Bayesian Estimations on the Burr Type XII Distribution Using Grouped and Un-grouped Data

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Russell Miles
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1 Austalia Joual of Basic ad Applid Scics, 5(6: , 20 ISSN Baysia Estimatios o th Bu Typ XII Distibutio Usig Goupd ad U-goupd Data Ima Mahdoom ad Amollah Jafai Statistics Dpatmt, Uivsity of Payam Noo, , Tha I.R of IRAN Abstact: I this pap, w obtaid Baysia stimatos fo th shap paamt of th Bu Typ XII distibutio usig goupd data u-goupd data, ad also cosid latioship btw thm. I Baysia stimatio, w cosid two typs of loss fuctios; th Squad Eo ad Pcautioay loss fuctios which a symmtic ad asymmtic, spctivly. I all cass, w cosidd both poit ad itval stimatios. Ths th poit ad itval stimatios a compad mpiically usig Mot-Calo simulatio. Ky wods: Bu Typ XII distibutio; Goupd data; Pcautioay loss fuctio; Baysia stimatio; Mot-Calo simulatio. INTRODUCTION I vaious filds of scic such as biology, giig ad mdici it is ot possibl to obtai th masumts of a statistical xpimt xactly, but is possibl to classify thm ito itvals, ctagls o disoit substs (Alodat ad Al-Salh (2002; ita (989; Suls ad Padgtt (200; Wu ad Ploff (2005; Pipp ad Ritz (2006. Fo xampl, i lif tstig xpimts, w obsv th failu tim of a compot to th ast hou, day o moth. Data fo which tu valus a ow oly up to substs of th sampl spac a calld goupd data. I gal goupd data ca b fomulatd as follows: Lt X, X2,..., X b a adom sampl fom th dsity f ( x;, x, ad, 2,..., b a patitio of th sampl spac χ ad N th umb of X ' s that fall i fo,2,...,. Th st of pais (, N,...,(, N is calld goupd data. Th goupd data poblm is to us ths data to daw ifcs about th paamt θ. Sic w do t hav complt ifomatio about th sampl, th th will b a loss i th ifomatio du to th goupig. Schvish (995, pag 4, shows th followig IX( IY( E IX Y( Y wh I ( ad I ( a th Fish's ifomatio umb obtaid fom X ad Y, spctivly, ad X Y E IXY ( Y is th coditioal sco fuctio. If w plac Y by th goupd sampl ( N, N2,..., N, th I ( I ( fo all θ, which mas that th ifomatio i th sampl about θ is ducd to I ( X bcaus of goupig. Kuldoff (96 cosidd o-baysia stimatio fom goupd data wh th data com fom omal ad xpotial distibutios. Alodat ad Al-Salh (2000 cosidd th Baysia stimatio fom goupd data wh th udlyig distibutio is xpotial. Alodat t al. (2007 obtaid Baysia pdictio itvals fom goupd data wh th udlyig distibutio is xpotial. Aludaat t al. (2008 obtaid th Baysia ad o-baysia stimatio fom goupd data wh th udlyig distibutio is Bu typ X. Also, Shadoh ad Pazia (200 obtaid th classical ad Baysia stimatio fom goupd ad u-goupd data wh th udlyig distibutio is Expotiatd X Cospodig Autho: Ima Mahdoom, Statistics Dpatmt, Uivsity of Payam Noo, , I.R of IRAN 525

2 Aust. J. Basic & Appl. Sci., 5(6: , 20 Gamma. Th Bu systm of distibutios was costuctd by Bu (942. Th Bu-XII distibutio is quit flxibl as a liftim modl. Th flxibility of th distibutio aiss fom th fact that it has a o-mooto hazad fuctio which mas it appopiat fo pstig th liftim fo may poducts, Solima (2002. Th pobability dsity fuctio of a Bu-XII distibutd adom vaiabl is giv by f x x x ( ( ; (, 0, 0 wh θ is scal paamt. Fo difft valus of θ, distibutio fuctio (.. cd f F ( x; ( x, x 0, 0 is; f( x; ( is a dcasig fuctio. Th cumulativ (2 I this pap, w cosid th u-goupd ad goupd data poblms wh th dsity f ( x; is Bu Typ XII (θ, th quatio (. This pap is ogaizd as follows: I Sctio 2, w obtai th Baysia stimatos of θ, ad costuct th cdibl itval fo θ fom u-goupd data. I Sctio 3, w obtai th Bays stimatos of θ ud th symmtic ad asymmtic loss fuctios usig goupd data. I sctio 4, w coductd a simulatio study to compa th stimatos. Ou coclusio is statd i Sctio 5. Baysia Poit ad Itval Estimatios basd o th U-goupd Data: I this sctio, w gt th Bays stimatos of θ ud th symmtic ad asymmtic loss fuctio ad compa ths stimatos basd o thi ma squad os (MSE's. Also, w pst th cdibl itvals fo θ. Lt X, X2,..., X b a adom sampl fom dsity (. Th lilihood fuctio is giv by L( ( xi i ( (3 I h, w wat to obtai th Bays stimatos of θ ud th impop pio distibutio. Cosid th impop pio distibutio (i.. 0 ( d. fo θ of th fom ( ; 0,, 0 Notic that this pio distibutio is th l of a Gamma distibutio wh α > 0. owv, such a stictio o α is ot cssay ad dcass th flxibility of th sultig stimato. Whas ( x L( (, thfo th postio distibutio of θ is x xi i ( ( xp log( xi i xp ( t t log ( x wh. Th postio distibutio of θ is pop wh + α > 0, i.. i i 526

3 Aust. J. Basic & Appl. Sci., 5(6: , 20 x ~ (,. I this cas, th Bays stimato of θ ud Squad Eo loss fuctio is th t postio ma, i.. ˆBS (4 T T log ( x wh. i i Now, w obtaid th Bays stimato of θ ud a asymmtic loss fuctio. Nostom (996 itoducd a altativ asymmtic pcautioay loss fuctio ad also pstd a gal class of pcautioay loss fuctio with quadatic loss fuctio as a spcial cas. Ths loss fuctios appoach ifiitly a th oigi to pvt udstimatio ad thus givig cosvativ stimatos, spcially wh low failu ats a big stimatd. Ths stimatos a vy usful wh udstimatio may lad to sious cosqucs. A vy usful ad simpl asymmtic pcautioay loss fuctio is ˆ 2 ˆ ( L(, ˆ Th Bays stimato ud this asymmtic loss fuctio is dotd by solvig th followig quatio, ˆBP (5 ad may b obtaid by ˆ 2 BP E ( x E ( x (6 Not: this spcial cas of th Pcautioay loss fuctio (5 ad th Etopy loss fuctio a th sam (fo th mo dtails s Nostom, (996, Pazia ad Shadoh (200. As said, ud th gamma pio distibutio, i.. ( ( ; 0, 0, 0 (7 th postio dsity of θ is gamma with th shap ad scal paamts as α + ad /(β+t, spctivly, thfo t E( x ( c, th Bays stimato of θ with spct to th pcautioay loss fuctio ud th gamma pio distibutio is as follows ( ( ˆBP T (8 wh T log ( x i i Th Baysia aalog to th cofidc itval is calld a cdibility itval. I gal, th itval Lx (, Ux ( is a 00( % cdibility itval fo θ if 527

4 Aust. J. Basic & Appl. Sci., 5(6: , 20 U( x P L ( x U ( x ( x d L( x 2 Sic x ~ (,, thby 2 ( T~ (2(, thus T 2 2 2( ( 2( ( P ( T 2( T Thfo, a 00( % Baysia cdibility itval fo θ is Lx (, Ux ( wh 2 2( ( Lx ( 2 2 log( xi i (9 ad 2 2( ( U( x 2 2 log( xi i (0 Baysia Estimatios Basd o th Goupd Data: I this sctio, w obtai th Bays stimatos of θ wh th data giv i Goups. Lt X, X2,..., X b a adom sampl fom Bu Typ XII (θ. Assum that th sampl spac of f ( x; is patitiod ito + qually-spacd itvals as follows. Lt I [(,,,..., ad I [,, 0. If N dots th umb of X ' s that fall i I,,2,...,, th N... N. Lt P P( P( X I P ( X ( ( ( ( fo,..., ad P ( ( P P X. A A A If w lt A log (, th P, fo,..., ad P. So th dsity of ( N, N2,..., N is giv by th multiomial distibutio as follows: 528

5 Aust. J. Basic & Appl. Sci., 5(6: , 20! f ( ; P... P!...! C A A A (9 wh C is a omalizig costat. I cotiu, w fid th Bays stimatos of θ ud th Gamma pio distibutio, quatio (7, with spct to th squad o ad pcautioay loss fuctios wh th data giv i Goups. Usig th Biomial thom, w wit th lilihood fuctio of th goupd data, quatio (9, as follows: A A A f( ; C ( 0... (,..., C ( 0 0 (0 wh (,..., A ( A A Combiig th lilihood ifomatio with th pio ifomatio yilds th postio distibutio of θ giv, i.., ( 0 f( ; ( f ( ; ( d ( 0 0 so w gt... (,..., ( ( ( ( ( wh (,...,. 529

6 Aust. J. Basic & Appl. Sci., 5(6: , 20 Th Baysia stimat of θ with spct to th squad o loss fuctio fom th Goupd data, say ˆBSG, is th postio ma, i.., ˆ BSG E (... ( ( 0 0 (2 Th Baysia stimat of θ with spct to th Pcautioay loss fuctio fom th Goupd data, say ˆBPG, is obtaid as follows: ˆ BPG E ( E ( ( ( ( (3 I xt sctio, w compa all ths stimatos i tms of Biass ad Ma Squad Eos (MSE's, usig Mot-Calo simulatio. Simulatio Study: Th stimatos ad a th Baysia stimatios of th shap paamt of th Bu Typ XII ˆBS ˆBP distibutio obtaid fom th u-goupd data. Mawhil, ad a th Bays stimatos of θ ud th Squad-o ad Pcautioay loss fuctios, spctivly, basd o th Goupd data. W also us th otatio BCL to dot th 95% Baysia Cdibility Lgth fo θ basd o th u-goupd data. Ou mai aim is to compa ths stimatos i tms of Biass ad MSE's. Th MSE's of th stimatos a valuatd basd o a Mot-Calo simulatio study of 000 sampls. W gatd ths sampls fom Bu Typ XII distibutio with θ = 2.5 by usig MATLAB. Th simulatio study was caid out with sampl siz =0, 5, 20, 25, 30 ad 50. W put ths sampls ito fiv itvals (=4 with δ=. Pio paamts a abitaily ta as α = 2 ad β =. All th sults a summaizd i Tabl. Coclusio: I this pap w obtaid Baysia stimatos fo th shap paamt of th Bu Typ XII distibutio basd o th goupd ad u-goupd data. W cosidd both poit ad itval stimatos. W divd th Bays stimatos ud symmtic ad asymmtic loss fuctios. Ou obsvatios about th sults a statd i th followig poits: Tabl shows that th Bays stimats ud th pcautioay loss fuctio hav th smallst stimatd MSE's as compad with th Bays stimats ud th Squad Eo loss fuctio. Ths a tu fo both u-goupd ad goupd data. It is immdiat to ot that MSE's dcas as sampl siz icass. O th ˆBSG ˆBPG 530

7 Tabl : Aust. J. Basic & Appl. Sci., 5(6: , 20 Biass ad Ma Squad Eos (MSE's of th Poit Estimats, ad Lgth of th Itval Estimat fom th u-goupd ad goupd data, wh =4, δ=, θ=2.5, α=2, β= ad τ=0.05 (Upp valu i ach cll fs to MSE ad low valu to Bias. ˆBS ˆBP BCL : th 95% Baysia Cdibility Lgth. BCL oth had th Bays stimats a ovstimatio. This is tu fo both u-goupd ad goupd data. Matim, th cofidc itval wo quit wll ulss th sampl siz is small, ad this is tu fo both u-goupd ad goupd data. Whas, th pfomac of th Bays stimats ud Pcautioay loss fuctio is btt tha th st, thus w suggst to us Bays appoach ud Pcautioay loss fuctio fo stimatig th shap paamt of Bu Typ XII distibutio ad this is tu fo both u-goupd ad goupd data. I gal, th Bays stimatos yild of th Goupd data wo vy wll. Thfo, w ca us th stimatos pstd wh th data giv i Goups, fo xampl i lif tstig xpimts. REFERENCES Alodat, M.T. ad M.F. Al-Salh, Baysia stimatio usig goupd data with applicatio to th xpotial distibutio. Soochow J. of Mathmatics, 26: Alodat, M.T., K.M. Aludaat ad T.T. Alodat, Baysia pdictio itvals fom goupd data: xpotial distibutio. Abhath Al-yamou, (Accptd. Aludaat, K.M., M.T. Alodat ad T.T. Alodat, Paamt stimatio of Bu typ X distibutio fo goupd data. Joual of Applid Mathmatical Scics, 2(9: Amad, A. ad B. Ayma, Itval stimatio fo th scal paamt of Bu typ X distibutio basd o goupd data. Joual of Mod Applid Statistical Mthods, 3: Bu,.W., 942. Cumulativ Fqucy Fuctios. A. Math. Stat., 3: ita, D., 989. Ifc fom goupd data: a viw (with discussio. Statistical Scics, 4: Kuldoff, 96. Estimatio fom goupd ad patially goupd sampls. Joh Wily, Ic. Nw Yo. Nostom, J.G., 996. Th us of pcautioay loss fuctio i is aalysis. IEEE Tas. Rliab., 45(3: Pazia,. ad A. Shadoh, 200. Compaiso of LINEX ad pcautioay Bays stimatos o th gamma distibutio usig csod data. Accptd i Joual of Statistics & Maagmt Systms. Pipp, C.B. ad C. Ritz, Chig th goupd data vsio of Cox modl fo itval-goupd suvival data. Scadiavia Joual of Statistics, 0: Schvish, M.J., 995. Thoy of Statistics. Spig-Vlag, Ic. Nw Yo. Shadoh, A. ad. Pazia, 200. Classical ad Baysia Estimatios o th Expotiatd Gamma Distibutio usig Goupd ad U-Goupd Data, Itatioal Joual of Statistics ad Systms, 5(2: Solima, A.A., Rliability Estimatio i a Galizd Lif Modl with Applicatio to th Bu-XII. IEEE Ta. o Rlibility, 5: Suls, J.G. ad W.J. Padgtt, 200. Ifc fo liability ad stss-lgth fo a scald Bu typ X distibutio. Liftim Data aalysis, 7: Wu, X. ad J.M. Ploff, "Chia s icom distibutio: Rviw of Ecoomitics ad Statistics, 87: Yamohammadi, M. ad. Pazia, 200. Miimax Estimatio of th Paamt of th Bu Typ XII Distibutio, Austalia Joual of Basic ad Applid Scics, 4(2: ˆBSG ˆBPG 53

Applied Mathematical Sciences, Vol. 2, 2008, no. 9, Parameter Estimation of Burr Type X Distribution for Grouped Data

Applied Mathematical Sciences, Vol. 2, 2008, no. 9, Parameter Estimation of Burr Type X Distribution for Grouped Data pplied Mathematical Scieces Vol 8 o 9 45-43 Paamete stimatio o Bu Type Distibutio o Gouped Data M ludaat M T lodat ad T T lodat 3 3 Depatmet o Statistics Yamou Uivesity Ibid Joda aludaatm@hotmailcom ad