( ) ( ) Weibull Distribution: k ti. u u. Suppose t 1, t 2, t n are times to failure of a group of n mechanisms. The likelihood function is

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1 Webll Dsbo: Des Bce Dep of Mechacal & Idsal Egeeg The Uvesy of Iowa pdf: f () exp Sppose, 2, ae mes o fale of a gop of mechasms. The lelhood fco s L ( ;, ) exp exp MLE: Webll 3//2002 page MLE: Webll 3//2002 page 2 L( ;, ) exp We wsh o choose vales of & whch maxmze L (o eqvalely, he logahm of L),.e., whch mae he obseved vales of as lage as possble! The log-lelhood fco s l L ;, l l + l The opmaly codos fo he maxmm of he loglelhood fco ae l L ( ;, ) 0 l L ( ;, ) 0 Ths gves s a pa of olea eqaos wo ows ( & ): ˆ + ˆˆ 0 ˆ l ˆ + l 0 ˆ ˆ MLE: Webll 3//2002 page 3 MLE: Webll 3//2002 page 4

2 B he lef sde of he fs eqao ca be facoed: ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ + ˆ ˆ 0 ˆ + Sce he fs faco cao be zeo, we se he secod faco eqal o zeo ad solve fo û ems of ˆ : ˆ Elmag û he secod eqao by sbsg he fs, we ge he followg olea eqao ˆ aloe: l + l 0 ˆ Ths ca ow be solved by, fo example, he seca mehod. ˆ Maxmm Lelhood Esmao wh cesoed daa Sppose ha a expeme was emaed a me τ afe oly of he s a lfees had faled. Ths s accoed fo by defg he lelhood as (, θ ) ( τ; θ ) ( ; θ) L F f The log-lelhood fco s heefoe ( θ ) ( ) ( θ ) + ( θ) l L ; l F ; l f ; MLE: Webll 3//2002 page 5 MLE: Webll 3//2002 page 6 Example: MLE of Webll paamees, gve cesoed daa The CDF of he Webll dsbo s F( ;, ) exp ad so he lelhood fco s τ L ;, exp exp exp ( ) + τ The log-lelhood fco s l L ;, l l + l + τ The opmaly codos fo a maxmm of he log-lelhood a ( ˆ, ˆ ) ( ˆ ˆ) l L ;, 0 l L ( ; ˆ, ˆ) 0 ae MLE: Webll 3//2002 page 7 MLE: Webll 3//2002 page 8

3 A esl smla o he cesoed case ca be deved: ad ˆ ˆ + τ ˆ ˆ l l + τ τ + τ + l 0 Ths secod eqao ca be solved fo ˆ by he seca mehod, ad he ˆ ca be sed o calclae û by he fs eqao. EXAMPLE: Twey devces ae esed smlaeosly l 500 days have passed, a whch me he followg fale mes ( days) have bee ecoded: Esmae he lfeme fo whch he devce s 90% elable. MLE: Webll 3//2002 page 9 MLE: Webll 3//2002 page 0 A plo of Y vs X, obaed by he asfomaos: Y log log whee R() s he obseved faco of he R () devces whch have svved l me, ad X log shold be a le f he Webll model wee o f he daa pefecly. LEAST SQUARES REGRESSION RESULTS: (scale paamee) (shape paamee) so ha mea sadad devao Noe: hs s deemed by mmzg he sm of he ( ) sqaed eos he leazed veso of F () e, amely y x l whee x l & y l l, R () ahe ha he ogal eqao! MLE: Webll 3//2002 page MLE: Webll 3//2002 page 2

4 If we se hese paamees fod by lea egesso, he elably fco wold have he vales: F() -F() Maxmm Lelhood esl: Solvg he olea eqao fo : l l + τ τ + τ g + l 0 Hece, accodg o hs model, 90% of he devces shold be opeag a 54.8 (appoxmaely 55) days. MLE: Webll 3//2002 page 3 MLE: Webll 3//2002 page 4 SECANT METHOD The seca jog he wo pos o he gaph of g coss he axs a We he epea, wh he 2 mpoved gesses 0.5 ad If o fs wo gesses a he vale of ae 0.5 ad 2.0, he we deeme ha g(0.5).3739 & ad g(2.0) MLE: Webll 3//2002 page 5 MLE: Webll 3//2002 page 6

5 SECANT METHOD RESULTS: eo E E 2 Oce we deeme he vale of ˆ whch maxmzes he lelhood fco, he he coespodg vale of he paamee û s fod by ˆ ˆ + τ ˆ ˆ MLE: Webll 3//2002 page 7 MLE: Webll 3//2002 page 8 MAXIMUM LIKELIHOOD RESULT: (scale paamee) , (shape paamee) F() -F() Accodg o hs model, he, 90% of he devces shold be opeag a (appoxmaely 74) days. MLE: Webll 3//2002 page 9