Bayesian and Maximum Likelihood Estimation for Kumaraswamy Distribution Based on Ranked Set Sampling

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1 Ameica Joual of Mathematic ad Statitic 04 4(): 0-7 DOI: 0.59/j.ajm Bayeia ad Maximum Lielihood Etimatio fo Kumaawamy Ditibutio Baed o Raed Set Samplig Mohamed A. Huia Depatmet of Mathematical Statitic Ititute of Statitical Studie ad Reeach (ISSR) Caio Uiveity Egypt Abtact I thi pape the etimatio of the uow paamete of the umaawamy ditibutio i coideed uig both imple adom amplig (SRS) ad aed et amplig (RSS) techique. The etimatio i baed o maximum lielihood etimatio ad Bayeia etimatio method. A imulatio tudy i made to compae the eultat etimato i tem of thei biae ad mea quae eo. The efficiecy of the etimate made uig aed et amplig ae alo computed. Keywod Bia Maximum lielihood etimato Mea quae eo Kumaawamy ditibutio Raed et amplig Simple adom amplig. Itoductio Maig ifeece about a populatio baed o a ample of data collected fom thi populatio i almot the mot impotat eeach aco mot o eve all id of ciece uch a agicultual biological ecological egieeig medical phyical ad ocial ciece. The appoach of collectig ample data that ae tuly epeetative to the populatio i a impotat ey to mae ucceful aalyi to the cietific quetio ude ivetigatio. The mot commo appoach to data collectio i the imple adom ample (SRS) appoach. A collectio of adom vaiable X... X i aid to be a imple adom ample (SRS) of ize fom a udelyig pobability ditibutio with pobability deity fuctio (pdf) f( x ) ad cumulative ditibutio fuctio (cdf) F( x ) if each X =... ha the ame pobability ditibutio a the udelyig populatio ad the adom vaiable X... X ae mutually idepedet. Fo fiite populatio coitig of a total of N obevatio a collectio of ample obevatio i aid to be a imple adom ample X... X if each of the N C poible ubet of obevatio ha the ame chace of beig elected a the adom ample[. A eiou dawbac of the SRS appoach i that thee i o guaatee that a pecific adom ample of uit elected fom the * Coepodig autho: maby@u.edu.a (Mohamed A. Huia) Publihed olie at Copyight 04 Scietific & Academic Publihig. All Right Reeved populatio i tuly epeetative of the populatio. Thi pecific ample might o might ot actually povide good ifomatio about the populatio. Becaue of that may attempt ad may appoache have bee uggeted to miimize the effect of thi poblem. Some of thee appoache ae ytematic amplig tatified amplig clute amplig ad quota amplig. Howeve oe of thee appoache ue exta ifomatio fom pecific uit i the populatio to guide thei each fo a tuly epeetative ample[. McItye[ itoduced the aed et amplig (RSS) appoach that utilize additioal ifomatio fom idividual populatio uit povidig a moe epeetative ample fom the populatio ude coideatio. A impotat advatage of thi appoach i that it impove the efficiecy of etimato of the populatio paamete. Fo example it impove the efficiecy of a ample mea a a etimato of the populatio mea i ituatio i which the vaiable of iteet i difficult o expeive to meaue but could be cheaply aed. Theoetical ivetigatio by Dell ad Clutte howed that egadle of aig eo the RSS etimato of a populatio mea i ubiaed ad i at leat a pecie a the SRS etimato with the ame umbe of quatificatio[4. David ad Levie ivetigated the cae whee aig i doe by a umeical covaiate[5. Futhemoe RSS povide moe pecie etimato of the vaiace[6 the cumulative ditibutio fuctio[7 ad the Peao coelatio coefficiet[8. Seveal autho have ued RSS fo paametic ifeece fo example Stoe[9 looed at the maximum lielihood ad bet liea ubiaed etimato of the locatio-cale paamete i locatio-cale family of ditibutio while Yu ad co-autho[0 developed a etimato fo the populatio vaiace of a

2 Ameica Joual of Mathematic ad Statitic 04 4(): 0-7 omal ditibutio baed o balaced ad ubalaced aed et ample. O the othe had eveal attempt wee made to impove the etimatio baed o RSS. Fom thoe deig fo optimal aed et amplig wee cotucted fo paametic familie of ditibutio[ ad bet liea ubiaed etimato baed o odeed aed et ample wee alo developed[. A modificatio of the RSS called movig exteme aed et amplig (MERSS) wa coideed fo the etimatio of the cale paamete of cale ditibutio[ ad a impoved RSS etimato fo the populatio mea wa obtaied[4. Oztu ha developed two amplig deig to ceate atificially tatified ample uig RSS[5. Reade ae ecouaged to peual at a hitoical pepective of the RSS appoach ee[6-5. I ode to obtai a adom ample of data of ize obevatio fom a populatio uig RSS appoach the followig poce i applied. (i) Radomly daw m adom et with m elemet Xim : i =... m i each ample (m i called the et ize ad i typically mall to miimize aig eo). (ii) Allocate the m elected uit a adomly a poible ito m et each of ize m. (iii) without yet owig ay value fo the vaiable of iteet a the uit withi each et baed o a peceptio of elative value fo thi vaiable. (iv) Chooe a ample fo actual aalyi by icludig the mallet aed uit i the fit et the the ecod mallet aed uit i the ecod et cotiuig util the laget aed uit i elected i the lat et. (v) Radomly daw othe m adom et with m elemet i each ample with a total of m ample uit ad epeat tep (ii) though (v) fo cycle util the deied ample X ( im :: ) j; i =... m j =... of ize = m i obtaied fo aalyi. I thi aticle the uow paamete of the Kumaawamy (Kw) ditibutio will be etimated ude both SRS ad RSS appoache. The etimatio i made uig maximum lielihood (ML) etimatio ad Bayeia etimatio method. The Kumaawamy' double bouded (Kw) ditibutio i a family of cotiuou pobability ditibutio defied o the iteval[0 diffeig i the value of thei two o-egative hape paamete ad [6. It i imila to the Beta ditibutio but much imple to ue epecially i imulatio tudie due to the imple cloed fom of both it pobability deity fuctio ad cumulative ditibutio fuctio. The Kw ditibutio pdf ad cdf ae give by f( x) = x ( x ) () F( x) = ( x ) () epectively whee 0< x < ad the hape paamete > 0. The et of the aticle i ogaized a follow. I Sectio ML ad Bayeia method of etimatio of uow paamete ae dicued ude SRS. I Sectio the ame method of etimatio ae dicued baed o RSS. Simulatio tudie ae caied out to illutate theoetical eult i Sectio 4. Fially cocluio ae peeted i Sectio 5.. Etimatio Uig SRS Appoach.. Maximum Lielihood Etimatio X X... X be a adom ample of ize Let daw fom the Kw ditibutio with hape paamete ad. The lielihood fuctio of ad fo the obeved ample i ( ; ) = ( ) = = () L data x x Theefoe the log-lielihood fuctio of ad will be log L = log + log + ( ) log x = x. (4) = + ( ) log[ The etimato ˆml ad ˆml of the paamete ad epectively ca be obtaied a the olutio of the lielihood equatio x log x = = x + log ( ) = 0 ( ) (5) + log[ x = 0. (6) = Fom Equatio (5) ad (6) we have ˆ ml = ˆml log[ x = (7) whee ˆml i the olutio of the oliea equatio ml ˆ ml = = ˆ ml log ˆ ml ˆ ml ( x ) log ( ˆ x + x ) = 0.(8) The ML etimato ˆml ad ˆml ae the olutio of the two oliea Equatio (7) ad (8). Thee etimato caot be obtaied i cloed fom theefoe umeical aalyi i ued to tudy thei popetie... Bayeia Etimatio I thi ectio the Baye etimato of hape paamete

3 Mohamed A. Huia: Bayeia ad Maximum Lielihood Etimatio fo Kumaawamy Ditibutio Baed o Raed Set Samplig ad deoted by ˆB ˆB epectively ae obtaied ude the aumptio that ad ae idepedet adom vaiable with pio ditibutio Gamma(a b ) ad Gamma(a b ) epectively with pdf' ad a b a b π( ) = e ; (9) Γ ( a ) a b a b χ π( ) = e ; (0) Γ ( a ) whee > 0 ad the hype-paamete a a > 0 ad b b > 0 ae aumed to be ow. Baed o the above aumptio ad the lielihood fuctio peeted i Equatio () the joit deity of the data ad ca be obtaied a + a + a b b = = K e e x ( x ) = = L ( data ) L ( data ; ) π ( ) π ( ) whee K i cotat ad = K Ψ. () a + a + b b + ( ) log[ x + ( ) log[ x. () Ψ= e = = ad Theefoe the joit poteio deity of the data ad give the data ca be obtaied a L( data ) π( / data ) = Accodig to that the poteio pdf' of ad ae π π L( data ) dd 0 ( / data ) = Ψ d Ψ d d 0 ( / data ) =. Ψ d Ψ dd = Ψ Ψ dd () (4) (5) epectively. Theefoe the Baye etimato fo the paamete ad deoted by ˆB ˆB ude quaed eo lo fuctio ae defied epectively a ad Ψ d d ˆ BS = E( / data) = Ψ dd (6)

4 Ameica Joual of Mathematic ad Statitic 04 4(): 0-7 Ψ dd ˆ BS = E ( / data ) =. Ψ dd Thee etimato caot be obtaied i cloed fom. Thu the popetie of thee etimato will be dicued uig imulatio tudie. (7). Etimatio Uig RSS Appoach.. Maximum Lielihood Etimatio Aume that (: ) ; im j X X (: im) j 0 < < i =... m ad j... = i a aed et ample with ample ize = m fom the Kw ditibutio whee m i the et ize ad i the umbe of cycle. Fo implificatio pupoe X (: im) j will be deoted a X. The pdf of the adom vaiable X i give by which i the cae of the Kw ditibutio will be m! i m i g( X ) = f( X )[ F( X ) [ F( X ) ; ( i )!( m i)! m! g( X ) X ( X ) ( i )!( m i)! ( m + i ) = The lielihood fuctio of ad fo the obeved ample i give by m ( m + i ) i ( ; ) = ( ( ) [ ( ) j= i= L data K X X X i X [ ( ). (8) i X Theefoe the log-lielihood fuctio of ad will be LogL = log K + mlog + mlog + ( ) log X m j= i= m j= i= + ( ( m i+ ) ) log[ X + ( i ) log[ ( X ) whee K i cotat. Thi implie that [ ( ) ). (9) m (0) j= i= ad m m X X + log X + ( ( m i+ ) ) j= i= j= i= X m log[ m X ( X ) log[ X j= i= ( X ) () + ( i ) = 0 m m ( m i ) log[ X m ( X ) log[ X ( i ) = 0 j= i= = = ( X ) j i. () ˆml ad ˆml ae the olutio of the two oliea Equatio () ad () ad umeical aalyi i ued to tudy thei popetie... Bayeia Etimatio

5 4 Mohamed A. Huia: Bayeia ad Maximum Lielihood Etimatio fo Kumaawamy Ditibutio Baed o Raed Set Samplig The Baye etimato of the hape paamete ad deoted by ˆB ad ˆB epectively ae obtaied imila to the pocedue ued i ectio (.). Let ad be idepedet adom vaiable with pio ditibutio give i Equatio (9) ad (0). Baed o thee aumptio ad the lielihood fuctio peeted i Equatio (9) the joit deity of the data ad ca be obtaied a m b b+ ( ) log[ X m+ a m+ a j= i= L ( data ) = L( data; ) π ( ) π ( ) = K e e m m ( ( m i+ ) ) log[ X + ( i ) log[ ( X ) j= i= j= i= Theefoe the joit poteio deity of the data ad give the data ca be obtaied a whee Λ= L( data ) π( / data ) = = L( data ) dd Λ m + m+ a m+ a j= i= e Λ dd b b ( ) log[ X. () (4) e m m ( ( m i+ ) ) log[ X + ( i ) log[ ( X ) j= i= j= i= (5) The Baye etimato fo paamete ad deoted by ˆB ˆB ude quaed eo lo fuctio ae defied epectively a ad Λ d d ˆ BS = E( / data) = Λ dd ˆ BS = E ( / data ) =. 4. Simulatio Study Λ dd Λ dd (6) (7) Numeical olutio ae ued to obtai the ML ad Baye etimato of the uow paamete of the umaawamy ditibutio ad to compae the pefomace of thee etimato baed o RSS ad SRS appoache. Mote Calo imulatio tudy i made uig MATHEMATICA oftwae ad i baed o 00 eplicatio. The imulatio ae made fo eveal combiatio of the paamete m ad value while the value of the hape paamete i equal to oe. The compaio i caied out though biae MSE of the etimato ˆml ˆml ˆB ˆB ˆml ˆml ˆB ad ˆB. Alo the efficiecy of the etimato that ae deived uig RSS with epect to thoe uig SRS ae computed whee the efficiecy of a etimato ˆ θ with epect to a etimato ˆ θ i give by eff ( ˆ θ ) MSE( ˆ θ ) = (0) MSE( ˆ θ) The lage the efficiecy the bette i ˆ θ i tem of MSE. The eult ae epoted i Table (Appedix A). Oe ca coclude fom thee eult that the etimate of ad baed o RSS have malle biae tha the coepodig etimate uig SRS. Biae ad MSE of the etimate made by both method deceae a et ize iceae. It i alo oted that biae ad MSE of the hape paamete iceae whe it populatio value iceae. Alo almot i all cae the biae ad MSE fo the Baye etimate of both paamete ad ae malle tha the coepodig value fo the ML etimate of ad epectively. A a eult ad fom table the ML ad Baye etimato of both paamete ad deived uig RSS ae moe efficiet of the coepodig etimato deived uig SRS. 5. Cocluio I thi aticle etimatio poblem of uow paamete of the umaawamy ditibutio baed o RSS wa coideed. ML ad Bayeia method of etimatio ae

6 Ameica Joual of Mathematic ad Statitic 04 4(): ued whee Baye etimate wee obtaied ude quaed eo lo fuctio. Baed o the imulatio tudy it i obeved that the Baye etimato pefom bette tha ML etimato elative to thei biaed ad MSE. Futhemoe biae ad MSE of the etimate fo the hape paamete ude RSS appoach ae malle tha the coepodig etimate computed ude the SRS appoach. Thi idicate that etimatio ude the RSS appoach i moe efficiet tha etimatio ude the SRS appoach. Appedix A Table. Biae of the etimato of the Kw ditibutio fo populatio paamete = ad the pio hype-paamete (a a b b ) = ( ) ˆml ˆB ˆml ˆB m ˆml ˆB ˆml ˆB Table. MSE of the etimato of the Kw ditibutio fo populatio paamete = ad the pio hype-paamete (a a b b ) = ( ) ˆml ˆB ˆml ˆB m ˆml ˆB ˆml ˆB

7 6 Mohamed A. Huia: Bayeia ad Maximum Lielihood Etimatio fo Kumaawamy Ditibutio Baed o Raed Set Samplig Table. Efficiecy of the etimato of the Kw ditibutio fo populatio paamete = ad the pio hype-paamete (a a b b ) = ( ) eff ( ˆ ) m ml eff ( ˆ ) B eff ( ˆ ) ml eff ( ˆ ) B REFERENCES [ D. A. Wolfe Raed Set Samplig: It Relevace ad Impact o Statitical Ifeece Iteatioal Scholaly Reeach Netwo ISRN Pobability ad Statitic Vol. 0 Aticle ID page. [ G.P. Patil Raed et amplig Ecyclopedia of Eviometic Vol. pp Joh Wiley & So Ltd Chichete [ G. A. McItye A method fo ubiaed elective amplig uig aed et Autalia Joual of Agicultual Reeach : [4 T. R. Dell ad J. L. Clutte Raed et amplig theoy with ode tatitic bacgoud Biometic 8: [5 H. A. David ad D. N. Levie Raed et amplig i the peece of judgmet eo Biometic 8: [6 S. L. Stoe Etimatio of vaiace uig judgmet odeed aed et ample Biometic 6: [7 S. L. Stoe ad T. W. Sage Chaacteizatio of a aed et ample with applicatio to etimatig ditibutio fuctio Joual of the Ameica Statitical Aociatio 8: [8 S. L. Stoe Ifeece o the coelatio coefficiet i bivaiate omal populatio fom aed et ample Joual of the Ameica Statitical Aociatio 75: [9 S. L. Stoe Paametic aed et amplig Aal of the Ititute of Statitical Mathematic 47: [0 P. L. H. Yu K. Lam ad B. K. Siha Etimatio of vaiace baed o balaced ad ubalaced aed et ample Eviometal ad Ecological Statitic 6: [ Z. Che ad Z. D. Bai The optimal aed-et amplig cheme fo paametic familie Sahya 0 Seie A 6: [ N. Balaiha ad T. Li Odeed aed et ample ad applicatio to ifeece Joual of Statitical Plaig ad Ifeece 8: [ W. Che M. Xie ad M. Wu Paametic etimatio fo the cale paamete fo cale ditibutio uig movig exteme aed et amplig Statitic & Pobability Lette 8(9) pp [4 N. Mehta ad V. L. Madowaa A bette etimato i Raed et amplig Iteatioal Joual of Phyical ad Mathematical Sciece 4() pp [5 O. Oztu Combiig multi-obeve ifomatio i patially a-odeed judgmet pot-tatified ad aed et ample Caadia Joual of Statitic 4()pp [6 M. Flige ad S. N. MacEache Nopaametic two-ample method fo aed et ample data Joual of the Ameica Statitical Aociatio 475: [7 J. Fey New impefect aig model fo aed et amplig Joual of Statitical Plaig ad Ifeece 7: [8 O. Oztu Statitical ifeece ude a tochatic odeig cotaied i aed et amplig Joual of Nopaametic Statitic 0: [9 O. Oztu Ditibutio-fee two-ample ifeece i aed

8 Ameica Joual of Mathematic ad Statitic 04 4(): et ample Joual of Statitical Theoy ad Applicatio 6: [0 O. Oztu Statitical ifeece i the peece of aig eo i aed et amplig The Caadia Joual of Statitic 5: [ O. Oztu Nopaametic maximum lielihood etimatio of withi-et aig eo i aed et amplig. Joual of Nopaametic Statitic : [ A. Helu M. Abu-Salih ad O. Alami Baye etimatio of Weibull ditibutio paamete uig aed et amplig Commuicatio i Statitic-Theoy ad Method 9: [ S. A. Al-Hadham Paametic etimatio o modified Weibull ditibutio baed o aed et amplig Euopea Joual of Scietific Reeach 44(): [4 A. S. Haa Maximum Lielihood ad Baye Etimato of the Uow Paamete Fo Expoetiated Expoetial Ditibutio Uig Raed Set Samplig Iteatioal Joual of Egieeig Reeach ad Applicatio (IJERA) (): [5 W. Abu-Dayyeh A. Ahai ad K. Ibahim Etimatio of the hape ad cale paamete of Paeto ditibutio uig aed et amplig Stat Pape 54: [6 P. Kumaawamy "A geealized pobability deity fuctio fo double-bouded adom pocee" Joual of Hydology 46 (-):