Bayesian Estimation and Prediction for. a Mixture of Exponentiated Kumaraswamy. Distributions

Transcription

1 Iteratioal Joural of Cotemporary Mathematial Siees Vol. 11, 2016, o. 11, HIKARI Ltd, Bayesia Estimatio ad Preditio for a Mixture of Expoetiated Kumaraswamy Distriutios Samia A. Adham Departmet of Statistis, Faulty of Siee Kig Adulaziz Uiversity Jeddah, Kigdom of Saudi Araia Afal A. ALgfary Departmet of Statistis, Faulty of Siee Qassim Uiversity Qassim, Kigdom of Saudi Araia Copyright 2016 Samia A. Adham ad Afal A. ALgfary. This artile is distriuted uder the Creative Commos Attriutio Liese, whih permits urestrited use, distriutio, ad reprodutio i ay medium, provided the origial work is properly ited. Astrat This paper deals with the Bayesia estimatio of the vetor of parameters of the fiite mixture of two-ompoet expoetiated Kumaraswamy distriutio, deoted y MEKum, whe osiderig omplete samples. I additio, Bayesia preditive desity futios of future oservatios from the MEKum distriutio, osiderig two ases, are otaied. Keywords: Mixture of distriutios; Bayesia estimatio; Bayesia preditio 1. Itrodutio The expoetiated Kumaraswamy distriutio is oe of the reetly ostruted distriutios. It has ee firstly appeared i Lemote et al. (2013) [12]. The expoetiated Kumaraswamy, deoted here y EKum, is a geeralizatio of the

2 498 Samia A. Adham ad Afal A. ALgfary Kumaraswamy distriutio, deoted y Kum. The three parameter EKum distriutio is a simple extesio of the two parameters Kum distriutio whih was firstly itrodued y Kumaraswamy i Although, the Kum distriutio is quite ew amog statistiias ad has ee little explored i the literature, the Kum distriutio was origially oeived to model hydrologial pheomea ad has ee used for this ad also for other purposes. [See, for example, Sudar ad Suiah (1989) [15], Flether ad Poamalam (1996) [6], Seifi et al. (2000) [14], Gaji et al. (2006) [7]]. Joes (2009) [10] explored the akgroud ad geesis of the Kum distriutio ad, more importatly, made lear some similarities ad differees etwee the eta ad Kum distriutios. The Kum distriutio has may of the same properties as the eta distriutio ut has some advatages. Moreover, the Kum distriutio argues that the eta distriutio does ot faithfully fit hydrologial radom variales suh as daily raifall, daily stream flow,, et. I additio, the Kumaraswamy s desities are uimodal, uiatimodal, ireasig, dereasig or ostat depedig i the same way as the eta distriutio o the values of its parameters. Further, Joes (2009) [10] highlighted several advatages of the Kum distriutio over the eta distriutio: the ormalizig ostat is very simple expliit formulae for the distriutio ad quatile futios whih do ot ivolve ay speial futios ad a simple formula for radom variate geeratio. However, the eta distriutio has the followig advatages over the Kum distriutio: simpler formulae for momets ad momet geeratig futio, a oeparameter su-family of symmetri distriutios, simpler momet estimatio ad more ways of geeratig the distriutio y meas of physial proesses. Studyig mixture of distriutios is of great iterests of may statistiias, as data sets a e osidered to e from mixture populatios. Some of the most importat referees that disussed differet types of mixtures of distriutios are the moographs y Everitt ad Had (1981) [5], Titterigto et al. (1985) [16], MLahla ad Basford (1988) [13]. The fiite mixture of two expoetiated Kumaraswamy distriutios has ee proposed y ALgfary (2015) [2]. She proved the idetifiaility of the MEKum distriutio, sie it is importat to prove the idetifiaility of the mixture distriutio efore disussig the prolem of estimatio. ALgfary (2015) [2] studied i details some of the MEKum distriutio statistial properties, as well as its speial ases, ad otaied the maximum likelihood estimates for the vetor of the parameters of the MEKum distriutio. The desity futio of the MEKum distriutio is give y: f(x) = p 1 β 1 γ 1 x 1 1 1) β 1 1 [1 1) β 1] γ 1 1 +q 2 β 2 γ 2 x 2 1 2) β 2 1 [1 2) β 2] γ 2 1. (1) where, i, β i ad γ i, i = 1,2 are positive parameters ad p is the mixig proportio, 0 p 1, q = 1 p. Let X 1, X 2,, X e a radom sample from the MEKum distriutio with desity futio give y Equatio (1), the the likelihood futio is

3 Bayesia estimatio ad preditio for a mixture 499 where, L(θ) = p 1 β 1 γ 1 x i 1 1 i 1 ) β 1 1 [1 i 1 ) β 1] γ (1 p) 2 β 2 γ 2 x i 2 1 i 2 ) β 2 1 [1 2 i ) β 2] γ 2 1, θ = ( 1, 2, β 1, β 2, γ 1, γ 2, p). (3) (2) I Setio 2, Bayesia estimatio of the vetor of parameters θ is osidered uder the squared error loss futio. Setio 3 deals with the preditive desities for two speial ases of the MEKum distriutio. Fially, olusios are disussed i Setio Bayesia Estimatio Let all the parameters i the vetor θ, i Equatio (3), are idepedet radom variales of eah others. A ojugate prior distriutio is assiged to eah of them. That is, a eta distriutio with kow parameters a 1 ad a 2 ad desity futio g 1 (p) p a 1 1 (1 p) a 2 1, 0 < p < 1. (4) is assiged as a ojugate prior distriutio for the mixig proportio parameter p. While, the prior distriutio used y AL-Hussaii ad Jahee (1992) [3], whih is a gamma desity with parameters ( 1j, 2j ), is assumed here to e the prior distriutio for the parameters β j, j = 1,2, with desity futio β j g 2j (β j ) β 1j 1 j e 2j, β j > 0. (5) Ad the gamma desity with parameters ( 1j, 2j ), j = 1,2, is take to e the prior distriutio of the parameters j with desity futio j (6) g 3j ( j ) 1j 1 j e 2j, j > 0. Whereas, the gamma desity with parameters (d 1j, d 2j ), j = 1,2, is assumed to e the prior distriutio of the parameters γ j with desity futio γ j d g 4j (γ j ) γ 1j 1 d j e 2j, γ j > 0. (7)

4 500 Samia A. Adham ad Afal A. ALgfary The the joit prior desity futio for θ = ( 1, 2, β 1, β 2, γ 1, γ 2, p) is the give y g(θ) p a 1 1 (1 p) a 2 1 β e 21 β 12 1 β 2 2 e 22 (8) e e 2 γ 1 22 d γ e d 21 d γ e γ 2 d 22. The joit posterior distriutio of the vetor of parameters θ is π(θ x) = L(θ;x) g(θ) θ L(θ;x) g(θ)dθ. (9) It follows, from (2), (8) ad (9), that the joit posterior desity futio is give y π(θ x) = δ{ p +a 1 1 (1 p) a e e 2 22 β e 21 β 12 1 β 2 γ 1 2 e 22 γ +d e d 21 γ d e i 1 ) β 1 1 [1 i 1 ) β 1] γ p a 1 1 (1 p) +a 2 1 β e γ 2 d 22 x i 1 1 where, e γ γ d e d 21 e x i 2 1 i 2 ) β 2 1 β γ 2 22 β 2 1e d 22 γ 2 +d e 1 21 [ 1 i 2 ) β 2] γ 2 1 ]}, (10) + δ = { eta(a 1 +, a 2 ) Г( + 11 )Г( + d 11 ) e (+d 11 )[l( 1 ) 1] l(x d i ) Г( ) ( 1 2 l(x i ) ) Г( 12 ) 12 d 22 Г( 12 ) d Г(d 12 ) + eta(a 1, a 2 + ) Г( + 12 )Г( + d 12 ) e (+d 12 )[l( 1 ) 1] l(x d i ) Г( 11 ) Г( 11 )d 21 d 11 Г(d 11 ) } 1 Г(m 12 + ) 12 + ) ( 1 2 l(x i ) 22 Uder the squared error loss futio, we a otai the Bayes estimators of the vetor of the parameters, θ = ( 1, 2, β 1, β 2, γ 1, γ 2, p ). Sie the parameters of

5 Bayesia estimatio ad preditio for a mixture 501 the MEKum distriutio are assumed as idepedet radom variales; hee the Bayesia estimator of eah parameter will e its posterior mea. That is the Bayesia estimator of the parameter p is p = E(p X = x) = p π(θ x) dp p 1 + = δ{eta(a , a 2 ) e e β 2 + β e 21 β e 22 +d γ e d γ 12 1 γ 2 2 e d 22 x 1 1 i γ 1 d 21 i 1 ) β 1 1 [1 1 i ) β 1] γ eta(a 1 + 1, a 2 + ) 2 β e e 22 d γ e β 2 e + 12 γ 2 22 β 2 1e γ 1 d 21 d 22 γ 2 +d e x i i ) β2 1 [1 2 i ) β 2] γ 2 1}. (11) Similarly, for l, j = 1,2, ad l j, the Bayesia estimators of the parameters j, β j ad γ j are give, respetively, y j = E( j X = x) = j π(θ x) d j j = δ(p +a1 1 (1 p) a2 1 1l 1 2 l e + 2l β 1j 1 j e β 1l 1 β γ l j l e 2l +d γ 1j 1 d j e 2j d γ 1l 1 γ l l e d 2l Г( 1j + + 1) x i [ 1 + (γ j β j 2) l(x i ) 2j ] 1j++1 β j +p a1 1 (1 p) +a 2 1 β 1j 1 j e 2j + 1l 1 l e l β j 2j 2l γ j d 1j 1 e γ j d 2j

6 502 Samia A. Adham ad Afal A. ALgfary β l e + 1l γ l 2l β l 1e x i l 1 d 2l γ l +d 1l 1 2j 1j +1 Г( 1j + 1) i l ) β l 1 [1 i l ) β l] γ l 1 }. (12) β j = E(β j X = x) = β j π(θ x) dβ j β j j = δ{p +a 1 1 (1 p) a j 1 j e 2j 1l 1 l e β l e γ j l 2l β 1l 1 β l l e 2l +d γ 1j 1 d j e 2j d γ 1l 1 γ l l e d 2l x j 1 i Г( 1j + + 1) 1j +1 2j Г( 1j + 1) j i ) [ 1 + (γ j 2) l j i ) 2j +p a1 1 (1 p) +a l 1 l l e 2l d γ 1j 1 j e j + 1l γ l 2l β l 1e d 2l +d γ 1l 1 1j 1 j e 2j l x i l 1 i l ) β l 1 [1 l i ) β l] γ l 1}. (13) j1 ++1 ] γ j d 2j γ j = E(γ j X = x) = γ j π(θ x) dγ j γ j = δ { p +a 1 1 (1 p) a 2 1 j + 1j 1 e j l 2j 1j 1 l e 2l

7 Bayesia estimatio ad preditio for a mixture 503 β j + β 1j 1 j e 2j β 1l 1 β l l e 2l d γ 1l 1 l e x i j 1 i j ) β j 1 Г( + d 1j + 1) {( 1 l[1 j d i )] 2j γ l d 2l β j} +d 1j +1 β j +p a 1 1 (1 p) +a 2 1 β 1j 1 j e 2j + 1l 1 l e + β 1l 1 γ l l e d 2l +d γ 1l 1 l 1j 1 j e j l β l 2l e 2l 2j d 2j d 1j +1 Г(d 1j + 1) x l 1 i l i ) βl 1 [1 l i ) β l] γl 1 }. i= (14) Equatios (11), (12), (13) ad (14) are o-liear system of equatios. Oe ould apply Newto-Raphso iteratio sheme to solve this o-liear system of equatios simultaeously. 3. Bayesia Preditio I may pratial prolems of statistis, oe wishes to use the iformative data to predit future oservatio from the same populatio. Oe way to do this is to ostrut a iterval, whih will otai these oservatio with a speified proaility. This iterval is alled preditio iterval. Preditio has ee applied i mediie, egieerig, usiess ad other areas as well. For details o the history of Bayesia preditio, aalysis ad appliatios, see for example, Aithiso ad Dusmore (1975) [1] ad Geisser (1993) [8]. AL-Hussaii (2003) [4] otaied Bayesia preditive desity futios whe the populatio desity is a fiite mixture of geeral ompoets. Jahee (2003) [9] osidered Bayesia preditio ouds for future oservatio from the fiite mixture of two ompoets of Gompertz distriutios ased o type-i esored samples. This setio is oered with Bayesia preditio of the MEKum distriutio. Havig θ a vetor of seve ukow parameters may ause prolems i otaiig the preditive desity. Hee, oly two ases are osidered here with the parameter γ 1 is assumed to e 1. I Case 1, the parameters p ad β 1 are assumed to e ukow; while the rest of the parameters are kow. Whereas, i Case 2, the parameters p, β 1 ad β 2 are assumed to e ukow ad the rest of the parameters are kow.

8 504 Samia A. Adham ad Afal A. ALgfary Case 1: (p ad β 1 are ukow) Let X 1, X 2,., X e sample radom sample of size draw from a populatio with desity futio f(x θ) give y Equatio (1). Let Y s = X +1 e a future oservatio from the same populatio. The, the Bayes preditive desity futio is give y: f(y s x ) = p π(p, β 1 β 1 x) f(y s p, β 1 ) dβ 1 dp, (15) where the joit posterior desity futio of p ad β 1 (whe γ 1 = 1) is give y where, π(p, β 1 x) = L(p, β 1 ; x) g(p, β 1 ) L(p, β 1 ; x) g(p, β 1 )dβ 1 dp p β 1 = δ 1 {p +a1 1 (1 p) a β e 21 x 1 1 i 1 i ) β1 1 + β e 21 p a1 1 (1 p) +a 2 1 x i 2 1 i 2 ) β 2 1 [1 2 i ) β 2] γ 1 }, δ 1 = { 1 x 1 1 i 2 β 2 γ Г( 11 + ) 1 i )[ 1 l i 1)] 21 eta(a 1 +, a 2 ) + 2 β 2 γ eta(a 1, a 2 + ) Г( 11 ) 11 + x i 2 1 i 2 ) β 2 1 (1 i 2 ) β 2) γ 1 } 1. Suh that, uder the assumptio that the parameters p ad β 1 are ukow ad idepedet radom variales ad the rest of the parameters are kow, the prior for the parameter p is assumed to e a eta distriutio with parameters a 1 ad a 2 ad give y g 1 (p) p a 1 1 (1 p) a 2 1, 0 < p < 1. (16)

9 Bayesia estimatio ad preditio for a mixture 505 The prior distriutio of β 1 is assumed to e a gamma distriutio with parameters ( 11, 21 ) ad desity futio give y The joit prior of p ad Equatio (17), g 2 (β 1 ) β 11 1 j e 21, β 1 > 0. (17) β 1 a e otaied y multiplyig Equatio (16) y g(p, β 1 ) p a 1 1 (1 p) a 2 1 β e 21. Ad the desity of Y s is f(y s p, β 1 ) = p 1 β 1 y s 1 1 (1 y s 1 ) β q 2 β 2 γ y s 2 1 (1 y s 2 ) β 2 1 [1 (1 y s 2 ) β 2] γ 1. Therefore, the Bayes preditive desity futio a e write as f(y s x ) = δ 1 { 1 +1 Beta(a , a 2 )Г( ) x 1 1 i 1 i )[ 1 l 1 i )] β 2 γ y 2 1 s (1 y 2 s ) β 2 1 [ 1 (1 y 2 s ) β 2] γ 1 Beta(a 1 +, a 2 + 1) Г( 11 + ) x 1 1 i 1 i )( 1 l 1 i ) β 2 γ 1 y 1 1 s Beta(a 1 + 1, a 2 + ) Г( ) ( 1 y 1 s ) ( 1 l(1 y 1 s )) x 2 1 i i 2 ) β i 2 ) β 2] γ Beta(a 1, a ) Г( 11 ) +1 2 β2 γ +1 y 2 1 s (1 y 2 s ) β2 1 [1 (1 y 2 s ) β 2] γ 1 ] x 2 1 i 2 i ) β2 1 [1 2 i ) β 2] γ 1 }. Case 2: (p, β 1 ad β 2 are ukow) Let X 1, X 2,., X e sample radom sample of size draw from a populatio with desity futio f(x θ) give y Equatio (1). Let Y s = X +1 e a future oservatio from the same populatio. The, the Bayes preditive desity futio is give y f(y s x ) = π(p, β 1, β 2 x) f(y s p, β 1, β 2 ) dβ 1 dβ 2 dp, p β 2 β 1 (18)

10 506 Samia A. Adham ad Afal A. ALgfary The prior distriutios of p ad β 1 are give y Equatio (16) ad Equatio (17), respetively. Ad the prior distriutios of β 2 is assumed to e a gamma desity with parameters ( 12, 22 ) give y β 2 g 3 (β 2 ) β e 22, β 2 > 0. (19) The, the joit prior of p, β 1 ad β 2 a e writte as g(p, β 1, β 2 ) p a1 1 (1 p) a2 1 β e 21 β e β (20) The joit posterior desity of the parameters p, β 1 ad β 2 is otaied from Equatios (2) ad (20), y where, ad, L(p, β 1, β 2 ; x) g(p, β 1, β 2 ) π(p, β 1, β 2 x) = L(p, β 1, β 2 ; x) g(p, β 1, β 2 )dβ 1 dβ 2 dp p β 2 β 1 = δ 2 {p a 1+ 1 (1 p) a β e 21 β e x 1 1 i x 2 1 i β β e β 2 22 i 1 ) β p a 1 1 (1 p) a β 2 21 e 22 γ i 2 ) β 2 1 [1 i 2 ) β 2] γ 1 }. δ 2 = [ Г( 12 ) 1 xi 1 1 eta(a 1 +, a 2 )Г( 11 + ) 1 i ) [ 1 l i 1)] γ e γ 1 Г( 12 + )eta(a 1, a 2 + ) Г(11 ) x 2 1 i 2 i ) [ ] 1. + (γ 2) l 2 i )] 22 f(y s p, β 1, β 2 ) = p 1 β 1 y 1 1 s (1 y 1 s ) β1 1 + q 2 β 2 γ y 2 1 s (1 y 2 s ) β 2 1 [1 (1 y 2 s ) β 2] γ 1. The, the Bayes preditive desity futio a e write as f (y s x) = δ 2 {Beta(a , a 2 ) Г( 12 ) 1 +1 y s 1 1 x 1 1 i Г( ) 1 i )(1 y 1 s )[ 1 l i 1) l (1 ys 1 )] y s 1 1 Beta(a 1 + 1, a 2 + )Г( ) ( 1 y s 1 )( 1 21 l(1 y s 1 )) Beta(a 1 +, a 2 + 1)

11 Bayesia estimatio ad preditio for a mixture a 2 γy s a 2 1 x 1 1 i Г( 11 + ) 1 i ) [ 1 l i 1)] 21 Г( ) (1 y s a 2 )[ 1 22 γ l(1 y s a 2 )] Beta(a 1, a ) Г( 11 ) +1 2 γ +1 a y 2 1 s x 2 1 i Г( ) i 2 )(1 y a s 2 )[ 1 γ l i 2 ) + l( 1 y a s Colusios )] I this paper a Bayesia estimatio of the vetor of parameters, θ, of the MEKum distriutio is studied uder the squared error loss futio ad ojugate priors. Moreover, the Bayesia preditive desity futios of the oservatios from the MEKum distriutio are otaied whe osiderig two ases. }. Referees [1] J. Aithiso ad I.R. Dusmore, Statistial Preditio Aalysis, Camridge Uiversity Press, Camridge, UK, [2] A.A. ALgfary, O Fiite Mixture of Expoetiated Kumaraswamy Distriutios, Master's thesis, Kig Adulaziz Uiversity, Jeddah-Kigdom of Saudi Araia, [3] E.K. AL-Hussaii ad Z.F. Jahee, Bayesia estimatio of the parameters, reliaility ad failure rate futios of the Burr type XII failure model, J. Statist. Comput. Simul., 41 (1992), [4] E.K. AL-Hussaii, Bayesia preditive desity of order statistis ased o fiite mixture models, J. Statist. Pla. Ifer., 113 (2003), [5] B. S. Everitt ad D. J. Had, Fiite Mixture Distriutios, Spriger, Lodo, New York, [6] S.C. Flether, K. Poamalam, Estimatio of reservoir yield ad storage distriutio usig momets aalysis, Joural of Hydrology, 182 (1996),

12 508 Samia A. Adham ad Afal A. ALgfary [7] A. Gaji, K. Poamalam, D. Khalili ad M. Karamouz, Grai yield reliaility aalysis with rop water demad uertaity, Stohasti Evirometal Researh ad Risk Assessmet, 20 (2006), o. 4, [8] S. Geisser, Preditive Iferee: A Itrodutio, Lodo, Chapma ad Hall, [9] Z.F. Jahee, Bayesia preditio uder a mixture of two-ompoet Gompertz lifetime model, Test, 12 (2003), o. 2, [10] M. Joes, Kumaraswamy s distriutio: A eta-type distriutio with some trataility advatages, Statistial Methodology, 6 (2009), o. 1, [11] P. Kumaraswamy, A geeralized proaility desity futio for douleouded radom proesses, Joural of Hydrology, 46 (1980), o. 1, [12] A. J. Lemote, W. Barreto-Souza ad G. M. Cordeiro, The expoetiated Kumaraswamy distriutio ad its log-trasform, Brazilia Joural of Proaility ad Statistis, 27 (2013), o. 1, [13] G. MLahla ad K. Basford, Mixture Models: Iferee ad Appliatios to Clusterig, Marel ad Dekker I., New York, [14] A. Seifi, K. Poamalam ad J. Vlah, Maximizatio of maufaturig yield of systems with aritrary distriutios of ompoet values, Aals of Operatios researh, 99 (2000), o. 1-4, [15] V. Sudar ad K. Suiah, Appliatio of doule ouded proaility desity futio for aalysis of oea waves, Oea egieerig, 16 (1989), o. 2, [16] D.M. Titterigto, A.F.M. Smith, ad U. E. Makov, Statistial Aalysis of Fiite Mixture Distriutios, Joh Wiley ad Sos, New York, Reeived: Novemer 15, 2016; Pulished: Deemer 27, 2016