On the Beta Mekaham Distribution and Its Applications. Chukwu A. U., Ogunde A. A. *

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1 Amrca Joural of Mathmatcs ad Statstcs 25, 5(3: DOI:.5923/j.ajms O th Bta Mkaham Dstruto ad Its Applcatos Chukwu A. U., Ogud A. A. * Dpartmt of Statstcs, Uvrsty Of Iada, Dpartmt of Mathmatcs ad Statstcs, Fdral Polytchc Ado-Ekt, Ngra Astract I ths papr, a w fv-paramtr gralzd vrso of th Gomprtz-Mkaham dstruto calld Bta Gomprtz-Mkaham (BGM dstruto s g troducd. It cluds som wll-kow lftm dstrutos as spcal su-modls. Th w dstruto s qut fll ad ca hav a dcrasg, crasg, ad athtu-shapd falur rat fucto dpdg o ts paramtrs makg t ffctv modlg survval data ad rlalty prolms. Som comprhsv proprts of th w dstruto, such as closd-form prssos for th dsty, cumulatv dstruto, hazard rat fucto, th rth ordr momt, momt ar provdd. Th mamum lklhood stmato of th BGM paramtrs as wll as osrvd Fshr s formato matr otad. At th d, ordr to show th capalty of BMG ovr ts su modls, a applcato to a ral datast llustrats ts pottalty. Kywords Bta grator, Gomprtz-Mkaham dstruto, Mamum lklhood stmato. Itroducto Modllg of trrlatoshp amog aturally occurrg phoma s mad possl y th us of dstruto fucto ad thr proprts. Thus a umr of statstcal dstrutos fucto ad thr gralzato hav proposd ad dfd th ltratur, most of whch ar foud to usful studyg aturally occurrg phoma partcularly whr th varals assur valus tw ad. May usful proprts of such dstruto ar rvald y trasformato of radom varals whl aothr approach s th covoluto tchqu whch s rarly usd. Amog th usful dstrutos applcal to ral lf data ad th logt of ta dstruto ar th ta-parto (Akst ad Low, 28; ta-laplac (Kozuowsk ad Nadarajah (28; ta-wull (Famoy, t al 25; ta-normal (Eug t al, 22; ta-potal (Nadarajah ad Kotz, 26; ad ta-guml (Nadarajah ad Kot, Gomprtz Mkaham Dstruto Th Gomprtz-Makham law stats that th dath rat s th sum of a ag dpdt compot (th Makham trm ad a ag-dpdt compot (th Gomprtz fucto whch crass potally wth ag. I a protctd vromt whr tral causs of dath ar rar (laoratory codtos, low mortalty coutrs, tc., th ag-dpdt mortalty compot s oft * Corrspodg author: Dz95@yahoo.com (Ogud A. A. Pulshd ol at Copyrght 25 Sctfc & Acadmc Pulshg. All Rghts Rsrvd glgl. I a protctd vromt whr tral causs of dath ar rar (laoratory codtos, low mortalty coutrs, tc., th ag-dpdt mortalty compot s oft glgl. I ths cas th formula smplfs to a Gomprtz law of mortalty. I 825, Bjam Gomprtz proposd a potal cras dath rats wth ag. Th Gomprtz Makham law of mortalty dscrs th ag dyamcs of huma mortalty rathr accuratly th ag wdow from aout 3 to 8 yars of ag. At mor advacd ags, som studs hav foud that dath rats cras mor slowly a phomo kow as th lat-lf mortalty dclrato ut othr studs dsagr. A radom varal X s dstrutd Gomprt-Makham f ad oly ts pdf satsfs th low λ ( f( + λ ( >, >, λ > Its cumulatv dsty fucto s λ 2 F( 3. Th Proposd Bta Gomprtz Mkaham Dstruto Th ta dstruto has wdly usd, appld, powrful ad wll kow as a proalty dstruto to admstr svral kds of prolms rlalty aalyss. I rct yars, dvlopmt focus o w tchqus for uldg magful dstrutos, cludg th us of th logt of ta (th lk fucto of th ta gralzd (2

2 38 Chukwu A. U. t al.: O th Bta Mkaham Dstruto ad Its Applcatos dstruto pord y Jos (24. Th logt of ta troducd y Jos has proalty dsty fucto (pdf. f( g( F( F( (3 Ba [ ] a [ ], whr ff( s th pdf of XX. If w put ( ad (2 (3 w hav λ a ( + ( λ λ Ba, λ g( ths smplfs to a λ ( + λ λ a, g a,,,, λ >... (4 3.. Statstcal Proprts I ths scto th statstcal proprts of th proposd dstruto wll vstgatd. Ths clud asymptotc havour, vrfcato of tru proalty fucto, cumulatv dsty ad hazard rat fucto. Asymptotc Bhavour W sk to vstgat th havour of th modl Equato (4 as If a, w hav a λ ( + λ λ ϕ ( a, lm ( λ lm + λ λ λ lm lm λ + ( + λ Ivstgato of Propr pdf Hr, w sk to vrfy hr whthr th proposd dstruto tgrat to uty Lt ( λ a, λ λ ( λ M a + d ( λ d ( λ

3 Amrca Joural of Mathmatcs ad Statstcs 25, 5(3: d d ( λ + ( λ + ( λ M a ( M ( λ M a, + λ ( a, M ( M a Lt P M, M P ( a, ( a, dp, dp a ( p p ( dp a (. QED.. p p dp ( Cumulatv Dsty Fucto Th dstruto fucto of th ta Gomprt Mkaham dstruto s drvd as follows Ths rducs to t a λt t ( t λ a, t P X + dt ( a, a λ ( As P M ad M Fally, ( p p dp ( P, a, ( a, ( λ {, a, } G ( a, (5

4 4 Chukwu A. U. t al.: O th Bta Mkaham Dstruto ad Its Applcatos 3.3. Hazard Rat Fucto Th hazard rat fucto s otad usg g h G a λ ( λ ( λ ( h ( + λ a, h ( ( λ, a, ( λ ( a, a λ ( λ ( λ ( + λ ( ( a, a, Th aov s Bta-Gomprtz Mkaham Mortalty Modl Wh a, w hav h ( h ( ( λ + ( λ ( λ λ + (7 whch s th hazard of Gomprtz-Mkaham dstruto, a Gomprtz-Mkaham Mortalty. (6 4. Estmato ad Statstcal Ifrc θ Lt, 2,... a radom varal dstrutd accordg (4 th lklhood fucto of vctor of paramtrs gv as ( a,,,, λ s λ λ L ( θ λ + ( a, Th ts log-lklhood fucto s λ ( (8 log l( θ log + λ λ + a log log a, Ths s smplfd as y otg that a

5 Amrca Joural of Mathmatcs ad Statstcs 25, 5(3: ( a, Γ( a Γ( Γ ( a + ( λ log log l θ λ λ a log + + log Γ( a Ths s th dffrtatd wth rspct to a,,, ad λ. If w lt φ ( K 5. Applcato log Γ ( + log Γ ( a+ (9 ( log L Γ( a Γ ( a+ a Γ a Γ a+ λ θ log + ( log L( θ Γ( Γ ( a+ λ + Γ( Γ ( a+ ( λ λ λ log L( θ + a λ λ + λ ( ΓK w hav Γ K λ log L θ log a + a+ a φ φ log L( θ λ φ + φ a+ λ λ λ λ + + a log L( θ λ ( I ths scto, w ft th Bta Gomprtz Mkaham dstruto ad ts su modls to a survval data st that was aalyzd y Fgl ad Zl(965. Th data rprst th survval tms, wks, of 33 patts suffrg from acutmylogous Lukama. Th data ca also foud at lrary SMIR of th R program( ar th followg: 65, 56,, 34, 6, 8, 2, 4, 39, 43,56, 26, 22,,, 5, 65, 56, 65, 7, 7, 6, 22, 3, 4, 2, 3, 8, 4, 3, 3, 4, 43. Th dstruto of th data s skwd to th rght. Blow s th summary of rsults (2

6 42 Chukwu A. U. t al.: O th Bta Mkaham Dstruto ad Its Applcatos Tal. Dscrptv Statstcs o Falur Tm o Codtoal Systm M Q Q 2 Ma Q 3 Ma Var skwss Shapro-Wlk ormalty tst data: Acutdata W.845, p-valu Tal 2. Log lklhood ad Alkak Iformato Crtro of Modls Dstruto Log-lklhood AIC Bta Gomprtz-Mkaham (Proposd Gomprtz-Mkaham Gomprtz Epotal

7 Amrca Joural of Mathmatcs ad Statstcs 25, 5(3: Coclusos Th Bta Gomprtz-Mkaham dstruto was dfd ad studd ths work. Varous proprts of th dstruto wr dscussd. Ths clud th asymptotc havors, Cumulatv dsty fucto, hazard fuctos amog othrs. Also w dscussd th stmato of paramtrs y th mthod of mamum lklhood. Ral lf applcato dcats that w dstruto apart from g mor fll has ttr rprstato of data tha ts su modls. REFERENCES [] Akst, A., Famoy, F. ad L, C. (28. Th ta-parto dstruto. Statstcs 42, [2] Chaudhry, M. A. ad Zuar, S. M. (22. O a Class of Icomplt Gamma Fuctos wth Applcatos. Chapma ad Hall, Boca Rato, Florda. [3] Cooray, K. ad Amada, M. M. A. (28. A gralzato of th half-ormal dstruto wth applcatos to lftm data. Commucatos Statstcs - Thory ad Mthods 37, [4] Cordro, G. M. ad Nadarajah, S. (2. Closd-form prssos for momts of a class of ta gralzd dstrutos. Brazla Joural of Proalty ad Statstcs 25, [5] Eug, N., L, C. ad Famoy, F. (22. Bta-ormal dstruto ad ts applcatos. Commucatos Statstcs - Thory ad Mthods 3, [6] Eto, H. (978. Hadook of Hyprgomtrc Itgrals: Thory, Applcatos, Tals, Computr Programs. Halstd Prss, Nw York. [7] Gupta, R. C., Gupta, P. L. ad Gupta, R. D. (998. Modlg falur tm data y Lhma altratvs. Commucatos Statstcs - Thory ad Mthods 27, [8] Gupta, R. D. ad Kudu, D. (999. Gralzd potal dstrutos. Australa ad Nw Zalad Joural of Statstcs 4, [9] Gupta, R. D. ad Kudu, D. (2. Epotatd potal famly: a altratv to gamma ad Wull. Bomtrcal Joural 43, 7-3.

Reliability of time dependent stress-strength system for various distributions

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