A note on finding geodesic equation of two parameter Weibull distribution


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1 Theoetical Mathematics & Applications, ol.4, no.3, 04, 435 ISSN: (pint), (online) Scienpess Ltd, 04 A note on finding geodesic eqation of two paamete Weibll distibtion William W.S. Chen Abstact The Weibll distibtion has eceied a geat deal of attention since 970. In Rssian statistical liteate, this distibtion is often efeed to as the Weibll Gnedenko distibtion. It has been applied to model a wide ange of data seced fom poblems sch as the yield stength of Bofos steel, the fibe stength of Indian cotton, the fatige life of ST37 steel, the states of adlt males bon in the Bitish Isles, and beadth of beans of Phaseols lgais. Many athos sed this distibtion in thei eliability and qality contol wok. Instead of sing the classical appoach by soling a pai of diffeential eqations, in this pape, we adopt the wellknown Dabox Theoy by soling a patial diffeential eqation to find the geodesic eqation of two paamete Weibll distibtions. Depatment of Statistics, The Geoge Washington Uniesity Washington D.C Aticle Info: Receied: Apil, 04. Reised: May 9, 04. Pblished online : Agst 3, 04.
2 44 Finding geodesic eqation of two paamete Weibll distibtion Mathematical Sbject Classification: 6E99 Keywods: Dabox Theoy, Diffeential Geomety; Gamma Distibtion; Geodesic Eqation; Patial Diffeential Eqation; Weibll Distibtion Intodction Swedish physicist, Waloddi Weibll, [,3] sed the Weibll Distibtion to descibe the beaking stength of mateial. By 95[4], thee wee a aiety of othe applications. Seeal basic examples of how to apply the Weibll Distibtion wee pesented in the abstact. In Rssian statistical liteate this distibtion is often efeed to as the WeibllGnedenko distibtion. It is one of the thee types of limit distibtion fo the sample maximm established by Gnedenko [5]. As his special, the Weibll distibtion may also inclde the exponential o the Rayleigh distibtion. When the shape paamete is less than, the haad fnction of the Weibll Distibtion is a deceasing fnction. When the shape paamete eqals, it is a constant. When the shape paamete is geate than, it is an inceasing fnction. Many athos hae sed this sitation in eliability and qality contol wok sch as Weibll[4], Kao[6,7], and Beetoni [8]. The Weibll Distibtion eceied the most attention in 970. This is eident fom the lage nmbe of efeences that can be fond fom the book of Johnson N.L. Kot S, and Balakishnan N [9]. Since Rao C.R. [0] pblished his fist pape, linking statistics with geometical popeties, nmeos athos hae expanded pon this aea. Fo example, Laiten S.L. [] deied the Gassian Cate, Geodesic Eqation of Gamma Manifold and Inese Gassian Manifold. Chen W. [] sing the Dabox Theoy deied a completed esion of the Gamma Geodesic Eqation. Chen W. [3] has sccessflly genealied the fomla to compte the Gassian Cate and claify its inticate mathematical concept. Uwe Jensen [4] has eiewed the deiation, calclation and simlation eslts of Rao Distance, applying it to the potfolio theoy. In this pape, we
3 William W.S. Chen 45 adopted the Dabox Theoem to sole a patial diffeential eqation. We fond this appoach wold be easie than the classical method. Dabox theoy and geodesic eqation In geneal, the distance between two points P and Q on a ce of twomanifold can be expessed as ds = E d + Fd d + G d (.) Howee, if we can tansfom the distance fnction (.) to the following simplified fom ds = d + σ d (.) it cold help s to find the Geodesic Eqation moe easily. The task of tansfoming eqation (.) is eqialent to asking how we can detemine two independent fnctions, =, ), and = (, ), sch that eqation (.) can ( be tansfomed into eqation (.). Since (,) is a fnction of (,), we know fom calcls that d = d + d, ds d = ( E ) d + ( F ) dd + ( G ) d. (.3) If we assme that (.) is alid, then it wold be necessay fo eithe the ight hand side of (.3) to be a pefect sqae, o fo the deteminant of (.3) to be eqal to eo. That is, Eqation (.4) can be ewitten as ( F ) ( E )( g ) 0. = (.4) E F + G EG F =. (.5) Fo conenience, we sally wite the left hand side of (.5) as Z =. Now, if we cold find an abitay soltion to (.5), then we cold ewite (.3) in the following fom:
4 46 Finding geodesic eqation of two paamete Weibll distibtion ds d = ( m(, ) d + n(, ) d), (.6) whee both mn, ae some known fnction of and. If we can fthe find an integation facto, sch that σ m(, ) d + n(, ) d = σ d, then the distance fnction ds cold be tansfomed into the fom (.). Smmaiing the aboe pocedes, we conclde that in ode to find the Geodesic Eqation, two steps mst be completed : Step : we mst find an abitay soltion of the patial diffeential eqation (.5); Step : we mst find an integation facto of eqation (.6). Dabox has poposed an impoed method to combine the two steps into one step; that method is stated as a theoem. Theoem : Assme the gien patial diffeential eqation Z = has fond an abitay soltion Z=Z(,,a), whee a is an abitay constant. Then Z (, ; a) = cons tan t a is the eqied Geodesic Eqation. Poof: Assme the distance between two points P and Q on a ce of twomanifold has the simplified fom (.). The total diffeential at a point (,) can be witten as: d = d + d, d = d + d (.7) Fthemoe, if we take the patial deiatie with espect to the constant a fo the aboe two eqations and we get d = d + d, a a a d = d + d, a a a and (.8)
5 William W.S. Chen 47 σ d d = σ d d + σ d a a a If (.8) wee te, then fom the thid eqation of (.8) we can conclde that d mst diide eenly on d d. This means that d can diide eithe d o a d a eenly. Concening the fist sitation that d can diide d eenly, then fom (.7) we hae = 0 (.9) Bt this means that and ae fnctionally dependent. This contadicts eqation (.), which assmes that and ae independent. Hence, the only case that can possibly be alid is that d can diide d a eenly. This means that = constant and cons tan t ae ces in the same families. This poes a = that the eqation = cons tan t is the eqied geodesic eqation. a 3 Finding the Geodesic Eqation of the Weibll btion Fom section 4, we hae calclated we sometimes also wite C E = whee C a b = +, π a = γ, and b=. 6
6 48 Finding geodesic eqation of two paamete Weibll distibtion γ = is known as Ele's constant. C F =, in this case C = a, (3.) Then Z = becomes G =. π ( a + b ). + a + = 6 In ode to find one of the soltions of eqation (3.), we make a tansfomation fom (, ) to (, ) as follow: Then, thogh the chain le, we get = = e,. = + =, o =, = + = e, o = e The gien patial diffeential eqation (3.) tns ot to be π (a + b ) + a + =, 6 Next, conside making anothe tansfomation to pola coodinates: = cos θ, = sin θ; then, thogh the chain le, we can find the following elation: (3.) = sin θ + cos θ θ, = cosθ sin θ Afte sbstitting, into eqation (3.), ecalling the tem and simplifying, we smmaie o eslts as follows: The coefficients of θ : θ
7 William W.S. Chen 49 this means ( a + b ) tan θ + a tan θ = 0 a tan θ = = a + b θ = , o θ =.8756 The coefficient of (a : + b )sin θ + a cosθ sinθ + cos θ = The coefficient of θ : Afte otating a + b cos θ + a( cosθsin θ) + sin θ =.8756, eqation (3.) becomes = = A θ We can sepaate the aiables of and θ, then sole this patial diffeential eqation as follows: On the othe hand, we can deie fom A =, so that = ± A ln = A = ± θ, θ we can now smmaie the aboe two eslts and wite one of the geneal soltions that we fond A A Z =± A ln ± θ (3.3).56 Fom peios elations, we know that (, θ ) and (, ) ae elated to + =, and tan θ =, hence, afte sbstitting into eqation (3.3) we get
8 50 Finding geodesic eqation of two paamete Weibll distibtion Making a fthe sbstittion: we can then easily find that A Z =± A ln + ± tan.,.56 = and = log + = + (log ), and = log. Finally, we find one of the geneal soltion of eqation (3.) A Z =± Aln ( + (log ) ) ± tan (log )..56 The Geodesic Eqation of the Weibll Distibtion can then be witten as o Z = B, A whee AB, ae abitay constants. Atan (log ) ± ln ( + (log ) ) ± = B A 4 List the fndamental tenso The pobability density fnction fo the Weibll Distibtion is gien by x x f( x:,, ) = exp( ( ) ) I(0, )( x); whee is the scale paamete and is the shape paamete. (4.) x ln f = ln + ( ) ln x ln ( ). Fom eqation (4.), we deie the metic tenso components fo the Weibll case
9 William W.S. Chen 5 () Γ + ln f( x) () E = E( ) =, ' ln f( x) Γ () F = E( ) =, ln f G = E( ) = ψ() = ψ() + Γ () = ψ () + ( Γ()) () ' ' = = = = ' = Γ () C = Γ () + ; C = Γ () C = C C = () ' 3 In the aboe deiation, we applied the following integal eslts x () ( ) Γ = = 0 x x x x x () E((ln( )) ( ) ) (ln ) ( ) e dx ; x ' ( ) Γ = = 0 x x x x x () E((ln( ))( ) ) (ln )( ) e dx we define the nth deiatie of the gamma fnction : ( n) t x n Γ ( x) = e t (ln t) dt, x > 0. 0 Refeences [] Stik, D.J., Lectes on Classical Diffeential Geomety, Second Edition, Doe Pblications, Inc, 96. [] Weibll, W., A statistical theoy of the stength of mateial, Repot 5, (939a), Ingenios Vetenskaps Akademiens Handliga, Stockholm. [3] Weibll, W., The phenomenon of pte in solids, Repot 53, (939b), Ingenios Vetenskaps Akademiens Hadliga, Stockholm.
10 5 Finding geodesic eqation of two paamete Weibll distibtion [4] Weibll, W., A statistical distibtion of wide applicability, Jonal of Applied Mechanics, 8, (95), [5] Gnedenko, B.V., S la distibtion limite d teme maximm d ne seie aleatoie, Annals of Mathematics, 44, (943), [6] Kao, J.H.K., Compte methods fo estimating Weibll paametes in eliability stdies, Tansactions of IREReliability and Qality Contol, 3, (958), 5. [7] Kao, J.H.K., A gaphical estimation of mixed Weibll paametes in lifetesting electon tbes, Technometics,, (959), [8] Beettoni, J.N., Pactical applications of the Weibll distibtion, Indstial Qality Contol,, (964), [9] Johnson, N.L., Kot, S., Balakishnan, N., Continos Uniaiate Distibtion, ol., Second Edition, by John Wiley and Sons, Inc [0] Rao, C.R., Infomation and accacy attainable in the estimation of statistical paametes, Blletin of Calctta Mathematical Society, 37, (945), [] Laiten S.L., Statistical Manifolds. In Diffeential Geomety in Statistical Infeence, (eds S.I. Amai, O.E. BandoffNielsen, R.E. Kass, S.L. Laiten and C.R. Rao), Institte of Mathematical Statistics, Haywad, CA, 0, (987), [] Chen W.W.S., A Note On Finding Geodesic Eqation of two Paamete Gamma Distibtion, Applied Mathematics, (03), aailable at [3] Chen W.W.S., On compting Gassian cate of some well known distibtion, Theoetical Mathematics and Applications, 3(4), (03), [4] Jensen, U., Reiew of The deiation and calclation of Rao distances with an application to potfolio theoy, in Adances in Econometics and Qantitatie Economics: Essays in Hono of C.R. Rao (eds. P. Maddala, G.S. Phillips, and T.Siniasan), Blackwell, Cambidge, (995),