Performance Rating of the Type 1 Half Logistic Gompertz Distribution: An Analytical Approach

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1 Amrica Joural of Mathmatics ad Statistics 27, 7(3): DOI:.5923/j.ajms Prformac Ratig of th Typ Half Logistic Gomprtz Distributio: A Aalytical Approach Ogud A. A. *, Osghal O. I., Audu A. T. Dpartmt of Mathmatics ad Statistics, Fdral Polytchic Ado-Ekiti, Dpartmt of Mathmatics ad Statistics, Josph Babalola Uivrsity, Ikji Arakji, Fdral School of Statistics Ibada, Nigria Abstract I this papr w dvlopd a thr paramtr distributio kow as Typ half logistic Gomprtz distributio which is quit flibl ad ca hav a dcrasig, icrasig ad bathtub-shapd failur rat fuctio dpdig o its paramtrs makig it mor ffctiv i modlig survival data ad rliability problms. Som comprhsiv proprtis of th w distributio, such as closd-form prssios for th dsity fuctio, cumulativ distributio fuctio, hazard rat fuctio ad a prssio for ordr statistics wr providd as wll as maimum liklihood stimatio of th Typ half logistic distributio paramtrs ad at th d a applicatio usig a ral data st was prstd. Kywords Typ half logistic Gomprtz distributio, Maimum liklihood stimatio, Bathtub-shap failur rat. Gomprtz Distributio Th Gomprtz (G) distributio is a flibl distributio which ca skw to th right ad to th lft. This distributio is a gralizatio of th potial (E) distributio ad is commoly usd i may applid problms, particularly i ral lif data aalysis (Johso, Kotz & Balakrisha 995, p. 25). Th G distributio is cosidrd to b usful i th aalysis of survival data, i som scics such as grotology (Brow & Forbs 974), computr (Ohishi, Okamura & Dohi 29), biology (Ecoomos 982), ad also i marktig scic (Bmmaor & Glady 22). Th hazard rat fuctio (hrf) of G distributio is a icrasig fuctio ad oft applid to dscrib th distributio of adult lif spas by actuaris ad dmographrs (Willms & Kopplaar 2). Th G distributio with paramtrs > ad > has th cumulativ distributio fuctio (cdf) giv as G =,, > () Ad th probability dsity fuctio is giv as g =,, > (2) A gralizatio basd o th ida of Gupta & Kudu (999) was proposd by El-Gohary & Al-Otaibi (23). This w distributio is kow as gralizd Gomprtz (GG) distributio which icluds th E, gralizd potial (GE), ad G distributios (El-Gohary & Al-Otaibi 23). * Corrspodig author: dbiz95@yahoo.com (Ogud A. A.) Publishd oli at Copyright 27 Scitific & Acadmic Publishig. All Rights Rsrvd 2. Gralizd Half Logistic Distributio (GHLD) A gralizatio of th half logistic distributio is dvlopd through potiatio of its cumulativ distributio fuctio ad trmd th Typ I Gralizd Half Logistic Distributio (GHLD). W dfi th cumulativ distributio fuctio (cdf) of th w typ I half-logistic (TIHL) family of distributios by F ; λ, ξ = log G,ξ 2λ λt + λt 2 dt = G ;ξ λ + G ;ξ λ (3) Whr G, ξ is th basli cumulativ distributio fuctio (cdf) dpdig o a paramtr vctor ξ ad λ > is a additioal shap paramtr. For ach basli G w ca grat th typ half logistic G "THL G" distributio is a widr class of cotiuous distributios. Th corrspodig probability dsity fuctio (pdf) to quatio (3) is giv by f ; λ, ξ = 2λg ;ξ G ;ξ λ + G ;ξ λ 2 (4) Whr g, ξ is th basli probability dsity fuctio (pdf). Equatio (4) will b most tractabl wh G, ξ ad g, ξ hav simpl prssios. 3. Th Typ Half Logistic Gomprtz Distributio (THLGD) Puttig quatio ( ito quatio (3) th cumulativ dsity fuctio of (TIHLGD) ca b obtaid as follows:

2 PDF PDF CDF CDF Ogud A. A. t al.: Prformac Ratig of th Typ Half Logistic Gomprtz Distributio: A Aalytical Approach F = λ + λ (5) CDF of THLGD CDF of GD Equatio (5) ca b simplifid as Usig th sris pasio, λ F = F = + λ (6) k= b k H k (7) whr, b k = i= i+k iλ. k k Ad H a = G() a dots th potiatd-g cumulativ distributio fuctio with paramtr a >, w grat a prssio for quatio (6) as follows: F = i+ λ k= (8) b k a If w cosidr th sris pasio, PDF of THLGD PDF of GD z m = j = j m j zj (9) Valid for z < ad m > quatio (8) ca b prssd as ral ad o-itgr, F = b k j a j j k= () j = A prssio for th probability dsity fuctio for th typ half logistic Gomprtz distributio ca b obtaid by isrtig quatio (2) i (4) 2λ f ; λ, ξ = + λ 2 λ This ca b simplifid as f ; λ,, = 2λ λ 2 () + λ Th dsity fuctio of X ca b prssd as a ifiit liar combiatio of p G dsitis giv as f = b k+ (k + ) k= (2) If w cosidr th sris pasio i quatio (8). Valid for z < ad m > ral ad o-itgr, quatio (2) ca b writt as, f = k k l b k+(k + ) (l+) k= l= (3) Th graph of th cumulativ dsity fuctio ad th probability dsity fuctio of th Typ half logistic Gomprtz distributio (THLD) wh = 2.8, =.5, λ =.5 is giv blow k Th graph abov clarly shows th flibility of th Typ Half logistic Gomprtz distributio ovr th Gomprtz distributio. 4. Th Asymptotic Proprtis Hr w ivstigat th asymptotic proprtis of th Typ half logistic Gomprtz distributio wh th valu of tds to 2λ lim + λ 2 = 2λ 4 = λ 2 λ

3 Th Hazard Rat Fuctio (h()) Th Hazard Rat Fuctio (h()) Amrica Joural of Mathmatics ad Statistics 27, 7(3): This implis that as tds to zro th probability dsity fuctio of THLD dpds oly o th two shap paramtrs λ ad. 5. Rliability Fuctio Giv a radom variabl, 2,... th rliability fuctio R is dfid as R = F() For THLGD, its rliability fuctio is giv as This givs R = R = λ + λ 2 λ + λ (4) h = + λ λ (6) Th quatio (6) ca b calld th Typ half logistic Gomprtz modl. 7. Hazard Graph Th THLGM hazard graph draw blow dpicts th flibility i th modl as its hibits both th proprtis of th bathtub ad th costat shap failur rat. b=.5,b2=.,c=2.5 b=.5,b2=.,c= Hazard Rat Fuctio Th hazard rat ca b obtaid usig, h = f() R() Substitutig quatio () ad (4) i (5), w hav h = 2λ λ + λ 2 This givs h = λ λ λ 2 λ + λ + λ Fially th hazard rat fuctio of THLGD is (5) Momt Gratig Fuctio Th momt gratig fuctio of a radom variabl is giv as M t = t f d Puttig quatio (3) i (4), w hav 5 2 (7) W hav, M t = solvig, = M t = j = k= t + t (j +) t + j + t k= j = j Γ(k + ) j! Γ(k j + ) b k+(k + ) j Γ(k + ) j! Γ(k j + ) b k+(k + ) t (j +) d (j +) d = t + (j +) d = t + j + = j + t (j +) d

4 96 Ogud A. A. t al.: Prformac Ratig of th Typ Half Logistic Gomprtz Distributio: A Aalytical Approach Thrfor, th momt gratig fuctio of THLGD is giv as M t = j = k= j Γ k+ j!γ k j + b k+(k + ) j + t (8) 9. Th Quartil Th quartil u of ordr u for th THLD distributio is giv by th solutio of u = u + l +u λ Proof Lt u = F = Th, λ + λ u + λ = λ Thrfor, This givs, u = λ (u + ) = u l λ + u Fially this producs th quatil fuctio of ordr u giv as, u = u + l +u λ (9) Spcial quartils may obtaid by quatio (6), for ampl wh u = 4 ; th uppr quartil, u = 2 ; th mdia, u = 3 4 ; th uppr quartils.. Ordr Statistics Th ordr statistics play a importat rol i rliability ad lif tstig. Lt X,..., X b a simpl radom sampl from THLG distributio with pdf as 3 ad cdf as rspctivly. Lt X i;,..., X ; dot th ith ordr statistics, say X i; dot th liftim of a i out of systm which cosist of idpdt ad idtical compots. Th pdf of X i; is giv by f i; =! i i f F i F i i =,2,..., (2) Sic, < F() < for >, th by usig th biomial sris pasio of F i, w obtai f i; =! i i Th substitutig for f ad F() i quatio 3 ad rspctivly, w obtai f i; =! i i i h= k=. Maimum Liklihood Estimatio i i h= h h f() F() h+i (2) h+j +l i k h l b k+(k + ) (l+) l= k= j = b a k j Th maimum liklihood stimatio (MLE) is o of th most widly usd stimatio mthod for fidig th ukow paramtrs. Lt, 2,..., b a idpdt radom sampl from THLD. Th total log-liklihood is giv by λ j h+i (22) l = log 2λ + log i i i= + λ log i i= 2 log + λ i I= (23)

5 Amrica Joural of Mathmatics ad Statistics 27, 7(3): If w lt, ψ =, thrfor, th liklihood fuctio ca b prssd as: l = log 2λ + Th scor vctor l = l λ, l, l log i+ψ i= + λ i= log ψ 2 I= log + (24) has compots ψ λ l = l = i= i +ψ i i +ψ i i i + + i= i +ψ l λ = λ λ i +ψ i= log ψ λ i ψ i= ψ + 2λ i i + ψ i= ψ ψ λ ψ i= (25) + ψ λ i ψλ + ψλ + 2λ i + ψλ + ψλ i (26) 2. Applicatio To illustrat th w rsults prstd i this papr, w fit th THLD distributio to a ral data. Th first ampl is a ucsord data st from Nichols ad Padgtt (26) cosistig of obsrvatios o brakig strss of carbo fibrs (i Gba). Th data ar as follows : 3.7, 2.74, 2.73, 2.5, 3.6, 3., 3.27, 2.87,.47, 3.,4.42, 2.4, 3.9, 3.22,.69, 3.28, 3.9,.87, 3.5, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53,2.67, 2.93, 3.22, 3.39, 2.8, 4.2, 3.33, 2.55, 3.3, 3.3, 2.85, 2.56, 3.56, 3.5, 2.35, 2.55, 2.59,2.38, 2.8, 2.77, 2.7, 2.83,.92,.4, 3.68, 2.97,.36,.98, 2.76, 4.9, 3.68,.84,.59,3.9,.57,.8, 5.56,.73,.59, 2,.22,.2,.7, 2.7,.7, 5.8, 2.48,.8, 3.5, 2.7,.69,.25, 4.38,.84,.39, 3.68, 2.48,.85,.6, 2.79, 4.7, 2.3,.8,.57,.8, 2.3,.6, 2.2,.89, 2.88, 2.82, 2.5, W shall compar th Typ half logistic Gomprtz modl with its sub- modl, th Gomprtz modl. Tabl givs th dscriptiv statistics of th data ad Tabl 2 lists th MLEs of th modl paramtrs for THLG ad G distributios, th corrspodig rrors (giv i parthsis) ad th statistics l(θ) (whr l(θ) dots th log-liklihood fuctio valuatd as th maimum liklihood stimats), Akaik iformatio critrio (AIC), th Baysia iformatio critrio (BIC), Cosistt Akaik iformatio critrio (CAIC) ad Haa-Qui iformatio critrio (HQIC). Also w provid total tim o tst plot. Tabl. Dscriptiv Statistics o Brakig strss of Carbo fibrs Mi Q Mdia ma Q 3 Ma kurtosis Skwss Variac Mod.39m TTT PLOT (TOTAL TIME ON TEST PLOT) FOR BREAKING STRESS OF CARBON

6 98 Ogud A. A. t al.: Prformac Ratig of th Typ Half Logistic Gomprtz Distributio: A Aalytical Approach Tabl 2. MLEs for th Brakig Strss of Carbo Data (stadard rrors i parthss) AIC, BIC, HQIC, CAIC Modl Estimats l(θ) AIC BIC HQIC CAIC THLG (,, λ) G, KP (a, b, θ, ) (.52) (49.552) (.45) ZBLL (a, θ, ).55 (.4).993 (.93) BF (a, b, θ, ) (.236) (3.552) (2.52) (.9) Coclusios Sic th Typ half logistic Gomprtz distributio provid a bttr fit tha its sub-modl, i modlig a ral lif data that hibits a bathtub shap failur rat by havig a smallr AIC, BIC, HQIC ad CAIC it should b cosidrd as a bttr modl. REFERENCES [] Aarst, M V. How to idtify bathtub hazard rat. IEEE Trasactios o Rliability, v. 36, p.6{8, 987. [2] Gauss M. Cordiro, Morad Alizadh ad Pdro Rafal Diiz Mariho (25). Th typ I half Logistic family of distributios, Joural of Statistical Computatio ad simulatio: DOI: :3233. [3] Akaik, H. A w look at th statistical modl idtificatio. Automatic Cotrol, IEEE Trasactios o, v. 9, p. 76{723, 974. [5] Ch, G, ad Balakrisha, N. A gral purpos approimat goodss-of-fit tst. Joural of Quality Tchology. v. 27. p. 54{6, 995. [6] Cordiro, Gauss M, Ortga, Edwi MM, ad Nadarajah, Sarals. Th Kumaraswamy Wibull distributio with applicatio to failur data. Joural of th Frakli Istitut, v. 347, p. 399{429, 2. [7] R Cor Tam. R: A Laguag ad Eviromt for Statistical Computig. R Foudatio for Statistical Computig, Austria, 26. [8] Cordiro, G. M. & Nadarajah, S. (2), Closd-form prssios for momts of a class of Bta gralizd distributios, Brazilia Joural of Probability ad Statistics 25(), [9] Ecoomos, A. C. (982), Rat of agig, rat of dyig ad th mchaism of mortality, Archivs of Grotology ad Griatrics (), [] El-Gohary, A. & Al-Otaibi, A. N. (23), Th gralizd Gomprtz distributio, Applid Mathmatical Modlig 37(-2), [4] Broyd, C G, Fltchr, R, Goldfarb, R, D. DF Shao J. Ist. Math. Appl, v. 6, p. 76, 97.