A new Family of Distributions Using the pdf of the. rth Order Statistic from Independent Non- Identically Distributed Random Variables

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1 Iteratoal Joural of Cotemporary Mathematcal Sceces Vol. 07 o HIKARI Ltd A ew Famly of Dstrbutos Usg the pdf of the rth Order Statstc from Idepedet No- Idetcally Dstrbuted Radom Varables Z. A. AL-Saary Departmet of Statstcs Faculty of Scece AL-Fasalah Kg Abdulazz Uversty Jeddah Saud Araba Copyrght 07 Z. A. AL-Saary. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted. Abstract The paper ams to propose a ew famly of dstrbutos usg the probablty desty fucto of the frst depedet ad o-detcally order statstcs (OS) whe sample sze =. Two dstrbutos from ths famly are deduced usg famous basele dstrbutos. We study some mathematcal propertes of these dstrbutos. Ther desty fuctos (pdf) ad cumulatve dstrbuto fuctos (cdf) are obtaed. Some useful characterzatos of these dstrbutos are also proposed. The method of mamum lkelhood s used to estmate the model parameters. We derve eplct epressos for the momet geeratg fucto (MGF) Graphs of pdf cdf ad hazard fucto (HF) are foud. Fally we geerate data usg (Mathematca Program Verso ) to kow the best dstrbuto usg Kolmogorov-Smrov Test. Keywords: No-detcally order statstcs Epoetal dstrbuto; Gamma dstrbuto; Ldley dstrbuto; Hazard fucto; Mamum lkelhood; Momets; Momet geeratg fucto Kolmogorov-Smrov Test. Itroducto Last recetly years may attempts have bee made to propose ew famles of dstrbutos the statstcal lterature by the use of some basele dstrbutos. Marshall ad Olk [] troduced the Marshall-Olk famly Eugee et al. [5]

2 9 Z. A. AL-Saary proposed the beta famly Kumaraswamy famly s foud by Cordero ad Castro [7]) ad the McDoald famly by Aleader et al. [8] Other well-kow famles are the epoetal half-logstc famly by Cordero et al. [0] the Zografos- Balakrsha-G famly of dstrbutos [] the geeralzed Cauchy famly by Ayma et al [] ad the geeralzed odd log-logstc famly of dstrbutos by Cordero et al. []. Eugee et al. [5] geerate a famly of beta-geerated dstrbutos used the beta dstrbuto. The cumulatve dstrbuto fucto (cdf) of a beta-geerated radom varable X s deoted as: F( ) G ( ) s ( t ) dt () 0 where s (t) s the probablty desty fucto ( pdf ) of the beta radom varable ad F() s the cdf of ay radom varable. The pdf correspodg to the betageerated dstrbuto s gve by: a b g ( ) f ( ) F ( )( F ( )). () B ( a b ) Ths famly of dstrbutos s geeralzed by Eugee et al. [5] ad Joes [6] usg the dstrbutos of depedet detcally order statstcs for the radom varable X. I ths paper we propose a ew famly of dstrbutos depedg o the dstrbutos of depedet but o-detcally r th order statstc whch deoted as: r f ( ) F ( ) f ( ) F ( ) ( ) r: a r c ( r -)! ( - r )! p a c r See (Davd ad Nagaraja [4]). (Vaugha ad Veables []) If = ad r = Eq () we obta: g ( ) f ( ) [F ()] f ( ) [ F ()] (4) Here f () ad F () be the (pdf) ad (cdf) of some famous dstrbutos. The cdf ad hazard fucto (hf) correspodg to () are gve by G () [ F ()] [ F ()] (5) ad

3 New famly of dstrbutos usg the pdf 9 f ( ) [ F ()] f ( ) [ F ()] h() [ F ()] [ F ()] (6). Some ew dstrbutos usg Equato (4): I ths secto we propose two ew dstrbutos usg basele dstrbutos as obta by equato (4).. Epoetal Gamma Dstrbuto (EG): Let the basele dstrbutos are epoetal dstrbuto ad gamma dstrbuto where X Ep ( ) wth pdf f ( ) e ; 0 0 ad the correspodg cdf s gve by F ( ) e ; 0 0. Ad X Gamma ( ) wth pdf f ( ) e ; 0 0 ad the correspodg cdf s gve by F ( ) e [ ]; The probablty desty fucto of EG Dstrbuto: Substtutg pdf ad cdf of Epoetal ad Gamma dstrbutos Eq (4) we obta: g ( ) e [ ]; 0 0 (7) Let ( 4 ) we get: g ( ) e [ ]; 0 0 (8) To prove that g () s probablty desty fucto t should be: ) g ( ) 0 sce 0 0 ) g( ). 0

4 94 Z. A. AL-Saary Proof: g ( ) d e [ ] d 0 0 [ e d e d 0 0 Usg substtutg y we get the result.... The cumulatve dstrbuto fucto of EG Dstrbuto: G ( ) e [ ]; 0 0 (9) G ( ) e [ ]; 0 0 (0)... Specal cases - Let (4) we obta: g ( ) e [ ]; 0 0 t s Sushlla dstrbuto wth ( ) see (Shaker et. [9]) - Let or (7) we obta: g ( ) e [ ]; 0 t s Ldley dstrbuto wth ( ) See (Ldley [])...4. Graphs of PDF & CDF of EG dstrbuto: Fgures ( & ) show that & are scale parameters they else show that the shape of pdf of EG s umodal for dfferet values of & the mode s equal zero. Fgures (&4) show the cdf of EG for dfferet values of scale parameters &

5 New famly of dstrbutos usg the pdf Fg. : Plots of pdf of EG dstrbuto for dfferet values of Fg. : Plots of pdf of EG dstrbuto for dfferet values of Fg. : Plots of c.d.f. of EG dstrbuto for dfferet values of

6 96 Z. A. AL-Saary Fg. 4: Plots of c.d.f. of EG dstrbuto for dfferet values of... The Hazard Fucto of EG Dstrbuto: [ ] ( ) h ; 0 0 e [ ] () [ ] h( ) ; e [ ] () Fg. 5: Plots of hazard fucto of EG dstrbuto for dfferet values of ad From fgure (5) we see that the HF of EG s decreasg for dfferet values of &.... MGF of EG dstrbuto: [ t ] M ( t) ; 0 0 [ t ] ()

7 New famly of dstrbutos usg the pdf Meda OF EG dstrbuto: The Meda of Epoetal Gamma dstrbuto s the soluto of the followg: G( M ) For M ad the same s obtaed usg Mathematca program Verso () as follows M.84.5 M or M M (4)..7 Mode of EG dstrbuto: Dfferetatg (8) wth respect to we get: g ( ) e ; 0 0. (5) Or g ( ) 8( ) e ; (6) Clearly g( ) 0 see Fgure (6 & 7) ; ths shows that g( ) s decreased fucto of ad hece = 0 s the mode of Epoetal gamma dstrbuto Fg. 6: Plots of g ( ) of EG dstrbuto for dfferet values of

8 98 Z. A. AL-Saary Fg. 7: Plots of g( ) of EG dstrbuto for dfferet values of..8. Row Momets of EG dstrbuto: The raw momets (.e. r th momet about org) of EG dstrbuto s obtaed as follows r r ( r ) r! r r! r... (7) r r r Estmato of parameters of EG dstrbuto:..9.. Mamum Lkelhood Estmato: Suppose X ( X X... X ) be depedet varables such that each X (... ) fucto for X L X follow EG-dstrbuto havg pdf Eq (8). The the lkelhood s gve by ( ) g ( ) (8) Puttg the value of g at from (8) (8) we get LX ( ) [ e ( )] (9) ( ) e ( )

9 New famly of dstrbutos usg the pdf 99 The log-lkelhood fucto for X s obtaed as l l L ( ) X l l ( ) (0) Hece the log-lkelhood equato for estmatg s gve by l 0 0 ( ) Above s a mplct equato hece we ca ot solve t aalytcally for &. We ca use Newto Raphso method to solve umercal soluto... Epoetal Ldley Dstrbuto (EL): Let the basele dstrbutos are epoetal dstrbuto ad Ldley dstrbuto where: X Ep ( ) wth pdf f ( ) e ; 0 0 ad the correspodg cdf F ( ) e ; 0 0. Ad X Ldley ( ) wth pdf s gve by f ( ) ( ) e ; 0 0 ad the correspodg cdf s gve by F ( ) e [ ]; The probablty desty fucto of EL Dstrbuto: Substtutg wth pdf & cdf of epoetal ad Ldley dstrbutos (4) we obta: g ( ) e [ ( )]; 0 0 () To prove that g () s probablty desty fucto t should be: ) g ( ) 0 trval sce 0 0 ) g( ). 0

10 00 Z. A. AL-Saary Proof: g ( ) d e [ ( )] d 0 0 e d [ e d e d ] Usg substtutg y we get the result Fg. 8: Plots of pdf of EL dstrbuto for dfferet values of The cumulatve dstrbuto fucto of EL Dstrbuto: G ( ) e [ ]; 0 0 () G ( ) e [ ]; 0 0 (4) Fg. 9: Plots of cdf of EL dstrbuto for dfferet values of Fgures (8 & 9) show that s scale parameters they else show that the shape of pdf of EL s umodal for dfferet values of the mode s equal zero.

11 New famly of dstrbutos usg the pdf The Hazard Fucto of EL Dstrbuto: ( ) h( ) ; 0 0 e [ ] (5) Fg. 0: Plots of hazard fucto of EL dstrbuto for dfferet values of I fgure (0) we see that the HF of EL s decreasg for dfferet values of...5. MGF of EL dstrbuto: M ( t ) ; 0 0 ( 6 ) t ( )( t ) ( )( t )..6. Meda OF EL dstrbuto: The Meda of Epoetal Ldley dstrbuto s the soluto of the followg: G( M ) For M ad the same s obtaed usg Mathematca program verso () as follows M l ( M ) 0 (7) t s a mplct equato M & t ca be solved aalytcally for M...7 Mode of EL dstrbuto: Dfferetatg () wth respect to we get:

12 0 Z. A. AL-Saary 4 g e ( ) ( ); 0 0. (8) Clearly g( ) 0 See (Fg ) ths shows that of ad hece = 0 s the mode of Epoetal Ldley dstrbuto. g( ) s decreased fucto Fg. : Plots of g ( ) of EL dstrbuto for dfferet values of..8. Row Momets of EL dstrbuto: The raw momets (.e. r th momet about org) of EL dstrbuto s obtaed as follows r r r! r... r r (9) ( ) 4 ( ) Estmato of parameters of EL dstrbuto:..9.. Mamum Lkelhood Estmator: X ( X X... X ) X (... ) Suppose be depedet radom varables such that each follow EL dstrbuto havg pdf Eq (). The the lkelhood fucto for X s gve by Eq (8) Puttg the value of g at from Eq () Eq (8) we get LX ( ) [ e [ ( )]] (0)

13 New famly of dstrbutos usg the pdf 0 The log-lkelhood fucto for X l l L ( ) X s obtaed as l ( ) l l () Hece the log-lkelhood equato for estmatg s gve by l 0 ( ) 0 () ( ) Above s a mplct equato hece we caot solve t aalytcally for. We used Mathematca Program verso () to solve t ad we get. θ = + = 4 = + ( Or θ = + = + 4 = + ( = ) = ). Geerate Data from Ldley Dstrbuto ad compare EG Dstrbuto wth kow dstrbutos usg the Geeratg Data: I ths secto we geerate data usg (Mathematca Program Verso ) from Ldley dstrbuto the test ths data usg Kolmogorov-Smrov Test (K-S) o some dstrbutos oe of them s Epoetal Gamma Dstrbuto we obta: Table (): p-value ad Statstc for LD ED ND ad EGD usg Geeratg DATA Dstrbuto p-value Statstc Ldley Dstrbuto Epoetal Dstrbuto Normal Dstrbuto Epoetal Gamma Dstrbuto From Table () we coclude that DATA are sutable for Ldley dstrbuto (0.787 > 0.05) Epoetal dstrbuto (95 > 0.05) Normal dstrbuto (0.9 > 0.05) ad Epoetal Gamma dstrbuto (0.94 > 0.05) Ths dcates acceptg the ull hypotheses Whch s that the data follows these dstrbutos. To kow the best dstrbuto that fts the data We fd the statstc value of the Epoetal Gamma

14 04 Z. A. AL-Saary dstrbuto (0.07) s the smallest value so Epoetal Gamma s the best (See Fgure ()). We estmate the mamum lkelhood estmator (MLE) ad the momet estmator (ME) usg our data. See table () Table (): the MLE & ME usg our DATA MLE ME MLE ME * EpoetalGamma EpmprcalDstrbuto LdleyDstrbuto NormalDstrbuto EpoetalDstrbuto Fg (): The Emprcal dstrbuto & CDF s of Epoetal Gamma Epoetal Ldley ad Normal Dstrbutos. Refereces [] D. V. Ldley Fducal dstrbutos ad Bayes Theorem Joural of the Royal Statstcal Socety Seres B 0 (958) [] R. J. Vaugha W. N. Veables Permaet epressos for order statstcs Destes J. R. Statst. Soc. Seres B 4 (97) [] A.N. Marshall I. Olk A ew method for addg a parameter to a famly of dstrbutos wth applcatos to the epoetal ad Webull famles Bometrka 84 (997) [4] H. A. Davd H. N. Nagaraja Order Statstcs Thrd Edto Joh Wley & Sos New York [5] N. Eugee C. Lee F. Famoye Beta-ormal dstrbuto ad ts applcatos Commucatos Statstcs - Theory ad Methods (00)

15 New famly of dstrbutos usg the pdf 05 [6] M.C. Joes Famles of dstrbutos arsg from the dstrbutos of order statstcs Test (004) 4. [7] G.M. Cordero M. de Castro A ew famly of geeralzed dstrbutos J. Stat. Comput. Smul. 8 (0) [8] C. Aleader G. M. Cordero E. M. M. Ortega ad J. M. Saraba Geeralzed beta-geerated dstrbuto Computatoal Statstcs ad Data Aalyss 56 (0) [9] Rama Shaker Shambhu Sharma Uma Shaker ad Rav Shaker Sushla Dstrbuto ad ts Applcato to Watg Tmes Data Iteratoal Joural of Busess Maagemet (0) -. [0] G.M. Cordero M. Alzadeh E.M.M. Ortega The epoetated half-logstc faml of dstrbutos: propertes ad applcatos J. Probab. Statstcs 04 (04) - Art.ID [] S. Nadarajah G.M. Cordero E.M.M. Ortega The Zografos- Balakrsha G famly of dstrbutos: mathematcal propertes ad applcatos Commu. Stat. Theory Methods 44 (05) [] Alzaatreh Ayma; Carl Lee Fel Famoye Idral Ghosh The geeralzed Cauchy famly of dstrbutos wth applcatos Joural of Statstcal Dstrbutos ad Applcatos (06) o [] G.M. Cordero M. Alzadeh G. Ozel B. Hosse E.M.M. Ortega ad E. Altu The geeralzed odd log-logstc famly of dstrbutos: propertes regresso models ad applcatos Joural of Statstcal Computato ad Smulato 87 (07) o Receved: September 07; Publshed: December 4 07