Global Joural of Pure ad Aled Mathematcs. ISSN 973-768 Volume 2, Number (26),. 95-964 Research Ida Publcatos htt://www.rublcato.com Beta Exoetated Mukherj-Islam Dstrbuto: Mathematcal Study of Dfferet Proertes Sabr Al Sddqu Deartmet of Mathematcs & Sceces, CAAS, Dhofar Uversty, Salalah, Sultaate of Oma. Shradha Dwved Ex-Lecturer Deartmet of Mathematcs, FP, Dhofar Uversty, Salalah, Sultaate of Oma. Peeyush Dwved Busess Studes, Salalah College of Techology, Salalah, Sultaate of Oma. Masood Alam Deartmet of Mathematcs, FP, Sulta Qaboos Uversty, Muscat, Sultaate of Oma. Abstract Mukherj-Islam dscussed a dstrbuto for the study of relablty through falure data, that dstrbuto s useful relablty aalyss. The same dstrbuto s used here to develo, a ew beta exoetated dstrbuto wth the ame Beta-Exoetated Mukherjee-Islam dstrbuto. It s a geeralzed form of the Mukherj-Islam dstrbuto. I ths artcle we have studed dfferet mathematcal roertes detal. We have obtaed quatle, momets, momets geeratg fucto, characterstc fucto of the dstrbuto ad the df of r th order statstcs. We have also obtaed maxmum lkelhood estmates of all arameters of the ew dstrbuto. A relato amog three shae arameters of the dstrbuto s also obtaed, whch gves a vew how oe shae arameter vares accordg to other shae arameter. I the last we have obtaed R ey ad q-etroes to measure the dsorder of the system. Keywords: Beta Exoetated Mukherj-Islam Dstrbuto (BEMI), Beta fucto, Momet geeratg fucto, Characterstc fucto, r th order statstcs, maxmum lkelhood estmato, R ey ad q-etroes.
952 Sabr Al Sddqu et al. Itroducto Mathematcal study of theoretcal robablty dstrbutos has bee a choce of may researchers, a umber of research workers have worked o develog ew dstrbutos to be used dfferet felds of alcatos real lfe data, few of them are, Webull (95), Folks ad Chkara(978), Mukherj ad Islam (983), Sddqu et al. (992, 994, 995), Cha (27), Madel ad Yaakov(2). Eugee et. al (22) oeered the class of beta geerated robablty dstrbutos. Eugee et al (22) used beta dstrbuto wth shae arameter α ad β to develo the beta geerated dstrbuto. The cumulatve dstrbuto fucto (cdf) of a beta geerated radom varable X s defed as, F(x) = G(x) b(t)dt Where b(t) s the robablty desty fucto (df) of the beta radom varable ad F(x) s the cdf of ay radom varable X. Eq. () ca be exressed the form, F(x) = G(x) B(a,b) wa ( w) b dw (2) Where a> ad b > are two addtoal arameters to troduce skewess ad to vary tal weght. Here, B(a, b) = w a ( w) b dw, s a beta fucto. Probablty desty fucto (df) of ths fucto wll be; g(x) f(x) = B(a,b) F(x)a { F(x)} b (3) If x s dscrete, the robablty mass fucto s gve by g(x) = G(x) G(x ) (4) The mortat roerty of ths class s that t s a geeralzato of the dstrbuto of order statstcs for the radom varable X wth cdf F(x) as rovded by Eugee et al. (22) ad Joes (24). Eugee el al. (22) defed ad studed ormal dstrbuto, o the bass of that Beta Frechet (Nadaraja ad Guta, 24), Beta Webull (Famoye et al. 25), Beta-Pareto (Aksete et al. 28), Beta-Brbaum-Sauders (Cordero ad Leomote, 2), ad Beta-Cauchy (Al-Showarbehetal, 22) aeared ths class. We have tred to elarge the class of exoetated dstrbutos ad develoed here a ew dstrbuto wth ts mortat roertes.. Short Detals of Mukherj-Islam Dstrbuto The cdf of Mukherj-Islam Dstrbuto (983) of a radom varable X s gve by, G(x) = x, where, >, x > Or = otherwse (5) Here ad are scale & shae arameters, the df of above cdf (5) wll be, ()
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 953 g(x) = x, Where,, >, x > (6) The relablty fucto R(x) ad hazard rate fucto h(x) of the dstrbuto are; R(x) = ( x ), Where,, >, x >, (7) h(x) = x ( x ), 2. Beta Exoetated Mukherj-Islam Dstrbuto (BEMI) Now we defe the cdf of the ew dstrbuto Beta Exoetated Mukherjee-Islam dstrbuto, F(x) = x B(a,b) wa ( w) b dw Where a, b,, >, ad x >, The df wll be from equ (6) x f(x) = )a x B(a,b) (x ( )b, x > () azard rate fucto wll be, h(x) = f(x) = x x ( a ) ( x b ) F(x) B(a,b)I x [ ] (a,b) where, F(x) = I G(x) (a, b) ad I y (a, b) = B(a,b) wa ( w) b dw The robablty desty fucto equato () does ot volve ay comlcated fucto. If X s a radom varable wth df (), we ca wrte X~BEMI(a, b,, ) We foud BEMI dstrbuto wth same secal case as follows If a = b =, we get back the ma Mukherjee-Islam dstrbuto. Now, comg sectos, we demostrate that BEMI desty fucto ca be exressed as a lear combato of the Exoetated Mukherjee-Islam dstrbuto. Result of above desty fucto s mortat gvg dfferet mathematcal roertes of BEMI dstrbuto. I secto 3 we dscussed some mortat statstcal roertes of BEMI as quatle, momets ad momet geeratg fucto ad the dea of order statstcs. Secto 4 deals wth the maxmum lkelhood estmates of the four arameters. I secto 5 the relatosh amog the arameters s show ad secto 6 deals wth the etroes. (8) (9) ()
954 Sabr Al Sddqu et al 2. Exaso for the Desty Fucto: Here, we wll show the two results of the cdf of BEMI dstrbuto, deedg o two cases: Case : f b s a real o-teger. It follows By Bomal exaso we kow ( z) b = ( ) j ( b j= x F(x) = B(a, b) wa ( w) b dw j ) Where z < Alg equato (2) cdf we get F(x) = (a+b) a j= z j = ( )j b j! (b ) zj (2) x F(x) = B(a, b) wa ( )j ᴦb j! (b ) wj dw j= x F(x) = b B(a, b) ( )j w j (b ) a+j dw ( ) j j= ( ) j a+j j! (b )(a+j) x Where B(a, b) = (ᴦa)(ᴦb), a beta fucto. ᴦ( a+b) Equato (3) shows that cdf of BEMI dstrbuto ca be exressed as a fte weghted sum of MI dstrbuto. Result ca be exressed as F(x) = ᴦ(a + b) ᴦa ( ) j G(x,,, (a + j)) j= Case 2:Ifb s a teger, Alyg bomal exaso cdf of BEMI we get F(X) = b B(a,b) (b j ) ( )j (a+j) (3) j= (4) Now, equato (4) holds the same roerty lke (3), but here sum s fte. If, we follow the same exaso method for df of BEMI we get two results, deeds o two cases, same as above. Whe b s a real o-teger:
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 955 f(x) = x B(a, b) G(x,, (a )) ( )j j! ᴦ(b j) G(x,, j), Where, x >. (5) 2. Whe b s a teger: f(x) = x B(a,b) b j= G(x,, (a )) j= (b ) ( )j G(x,, j), x > (6) j 3. Statstcal Proertes Statstcal roertes clude study of quatle fucto, momets ad mgf of BEMI dstrbuto. 3. Quatle Fucto: Quatle fucto x q s defed as F(x q ) = I x (a, b) [ ] (x q ) BEMI = F (u) 3.. Momets: Momets are used to study the tedecy, dserso, skewess ad kurtoss, whch are the most mortat features ad characterstc of a dstrbuto. If X be a radom varable wth desty fucto x a f(x) = B(a, b) (x ) ( x b ), x > The r th momet of the BEMI dstrbuto s gve by, = By usg bomal exaso, we have, j= = μ r (x) = E(x r ) = x r f(x)dx B(a, b) xr+ ( x a ) ( x b ) B(a, b) xr+ ( x ) a = B(a, b) (b ) ( ) j x r+ ( x a+j j ) dx ( b ) j j= dx ( ) j ( x ) j dx = B(a, b) (b ) ( ) j x r+ ( x a+j x r+(a+j) j ) (a+j ) dx j=
956 Sabr Al Sddqu et al μ r (x) = E(x r ) = ( b ) ( ) j r j r + (a + j), B(a, b) j= Or we ca wrte, μ r (x) = E(x r ) = t j r (7) Where, t j = ( b ) ( ) j r j + (a + j), B(a, b) j= Eq. (7) gves the momets of BEMI dstrbuto. By uttg r= ad r=2 Eq. (7), we ca get the mea ad varace by the followg way, μ = t j ad μ 2= t j 2 where, So mea = t j ad t j = ( b ) ( ) j r j + (a + j), B(a, b) j= Varace = μ 2 (μ ) 2 = t j 2 t j 2 2 = t j 2 ( t j ) Based o the frst four momets of the BEMI dstrbuto, the measures of skewessa(φ) ad Kurtoss K(Φ) of BEMI dstrbuto ca be obtaed as: A(Φ) = μ 3 () 3μ ()μ 2 ()+2μ 3 () [μ 2 () μ 2 3 ()] 2 Ad K(Φ) = μ 4 () 4μ ()μ 3 ()+6μ 2 ()μ 2 () 3μ 4 () [μ 2 () μ 2 ()] 2 3... Momets Geeratg Fucto(mgf) Momet geeratg fucto s a mortat statstcal roerty of a robablty dstrbuto as t rovdes the bass of a alteratve route to aalytcal results comared wth comared to workg drectly wth df or cdf. If some radom varable X has BEMI dstrbuto, the the momet geeratg fucto wll be accordg to the defto of Mgf, as follows: M x (t) = E(e tx ) = e tx f BEMI (x)dx, where,, >, x >, So we wll have Mgf of BEMI dstrbuto M x (t) = E(e tx ) = e tx x a B(a, b) (x ) ( x b ) = (a+j) B(a, b) (b ) ( ) j e tx x (a+j) dx j j=
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 957 = (a+j) B(a,b) j= (b j ) ( ) j [(a + j), t] (a+j) ( t) (a+j) M x (t)= [(a + B(a,b) (b j ) j= ( ) j j), tx)] ( t) (a+j) (8) Whch s the desred Momet geeratg fucto. 3... Characterstc fucto (cf) Φ x (t) = E(e tx ) = e tx f BEMI (x)dx Where,, >, x >, Above defto of characterstc fucto gves us the result as ; Φ x (t)= B(a,b) (b j ) j= ( ) j [(a + j), tx)]( t) (a+j) (9) 3... Dstrbuto of order statstcs Let X, X 2. X be a smle radom samle from BEMI dstrbuto wth df ad cdf gve by Eqs. (9) ad () cosecutvely, x F(x) = B(a, b) wa ( w) b dw x a f(x) = B(a, b) (x ) ( x b ), x > Let X, X 2. X deote the order statstcs obtaed from ths samle. The robablty desty fucto of X r: s gve by f r: (x, ) = [F(x, B(r, r+) )]r [ F(x, )] r f(x, ) (2) Where F(x, ) ad f(x, ) are the cdf ad df of BEMI order statstcs. Here, < F(x, ) < for x >, bomal seres exaso of [ F(x, )] r wll gve us, [ F(x, )] r =( r j ) [F(x, )]j j= ( ) j Puttg ths Eq. (2) we wll get, f r: (x, ) = B(r, r + ) [F(x, )]r+j f(x, ) Substtutg cdf ad df of BEMI to above equato we ca exress k th ordary momet of the r th order statstcs X r: say E(X k r: )as a lear combato of the k th momets of the BEMI dstrbuto wth dfferet shae arameter. Therefore, the measures of skewess ad kurtoss of the dstrbuto of X r: ca be calculated.
958 Sabr Al Sddqu et al 4. Estmato 4. Maxmum Lkelhood Estmato (mle): Let X, X 2. X be a radom samle of sze from BEMI (,, a, b) the lkelhood fucto for the vectors of arameters, ɸ=(,, a, b) ca be wrtte as, L(x, Φ) = f(x, Φ) x a = B(a, b) (x ) ( x ) b = ( x ) [B(a, b)] ( (a ) ) x (a ) ( x b ) Takg log-lkelhood of above for the vector of arameters ɸ=(,, a, b), we get, log L = log log + log (a + b) log a log b (a ) log + log x a + (b ) log ( x ) The log-lkelhood ca be maxmzed ether drectly or by solvg the o-lear lkelhood equatos obtaed by dfferetatg above equato The comoets of the score vectors s gve by, log L = α + (b ) x [ x ] The mle of wll of be, = [ a [ b x ] ] log L x x log[ ] [ x ] = a log + a log x (b ) (23) The mle of, wll be, = d log L da a log a log x +(b ) x log[ x ] [ x ] (2) (22) (24) = Ψ(a + b) Ψ(a) log + log x (25) Ψ(a) = {Ψ(a + b) log } + log x (26) d log L db = Ψ(a + b) Ψ(b) + log ( x (27) Ψ(b) = Ψ(a + b) + log ( x (28) Where Ψ(. )s a dgamma fucto. ) )
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 959 5. Relato Betwee all three Shae Parameters(, a, b) We used three shae arameters BEMI dstrbuto. Relato betwee all three shae arameters, a, b ca be establshed as follows, The df of BEMI dstrbuto s, x a f(x) = B(a, b) (x ) ( x b ), x > Takg logarthm of both sde of above equato of df of BEMI dstrbuto, log f(x) = log + ( ) log x log log B(a, b) +(a )[log x log ] (b )[log x log ] (29) Takg logarthm of both sdes ad equatg t to zero we get, d log L ( ) (a ) (b ) = + = dx x x x = (3) (+a b) Above relato shows how three shae arameters vares accordg to each other. 6. R ey ad q-etroes 6. R ey Etroy: The etroy of a radom varable X s a measure of the ucerta varatos the system. The R ey etroy s defed by, I R (δ) = log I(δ) (3) δ Where I(δ) = f δ (x) dx forδ > adδ R f δ (x) = δ (x ) δ δ(a ) ( ) δ B δ (a, b) (x ) ( x δ(b ) ), x > = gδ (x) * B δ (a,b) G(x)δ(a ) ( G(x)) δ(b ) Alyg the ower seres used before, we obta δ(b ) f δ (x) = gδ (x) B δ (a, b) G(x)δ(a ) ( ) k δ(b ) ( ) (G(x)) k δ(b ) = gδ (x) B δ (a, b) δ(b ) ( )k ( ) (G(x)) k = gδ (x) B δ (a, b) δ(b ) ( )k + k ( ) k k 2 = k = k k k +δ(a ) k ( k k 2 2 + δ(b ) ) (G(x)) k 2
96 Sabr Al Sddqu et al Now, Where, Now, I(δ) = f δ (x) R f δ (x) = v k,k 2 g δ (x)(g(x)) k 2 v k,k 2 = B δ (a, b) ( ) k δ(b ) + k ( ) k k 2 = k = dx = v k,k gδ (x)(g(x)) k2 2 ( k 2 + δ(b ) k 2 ) Sce for BEMI dstrbuto lmt for x vares lke x >. Now uttg values of cdf ad df of BEMI dstrbuto Eq. (32), we get f δ (x) R dx = v k,k 2 f δ (x) dx = v k,k 2 R δ δ (x ) δ ( ) δ B δ (a, b) ( x ) k2 (k2+δ) B δ (a, b) x(k 2+δ) δ dx I(δ) = f δ (x) dx = v k,k 2 R B δ (a, b) (k 2 + δ) δ + Hece R ey etroy reduces to I R (δ) = δ log v k,k 2 δ B δ (a,b) δ Where, v k,k 2 = B δ (a,b) k k ( )k + k ( δ(b ) 2 = = 6.. q-etroy: q-etroy say H q (f) s defed by, δ δ (32) (k 2 +δ) δ+ (33) k ) ( k 2+δ(b ) k 2 ) H q (f) = q log I q(f) Where I q (f) = f q (x) dx for q > ad q R Same lke above Eq. (33), we ca easly obta, H q (f) = q log q q v k,k 2 B q (a, b) (k 2 + δ) q + Where, v k,k 2 = B q (a,b) k k ( )k + k ( q(b ) 2 = = k ) ( k 2+q(b ) k 2 )
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 96 Cocluso I ths aer, we roosed a ew four-arameter Beta Exoetated Mukherjee-Islam (BEMI) dstrbuto. We studed some of ts structural roertes cludg a exaso for the desty fucto ad exlct exressos for the quatle fucto, momets geeratg fucto ad characterstc fucto. We dect a relato betwee three shae arameters to gve a cture of ther varato accordg to each other. The maxmum lkelhood method s emloyed for estmatg the model arameters. Two etroes R ey ad q-etroy are also obtaed. We also obta observed formato matrx. Aedx The elemets of the 4 4 observed formato matrx J(Θ) = {BM m, } = {form, =,, a, b}are gve by, BM = 2 (b ) log ( x BM = a (b ) x log ( x ( x ) BM a = ) ) (b ) (x ) 2 log ( x 2+ ( x 2 ) log x log, BM b = x x [ x 2 ] x + (b ) ( ) 2 ( x ) ( x ) ) log ( x ) ( x ) BM P = a (b ) x log ( x ( x ) x + (b ) ( ) 2 ( x ) ( x ) (b ) (x ) 2 log ( x 2+ ( x 2 ) ) )
962 Sabr Al Sddqu et al BM = a 2 (b )( + ) x +2 ( x ) 2 (b ) (x ( + ) 2 x BM a = BM b =, BM a = log + log x + ( x ) BM a = ; BM aa = Ψ (a + b) Ψ (a) Refereces BM ab = Ψ (a + b) BM b = x log ( x ) BM b = ( x ) BM ba = Ψ (a + b) BM bb = Ψ (a + b) Ψ (b) x + ( x ) [] Aksete, A., Famoye, F. ad Lee, C. :The Beta-Pareto dstrbuto, Taylor& Fracs, (28). [2] Alexeder, C. Cordero, G. M ad Ortega, E. M. M: Geeralsed Beta- Geeralsed dstrbutos, Statstcs & Data Aalyss, Elsever, (22). [3] Cha, J. H, ad M, J. :Study of a Stochastc falure model a radom evromet, Joural of Aled robablty, Vol. 44, No.,. 5-63, (27). [4] Alshawarbeh, E, Lee, C. ad Famoye, F. : The Beta-Cuchy dstrbutos-joural of Probablty ad Statstcal Scece, (22). [5] Famoye, F., Lee, C. ad Olumolade, O. : Joural of Statstcal Theory ad Alcato, (25). [6] Folks, J. F. ad Chkara, R. S. :The verse Gaussa dstrbuto ad tsstatstcal alcato-a Revew, Joural of Royal Stat. Soc., Vol. B4,. 263-289, (978). [7] Gauss M. Cordero, Caral Alexeder Ortega, Saraba: Geeralsed Beta- Geeralsed Dstrbutos, HENELEY Uversty of readg, (2). [8] Joes, M. C. : Famles of dstrbutos arsg from dstrbutos of order statstcs; Srger, Volume 3, Issue, -43, (24). [9] Madel, M. ad Rtov Yaakov. :The accelerated falure tme model uder based samlg, Bometrcs, Vol. 66, No. 4.. 36-38, (2). [] Mukheerj, S. P. ad Islam, A. :A fte rage dstrbuto of falures tmes, Naval Research Logstcs Quarterly, Vol. 3,. 487 49, (983). ) 2
Beta Exoetated Mukherj-Islam Dstrbuto: Mamthematcal Studyo 963 [] Eugee, Ncolas, Carl Lee ad Felx Famoye: Beta Normal Dstrbuto ad ts alcatos, ; Taylor-Fracs Ole, Commucato Statstcs-Theory & Methods, Volume 3, Issue 4, 497-52, (22). [2] Nadarajeh, ad Guta, : The Beta-Frechet dstrbuto: East joural of theoretcal Statstcs, (24). [3] Sddqu, S. A. ad Subharwal, Mash: O Mukherjee Islam falure model, Mcroelectro& Relablty, Vol. 32, No-7,. 923-924, (992). [4] Sddqu, S. A., Deok, N., Guta, S. ad Sabharwal, M. : A ew creasg rate falure model for lfe tme data, Mcroelectro Relablty., Vol. 35., No.,. 9-, (995). [5] Sddqu, S. A., Sabharwal, M, Guta, S. ad Balkrsha: Fte rage survval model, Mcroelectro Relablty., Vol. 34., No. 8,. 377-38, (994). [6] Webull, W. : A statstcal dstrbuto of wde alcablty, Joural of Aled Mechacs, Vol. 8,. 293-297, (95).
964 Sabr Al Sddqu et al Authors Profles: Dr. Sabr Al Sddqu Got hs Ph. D. degree from Agra Uversty, Ida (99), resetly workg as Assstat Professor, Dhofar Uversty, Salalah, Sultaate of Oma. Major cotrbuto develog ew dstrbutos, some work o Bayesa aalyss. Mrs. Shradha Dwved Ex lecturer of Mathematcs, FP, Dhofar Uversty, Salalah, Sultaate of Oma. Dr. Peeyush Dwved Got hs PhD Degree from CMJ Uversty, Ida 23, resetly workg as a lecturer Salalah College of Techology, Salalah, Sultaate of Oma. Masood Alam Lecturer of Mathematcs, FP, Sulta Qaboos Uversty, Muscat, Oma.