ 2, >, λ >. Cas6. Puttig α =, =, = 2 ad multiplyig by (), this modl givs th ivrs Rayligh distributio (IRD)giv as: f, λ = 2 2λ 3 2λ 2, >, λ >. Cas7. Puttig α =, =, th rsultig distributio is lgth biasd wightd wibull distributio (LBWWD)giv as: f, λ, = λ + λ Γ +, >, λ >, >. Cas8. Puttig =, = 2, α =, th rsultig distributio is lgth biasd wightd Rayligh distributio (LBWRD)giv as: f, λ = 2λ3 2 2 λ 2, >, λ >. Γ 3 2 Cas9. Puttig =, α =, th rsultig distributio is wibull distributio (WD)giv as: f, λ, = 2λ 2λ, >, λ >, >. Cas2. Puttig =, =, α =, th rsultig distributio is potial distributio (ED)giv as: f, λ = 2λ 2λ, >, λ >. Cas2. Puttig =, = 2, α =, th rsultig distributio is Rayligh distributio (RD)giv as: f, λ = 2 2λ 2λ 2, >, λ >. Cas22. Puttig =, =, α =, th rsultig distributio is potial distributio (ED)giv as: f, λ = λ λ, >, λ >. Cas23. Puttig =, th rsultig distributio is siz biasd wightd trasmutd potial distributio (SBWTED) giv as: f, λ, α, = λ+ λ α + 2α λ Γ + α + α 2, >, λ >, >, α Cas24. Puttig = 2, th rsultig distributio is siz biasd wightd trasmutd Rayligh distributio (SBWTRD) giv as: f, λ, α, = 2λ λ 2 α + 2α λ 2, >, λ >, >, α Γ + α + α V. THE STATISTICAL PROPERTIES OF SBWTWD I this stio, w prst som basi statistial proprtis of SBWTWD iludig, th umulativ distributio futio (CDF), rliability futio, hazard futio ad th rvrs hazard futio, rt momt, th ma, varia ad ordr statistis as follow: i. Th CDF of SBWTWD is dfid as: Thrfor, Th CDF of SBWTWD is giv as: F SBWTWD = f SBWTWD t dt. F SBWTWD, λ, α,, = 27, IJARCSSE All Rights Rsrvd Pag 32 γ +, Γ +, whr, γ +, is th lowr iomplt gamma futio dfid as: γ s, = t s t dt, >.

5 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp Th, γ +, = t + t dt, >. ii. Th rliability futio of SBWTWD is dfid as: R SBWTWD, λ, α,, = F SBWTWD, λ, α,,. Thrfor, th rliability futio of SBWTWD is giv as: γ +, R SBWTWD, λ, α,, = Γ +. iii. Th hazard futio is mathmatially giv by: f H = F. Thrfor, th prssio for th hazard futio of th SBWTWD is dfid by: H SBWTWD, λ, α,, = λ + + α + α 2 λ λ α + 2α Γ + γ +,, > iv. Th rvrsd hazard rat futio is mathmatially rprstd by: f H = F. Thrfor, th prssio for Rvrsd hazard rat of SBWTWD is giv as: H SBWTWD, λ, α,, = λ + + λ λ α + 2α γ +, α + α 2 v. Th rth momt of th radom variabl X w follows SBWTWD is giv as: M r SBWTWD, λ, α,, = Γ r+ + Γ + α + α 2 λ r α + α r+ 2 vi. Th ma of th radom variabl X w follows SBWTWD is giv as: Γ + + SBWTWD M, λ, α,, = Γ + α + α λ 2 vii. Th varia is mathmatially dfid as: σ 2 = M 2 M 2. Th, th varia of th radom variabl X w follows SBWTWD is giv as: σ 2SBWTWD = Γ +2 + α + α Γ + α + α λ 2 Γ + + Γ + α + α 2, >, r =,2,3,.. α + α viii. Th mod is th valu of th radom variabl whih maks th pdf is a maimum. Takig logarithm of th pdf of SBWTWD as: log f SBWTWD, λ, α,, λ 2 + α + α. 2 + = + log λ + log + + log λ + log α + 2α λ log Γ + 2. log α + α 2. log f SBWTWD, λ, α,, + = λ 2αλ λ α + 2αλ. (5) Th mod of th SBWTWD is obtaid by solvig th quatio (5)with rspt to. + λ 2αλ λ =. (6) α + 2α λ 27, IJARCSSE All Rights Rsrvd Pag 32

6 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp By solvig th oliar quatio (6), a b alulatd th mod of th SBWTWD. i. Th ordr statistis hav grat importa i lif tstig ad rliability aalysis. Lt X, X 2,, X b radom variabls ad its ordrd valus is dotd as, 2,,. Th pdf of ordr statistis is obtaid usig th blow futio:! f s:, = s! s! f F s F s. (7) To obtai th smallst valu i radom sampl of siz put s = i (7), th th pdf of smallst ordr statistis is giv by f :, = f F. Thrfor, th pdf of smallst ordr statistis for th SBWTWD is: f :, = λ + + λ λ α + 2α, λ, >, >. Γ + α + α 2 γ +, Γ + To obtai th largst valu i radom sampl of siz put s = i (7), th th pdf of ordr statistis is giv by: f :, = f F. Thrfor, th pdf of largst ordr statistis for th SBWTWD is: f :, = λ + + λ λ α + 2α γ Γ + α + α +,, > 2 VI. MAXIMUM LIKELIHOOD ESTIMATION OF THE SBWTWD Lt, 2,, b a idpdt radom sampl from th SBWTWD, th th liklihood futio, L ; λ,, α,, of SBWTWD is giv by: L ; λ,, α, = Substitutig from (4)ito (8), w hav, f SBWTWD, λ, α,, λ + L ; λ,, α, = + i λ α + α Γ + 2 So, logarithm liklihood futio log L ; λ,, α,, is giv as: log L ; λ,, α, = log λ + log λ log L ; λ,, α, λ + log log α + α 2 + log α + 2α λ i. (8) i log Γ log. (9) Diffrtiatig (9) with rspt to λ,, α,ad, rsptivly, as follows: = λ + λ i (2α i ) λ i log L ; λ,, α, = 2 log λ λ i l i α l α + α 2 2αλ λ i whr, ψ + is th digamma futio. α + 2α λ i. i α + 2α λ, () i i l i α + 2α λ i + 2 ψ + + log i, () λ i log L ; λ,, α, α = 2 α + α 2 ( 2 λ i ) α + 2α λ i, (2) 27, IJARCSSE All Rights Rsrvd Pag 322

7 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp log L ; λ,, α, = log α l 2 λ + 2 α + α ψ log i. (3) Sttig th quatios(), (), (2)ad (3)qual to zro, w hav th followig quatios: (2α i ) λ i λ + λ α + 2α λ i =, (4) i 27, IJARCSSE All Rights Rsrvd Pag α + α 2 ( 2 λ i ) α + 2α λ i =, (6) log α l 2 λ + 2 α + α 2 + log i =. (7) ψ + W a gt MLEs of th ukow paramtrs by solvig th quatios(4),(5), (6)ad (7)to stimat th paramtrs λ,, α ad usig umrial thiqu mthods suh as wto Raphso mthod baus it is ot possibl to solv ths quatios aalytially. By takig th sod partial drivativs of (), (), (2)ad (3) th Fishr s iformatio matri a b obtaid by takig th gativ ptatios of th sod partial drivativs. Th ivrs of th Fishr s iformatio matri is th varia ovaria matri of th maimum liklihood stimators. VII. APPLICATION I this stio, w provid a appliatio of th proposd distributio to show th importa of th w modl. Th data st (gaug lgths of mm) from Kudu ad Raqab [2]. This data st osists of, 63 obsrvatios:.9, 2.32, 2.23, 2.228, 2.257, 2.35, 2.36, 2.396, 2.397, 2.445, 2.454, 2.474, 2.58, 2.522, 2.525, 2.532, 2.575, 2.64, 2.66, 2.68, 2.624, 2.659, 2.675, 2.738, 2.74, 2.856, 2.97, 2.928, 2.937, 2.937, 2.977, 2.996, 3.3, 3.25, 3.39, 3.45, 3.22, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.48, 3.435, 3.493, 3.5, 3.537, 3.554, 3.562, 3.628, 3.852, 3.87, 3.886, 3.97, 4.24, 4.27, 4.225, 4.395, 5.2. This data st is prviously studid by Afify t al [] to fit th trasmutd wibull loma distributio. W fit both trasmutd wibull (TW) ad siz biasd wightd trasmutd wibull (SBWTW) distributios to th subjt data. W also stimat th paramtrs λ, α, ad usig Nwto-Raphso mthod by takig th iitial stimats λ =.5, =.5, α =.99 ad =.99 ad th stimatd valus of th paramtrs a b show i tabl. To s whih o of ths modls is mor appropriat to fit th data st, w alulat Akaik Iformatio Critrio (AIC), th Cosistt Akaik Iformatio Critrio (CAIC) ad Baysia Iformatio Critrio (BIC). Th bst distributio orrspods to lowr for 2 log-liklihood, AIC, BIC, ad CAIC statistis valus, whr, AIC = 2l + 2k, CAIC = 2l + 2k k, ad, BIC = 2l + k l, whr l dots th log-liklihood futio valuatd at th maimum liklihood stimats, k is th umbr of paramtrs ad is th sampl siz. Ths umrial rsults ar obtaid usig th MATH- CAD PROGRAM.

8 Mobarak t al., Itratioal Joural of Advad Rsarh i Computr Si ad Softwar Egirig 7(3), Marh- 27, pp Tabl I Th Estimatd Valus of th Paramtrs Paramtrs stimats valus Modl λ α TW SBWTW Tabl I shows th stimatd valus of th paramtrs for th (TWD) ad SBWTWD. Tabl II Th Statistis 2l, AIC, BIC ad CAIC for Gaug Lgths of MM Data St. Modls 2l AIC BIC CAIC TW SBWTW Tabl II shows th valus of 2l, AIC, BIC ad CAICstatistis. W ot that th SBWTW modl givs th lowst valus for 2l, AIC, BIC ad CAIC statistis so SBWTWD lads to a bttr fit to ths data tha TWD. VIII. CONCLUSION I this papr w propos a w four-paramtr modl, alld siz biasd wightd trasmutd wibull distributio whih is a gralizatio of trasmutd wibull distributio. W prst som of its statistial proprtis. Th w distributio is vry flibl modl that approahs to diffrt lif tim distributios wh its paramtrs ar hagd. W disuss maimum liklihood stimatio. W osidr Akaik Iformatio Critrio (AIC), th Cosistt Akaik Iformatio Critrio (CAIC) ad Baysia Iformatio Critrio (BIC) statistis to ompar th modl with trasmutd wibull modl. A appliatio of th siz biasd wightd trasmutd wibull distributio to ral data shows that th proposd distributio a b usd quit fftivly to provid bttr fits tha th trasmutd-wibull distributio. REFERENCES [] Azzalii, A. (985). A lass of distributios whih iluds th ormal os, Sadiavia Joural of Statistis,2, [2] Azzalii, A. ad Dalla Vall, A. (996). Th multivariat skw-ormal distributio. Biomtrika, 83, [3] Arold, B. C. ad Bvr, R. J. (2). Th skw Cauhy distributio. Statistis & Probability Lttrs, 49, [4] Al-Salh, J. A. ad Agarwal, S. K. (26). Etdd Wibull typ distributio ad fiit mitur of distributios. Statistial Mthodology, 3, [5] Aryal, G. R ad Toskos, C. P. (2). Trasmutd Wibull Distributio: A Gralizatio of th Wibull. Europa Joural of Pur ad Applid Mathmatis, 4, [6] Alm, M., Sufya, M. ad Kha, N. S. (23). A lass of modifid wightd Wibull distributio ad its proprtis. Amria Rviw of Mathmatis ad Statistis,, [7] Al-Kadim, K. ad Hatoosh, A. F. (23). Doubl wightd distributio & doubl wightd potial distributio. Mathmatial Thory ad Modlig, 3, [8] Al-Kadim, K. A. ad Hatoosh, A. F. (24). Doubl wightd ivrs Wibull Distributio. Pakista Publishig Group, [9] Al-Kadim, A. K. ad Hussi, A. N. (24). Nw proposd lgth-biasd wightd Epotial ad Rayligh distributio with appliatio. Mathmatial Thory ad Modlig, 4, [] Ahmad, A., Ahmad, S. P. ad Ahmad, A. (24). Charatrizatio ad stimatio of doubl wightd Rayligh distributio. Joural of Agriultur ad Lif Sis,, [] Afify, A. Z., Nofal, Z. M., Yousof, H. M., El-Gbaly, Y. M. ad Butt, N. S. (25). Th trasmutd wibull loma distributio: Proprtis ad Appliatio. Pak. J. Stat. Opr. Rs.,, [2] Bashir, S. ad Rasul, M. (25). Som proprtis of th Wightd Lidly distributio. Itratioal Joural of Eoomi ad Busiss Rviw, 3, [3] Das, K. K. ad Roy, T. D. (2) Appliability of lgth biasd wightd gralizd Raylig distributio. Advas i Applid Si Rsarh, 2, [4] Fishr, R. A. (934).Th fft of mthods of asrtaimt upo th stimatio of frquis. Th Aals of Eugis, 6, [5] Fathizadh, M. (25). A w lass of wightd Lidly distributios. Joural of Mathmatial Etsio, 9, [6] Gupta, R. C. ad Katig, J. P. (985). Rlatios for rliability masurs udr lgth biasd samplig. Sadaavia Joural of Statistis, 3, [7] Gupta, R. C. ad Kirmai, S. N. U. A. (99). Th rol of wightd distributios i stohasti modlig. Commuiatios i Statistis Thory ad Mthods, 9, [8] Gupta, R. D. ad Kudu, D. A. (29). A w lass of wightd potial distributios. Statistis, 43, , IJARCSSE All Rights Rsrvd Pag 324

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