On Po Selecton fo the Mxtue of Rylegh Dstbuton usng Pedctve Intevls Muhmmd Sleem Deptment of Sttstcs Athens Unvesty of Economcs nd Busness
Slent fetues of ths wo Mxmum Lelhood nd Byes estmton of the Rylegh mxtue wth fxed censo tme Po Selecton fo Rylegh mxtue usng Pedctve Intevls Mxmum Lelhood nd Byes estmton of the Rylegh Suvvl tme wth Rndom censo tme Rylegh
Composton Pt One Pt Two Fxed censo tme The ML nd Byes estmton The Byesn pedctve ntevls Advntges of the hype-pmetes tend Rndon censo tme The mxmum lelhood nd Byes estmton Cedble Intevls nd HPD s Pedctve Dstbuton
Pt One
Why Rylegh? Rylegh model s especlly sutble fo the lfe-testng of the poducts tht ge wth tme. Why Mxtue? A fnte mxtue of some sutble pobblty dstbuton s ecommended to study populton tht s supposed to compse numbe of subpopultons mxed n n unnown popoton. tj tj f t = p t exp + q t exp j j
The Lelhood Functon Among the ded objects, e dentfed to be fom the fst subpopulton, e dentfed to be fom the second subpopulton nd the emnng n- undentfed objects of the populton suvve fte the fxed censo tme T s ove. n- L,, p { p f t } { j q f t j } { F T } t j= j= Fo MLE, t s convenent to wo wth ths veson of the lelhood,, exp{ }{ exp exp } + + t t n p q T T L p t p q Applyng Bnoml theoem to the lst fcto of the lelhood, fclttes the devton of the closed fom expessons fo the Byes estmtos n L p p p n n +,, t exp exp = 0 A A whee t j A = t + { n + } T, t =, =, j =
Mxmum Lelhood Estmtos The second tem on the left hnd sde vnsh s n tends to, mng closed fom expessons fo the ML estmtos possble. exp / p T + q T t T n p q T + =, =, exp / exp / p n exp T / exp T / + = exp / + exp / p p p T q T Some tetve numecl method s equed fo the soluton of the bove system of nonlne equtons fo the evluton of the Mxmum lelhood estmtes.
Vnces of the MLE Appoxmte vnces of the Mxmum Lelhood estmtes ˆ, I - N A well nown popety of MLE whee =,, p Infomton mtx s the Expectton of the Negtve Hessn mtx l l l p l l l l I = E = E p l l l p p p
Elements of the Infomton mtx The elements on the mn dgonl tend to become lge s n tends to s the second tem on the RHS vnsh. Ths would educe the vnces when dgonl mtx wll be nveted. T T / + / l 6 T / T / pe qe 4 T n p q { q p T 3 e 3 p q e } E =, =, + T / / T / T l n e e T / / E = + + p p q T pe qe + The off-dgonl elements tend to zeo s tends to n.e., when censo tme s lge enough to ncopote ll the obsevtons nto the nlyss nd wll tun the Infomton dgonl s dgonl mtx. T 4 T / + / l 4 T n pqe l E = = E 3 3 T / T / pe + qe T T / + / l T n e l E = = E, =, 3 T / T / p pe + qe p
Byes estmtos ssumng Unfom Po ˆ n = x Β,, =, d A A Unle Mxmum lelhood estmtes, no tetve numecl pocedue s equed fo the evluton of the Byes Estmtes pˆ x = Γ Γ Β n +, 0.5 0.5 A A n Β, 0.5 0.5 A A The vlblty of the closed fom expessons mes the evluton of the Byes estmtes much moe convenent s comped to tht of the MLE s. t j n Γ 0.5 Γ 0.5 = = Β, 0.5 05 = A A j = = n + + +, A = t + n + T { }, t, d,,
Vnces of Byes estmtos ssumng Unfom Po The closed fom lgebc expessons lso fclttes the evluton of vnces. ˆ Γ V 3 3 Γ n = Β, 3 3 d A A x d Γ n Β,, =, A A Γ V pˆ x n n Β +, Β +, 0.5 0.5 0.5 0.5 A A A A = n n Β, Β, 0.5 0.5 0.5 0.5 A A A A t j n Γ 0.5 Γ 0.5 = = Β, 0.5 05 = A A j = = n + + +, A = t + n + T { }, t, d,,
Byes estmtos ssumng Jeffeys po Closed fom expessons e lso vlble fo the ByesJeffeys estmtos ˆ Γ Γ = Β, =, n x, d A A pˆ x = Β n +, A A n Β, A A Jeffeys po s pefeed becuse of ts nvnt popety nd beng popotonl to the sque oot of the detemnnt of the Fshe Infomton mtx. Γ Γ n A t n T d n = + + +, = + { + }, = Β, A A
Vnces of the Byes estmtos ssumng Jeffeys po Beng bsed on the nfomton mtx, Byes Jeffeys hve slghtly lesse vnces thn ts unfom nd mle countepts. ˆ n Γ Γ V = x Β, A A d V pˆ x d Γ n Β,, =, A A n n Β +, Β +, A A A A = n n Β, Β, A A A A Γ Γ Γ n A t { n } T, d n = + + +, = + + = Β, A A
The Byes Estmtos ssumng the IC Po The functonl fom of the Inveted Ch dstbuton s comptble wth the lelhood of the Rylegh mxtue to yeld n Inveted Ch mxtue s posteo dstbuton. The Byes estmtos e the expectton of the espectve mgnl posteo dstbuton of ech pmete unde squed eo loss functon. The Byes estmtos hve closed foms s well, n ˆ { n Β t = Γ + },, = d + 0 + = A + 0.5 A + 0.5, n n B + = 0 A + 0.5 A + 0.5 pˆ t = A 0.5 0.5 n Β, + A + + +, n n = + + +, = + { + }, =Γ n A t n T d + + 0 + = Β A + 0.5 A + 0.5
Vnce of the Byes Estmtos ssumng the IC Po V n = Γ, n ˆ [ { Β + t }] d + + = 0 A + 0.5 A + 0.5, n [ { n Β Γ + }],, = d + 0 + = A + 0.5 A + 0.5 V pˆ t n n Β n +, n +, { } { } + 0 + + + = = 0 A+ 0.5 A+ 0.5 A+ 0.5 A+ 0.5 = n, n n Β n, { } Β { A+ 0.5 A+ 0.5 A+ 0. 5 A + 0.5 + 0 + = = 0 + + Β } Closed fom expessons e vlble fo the evluton of the vnces of the Byes estmtos., n n = + + +, = + { + }, =Γ n A t n T d + + 0 + = Β A + 0.5 A + 0.5
Byes estmtos ssumng IR po The Inveted Rylegh s nothe stndd pobblty dstbuton tht hs functonl fom comptble wth the Rylegh mxtue to poduce closed fom exessons fo the Byes estmtos of the pmetes. + Β, n ˆ Γ n t = { }, =, d 0 + + = A + A + pˆ t =, n n { Β + } + + A + A + = 0 n, n Β { } + + A + A + = 0, n n, = + + + = + { + }, + =Γ = 0 A+ A+ n A t n T d Β + +
Vnces of the Byes estmtos ssumng IR po V n V n, n ˆ Β t = { Γ } + + d A + A + pˆ = 0 Γ + Β, n [ { }],, d 0 + + = t A + A +, n n Β + { } + + = 0 A + A + = n n Β, { } + + A + A + = 0, n n Β + { } + + = 0 A + A + n n Β, { } = 0 A + + A + + Agn, the closed fom expessons fo the vnces e lso vlble n, n Β, = n + + + A = t + { n + } T, d + =Γ A + A + = + + = 0
Byes estmtos ssumng SRIG po Sque oot Inveted Gmm s nothe dstbuton mong the stndd pobblty dstbutons tht cn be used s conjugte po fo the Rylegh mxtue. Closed fom expessons fo the Byes estmtos e vlble Squed eo loss functon s ssumed gn, n n Β { Γ + } 0 + + = A + b A + b ˆ t =, =, n, n Β { Γ + } + + A + b A + b = 0, n n Β + { } + + = 0 A ˆ + b A + b p t = n n Β, { } + + A + b A + b = 0, b Β n A t { n } T, d { } n n, = + + + = + + + = Γ = 0 A B + +
Vnces of Byes estmtos ssumng SRIG po V V pˆ t n + Β, n ˆ Γ t = { } + + d A + b A + b n = 0 + Β, Γ = d A + b A + b n { n },, 0 + + = n Β n +, n +, { } { } + + + + = 0 A + b A + b = 0 A + b A + b = n, n n Β n Β, { } { = 0 A + b A + b = 0 A + b A + b + + + + Β } Closed fom estmtos fo the vnces of the Byes estmtos, b Β n A t { n } T, d { } n n, = + + + = + + + = Γ = 0 A B + +
Compson of the Mxmum Lelhood nd the Byes Unnfomtve estmtes Pmetes Byes Estmtes Unfom Byes Estmtes Jeffeys ML Estmtes Byes Vnce Unfom Byes Vnce Jeffeys ML Vnces =8 7.9845 8.04847 8.068367 0.958 0.9006 0.4065 =.05443.03305.0656 0.05504 0.043 0.88 p = 0.375 0.3780 0.377944 0.376005 0.000638 0.000637 0.00004 Compson of the Mxmum Lelhood nd the Byes nfomtve estmtes Pmetes =8 = p=0.375 Byes Estmtes Infomtve 6.44773 VnceBE Infomtve 0.048303 0.058 0.090884 0.367738 0.000583 Equlty of ll the Byes Unnfomtve estmtes nd the MLE s mmedte Byes Infomtve estmtes e undeestmted but the vnces e lesse. The vnces cn futhe be educed f moe ppopte po mfomton s thee.
Byes Unfom estmtes s functon of T 0.5 0.4 0.3 0. 0. VnceOf.5 5 7.5 0.5 5 7.5 30 Th. Th.
Pedctve ntevls fo the Rylegh mxtue ssumng IC po Vyng the vlues of the hype-pmetes shows tht the IC po nfomton cn mpove the effcency of the pedcton but no s much s the SRIG cn do. Howeve, t s bette thn the IR po. = 0 = 0 = 40 = 70 = 00 = 50 L =.585855 U =.67399 δ =.08837 L =.5765 U =.8606 δ =.9886 L =.477658 U =.90983 δ =.4355 L =.434378 U =.958885 δ =.54507 L =.37958 U = 3.00 δ =.630 = 40 L =.567585 U =.78478 δ = 0.3893 L =.508695 U =.958387 δ = 0.44969 L =.45995 U =.0568 δ = 0.5973 L =.47656 U =.0694 δ = 0.68986 L =.356888 U =.5473 δ = 0.797385 = 70 L =.5577 U = 0.9675 δ = 9.40403 L =.4947, U =.73564 δ = 9.68093 L =.44335 U =.8056 δ = 9.83747 L =.40654 U =.340768 δ = 9.9394 L =.34373 U =.39575 δ = 0.0500 = 00 L =.54030 U = 0.89093 δ = 8.648783 L =.47563 U = 0.456404 δ = 8.98077 L =.473 U = 0.579458 δ = 9.545 L =.386349 U = 0.64656 δ = 9.6067 L =.38387 U = 0.70384 δ = 9.374897 = 50 L =.555455 U = 9.89 δ = 8.66764 L =.453654 U = 9.3759 δ = 7.97605 L =.40766 U = 9.539050 δ = 8.3684 L =.36360 U = 9.6058 δ = 8.5868 It hs mnged to educe the sped of pedctve ntevl fom.09 to 8.38 L =.30694 U = 9.686750 δ = 8.380556
Pedctve Intevls fo Rylegh mxtue ssumng IR po Hee t s evdent tht lthough the IR nd the Rylegh mxtue hve the sme fucntonl foms but the IR beng f wy fom the lelhood s unble to educe the pedve ntevl lengths. Fom the tble t s cle tht IR cn mnge to educe the length of the pedctve ntevl fom.688 to.596 only whch s dculous. = 0 = 0 = 40 = 70 = 00 = 50 L =.609639 U =.87097 δ =.688 L =.683 U =.86357 δ =.5704 L =.63980 U =.85609 δ =.4049 L =.6640 U =.848464 δ =.334 L =.6975 U =.835686 δ =.596 = 40 = 70 = 00 L =.609883 U =.88947 δ =.73064 L =.607 U =.894957 δ =.8483 L =.60370 U =.906956 δ =.96586 L =.6057 U =.875558 δ =.6350 L =.630 U =.887589 δ =.7588 L =.6545 U =.899609 δ =.87064 L =.644 U =.86809 δ =.53868 L =.64469 U =.88044 δ =.65675 L =.6473 U =.8985 δ =.7747 L =.66385 U =.860549 δ =.4464 L =.6669 U =.876 δ =.55993 L =.66874 U =.884685 δ =.678 L =.69970 U =.847808 δ =.7838 L =.605 U =.85999 δ =.39704 L =.60460 U =.8708 δ =.5558 = 50 L =.60776 U =.9693 δ =.3655 L =.695 U =.9969 δ =.306668 L =.659 U =.930 δ =.97 L =.678 U =.904766 δ =.87485 L =.60869 U =.8959 δ =.79
Pedctve Intevl fo Rylegh mxtue ssumng SRIG po Among the vlble stndd conjugte pos, SRIG po hs poved tself the best. SRIG mnged mxmum educton n the pedctve ntevls by chngng t fom 0.87 to 6.3. So the SRIG po s pefeed. b = 0 b = 0 = 0 = 0 L =.559359 U =.433777 δ = 0.87448 = 40 = 40 L =.439907 U =.06663 δ = 9.6463 = 70 = 70 L =.360 U = 9.87959 δ = 8.550357 = 00 = 00 L =.399 U = 8.840033 δ = 7.600 = 50 = 50 L =.33306 U = 7.4904 δ = 6.78598 b = 40 b = 40 L =.56678 U =.439364 δ = 0.877686 L =.46038 U =.07408 δ = 9.648043 L =.33459 U = 9.880053 δ = 8.556594 L =.464 U = 8.84843 δ = 7.60660 L =.34858 U = 7.49886 δ = 6.8508 b = 70 b = 70 L =.56399 U =.444904 δ = 0.88093 L =.48080 U =.08505 δ = 9.65345 L =.353 U = 9.8885 δ = 8.5683 L =.43356 U = 8.856438 δ = 7.6308 L =.36409 U = 7.47860 δ = 6.945 b = 00 b = 00 b = 50 b = 50 L =.566300 U =.450397 δ = 0.884097 L =.57035 U =.459449 δ = 0.88934 L =.4309 U =.088896 δ = 9.658777 L =.433508 U =.037 δ = 9.66769 L =.376 U = 9.89673 δ = 8.5690 L =.33034 U = 9.909537 δ = 8.579303 L =.45067 U = 8.86467 δ = 7.6955 L =.479 U = 8.8784 δ = 7.630303 L =.37956 U = 7.43584 δ = 6.97868 L =.4057 U = 7.449078 δ = 6.30855
SRIG helps moe effcent pedctons 0 9 8 7 6 Pos 3 4 5 IR IC SRIG
The possble effect of po dstbuton on the pedctve ntevls detemnes ts selecton s sutble po To choose mong the po dstbutons ech hvng functonl fom comptble wth the lelhood, we cn judge the espectve cpcty to ncese the pedctve pecson. Advntges of the tend obseved n tems of the hype-pmetes The tend obseved n tems of the hype-pmetes cn dd objectvty to the subjectve po nfomton. Let the po nfomton fom n expets be pocessed by some sutble method of po elctton to yeld the coespondng n sets of hype-pmetes. Let [,,b,b, =,,3,...,n] be the n sets of hype-pmetes, pmetes, one set fo ech expet. Hee the vlble po nfomton ves fom expet to expet but the sd tend obseved my help us ech consensus by choosng sngle set of hype-pmetes pmetes s [,,b,b ]=[ =mx, =mx,b =mn b,b =mn b ] Hence the subjectve po nfomton cn be tuned nto n objectve po nfomton wth the help of the sd tend obseved n tems of the hype-pmetes. pmetes. Secondly, ths choce of hype-pmetes pmetes would obvously esult n the effcent estmton nd pedcton. Thdly, s the sd tend obseved nows the possble spce of the unnown hype- pmetes, t cn be clled sot of ptl po elctton.
Pt Two The Rylegh dstbuted suvvl tme wth Rylegh dstbuted censo tme The mxmum lelhood estmton The Byes estmton unnfomtve po The Byes estmton nfomtve po Cedble Intevls Hghest Posteo Denstes HPD Pedctve Dstbuton
The Populton, the Model nd Smplng fo Rylegh suvvl tme nd Ryleth censo tme It s ssumed tht the suvvl tme,x, nd the censo tme,t, ndependently follow the Rylegh dstbutons wth unnown pmetes nd espectvely. The denstes e: λ Let A smple of x - x f x = e, x> 0, > 0 nd t t λ f t λ = e, t > 0, λ > 0 λ n ndomly censoed, ndependently nd dentclly dstbuted obsevtons Y, D, =,,..., n e consdeed wth Y = Mn X, T nd D =, f X T 0, othewse Mgnl of Y s :, λ, λ λ d = 0 λ + y λ, ; 0,, 0 f y = f y d = y + e y > > Mgnl of D s : λ d d P D = d =, d 0, = + λ + λ
Mxmum Lelhood nd Byes Unfom estmtos Pmetes Byes Estmtes Unfom y Γ d Γ d ML Estmtes y d Γ Byes Vnce Unfom y 3 d d d d { Γ Γ Γ } ML Vnces y 4 d λ Pmetes y Γ n d Γ n d n y d Byes estmtes Sque Root Inveted Gmm y + b Γ d+ Γ d + y 3 Γ n d Γ n d Γ n d { } Γ n d Byes Vnces Sque Root InvetedGmm y 4 n d y + b Γ d+ Γ d+ Γ d+ d Γ + { } λ y + b Γ n d+ Γ n d + y + b Γ n d+ Γ n d+ Γ n d+ n d Γ + { } The Jont densty of Y,D: The lelhood s : d n d λ y y L, λ e λ e z y λ d + λ d fyd, λ, = y + e ; y 0,, 0 > λ> λ + λ + λ
Byesn Cedble Intevls nd HPD s A Byesn cedble ntevls e obtned usng the mgnl dstbuton of the espectve pmete of nteest. SRIG Po y +b y +b χ χ α α,, d+ d+ y +b y +b λ χ χ α α, n d, n d + + Unfom Po y y χ χ α α,, d d y y λ χ χ α α, n d, n d p z d = α nd p z = p d + + ln y + b = 0 The hghest posteo denstes e obtned by solvng these two equtons tetvely. Γ + Γ + Γ + = y + b y + b α d, d, d 0 z
ML estmtes, Byes estmtes ssumng Unfom nd SRIG pos Pmetes Byes estmtes Unfom Byes estmtes Sque Root Inveted Gmm ML estmtes Byes Vnces Unfom Byes Vnces Sque Root InvetedGm m ML Vnces = 50 5.466 40.8 50.04 96.80 76.869 94.0054 λ = 00 00.696 94.377 00.36 8.40 3.3754 7.8534 Supemcy of the SRIG po s mmedte becuse of ts lesse vnces. Howeve ByesUnfom nd ML estmtes e lmost eqully effcent. Pmetes Cedble Intevls Unfom po HPD Intevls Unfom po Cedble Intevls Sque Root Inveted Gmm HPD Intevls Sque Root Inveted Gmm =50 33.39,7.948 3.63,7.03 3.678,57.846 3.0,57. λ =00 93.583,5.044 9.9,4.63 87.398,06.957 87.0,06.59 Itetve numecl pocedue ws equed fo the evluton of HPD s. HPD s e bt shned thn the coespondng Cedble Intevls, becuse the posteo dstbuton s slghtly postvely sewed.
The Posteo Pedctve Dstbuton The posteo pedctve dstbuton of the futue obsevton, s defned s p y p, λ f y d dλ z = 0 0 f y λ p, λ z z, λ whee, nd e dt model nd posteo espectvely. d+ n d+ d+ n d+ py z = { + }; y > 0 y b y y b y y y + b y + b d+ n d+ + y b + + + + + y y + b+ y = = wth nd b = b = b, t futhe educes to n+ y n + y + b p y z = ; y > 0 n+ + y + b + y Unle the posteo pedctve dstbuton n pt one, the cedble ntevls cn be used hee to locte those vlues of the hype-pmetes whch leds towds effcent estmton. They cn lso be used to compe numbe of conjugte pos to loo fo the most sutble one.
Concluson The ML estmtes nd the Byes estmtes ssumng the Unfom nd the Jeffeys pos e dentcl nd e lmost eqully effcent fo lge smples. The ML estmtes tend to be ndependent wth the ncese n the test temnton tme. Vnces e epoted s functon of test temnton tme to decde the sutble test temnton tme. An ncese n the test temnton tme s ecommended f t s ecpocted by hghe eltve ncese n the effcency of the estmtes. The SRIG po cn poduce moe pecse estmtes nd pedctve ntevls thn ts compettos, IC nd IR.
Concluson the ML nd the Byes estmtes ssumng the Unfom po e lmost dentcl nd so e the vnces. The Byes estmtes ssumng the Sque Root Inveted gmm po hve the lest vnces. The Cedble ntevls ssumng the Sque Root Inveted Gmm po e much nowe thn the cedble ntevls ssumng the Unfom po. The HPD ntevls ssumng the Sque Root Inveted Gmm po e moe pecse thn the HPD ntevls ssumng the stte of gnonce. As the mgnl posteo denstes e postvely sewed, so the HPD ntevls e slghtly left lgned s comped to the coespondng cedble ntevls. Also, the lengths of the HPD ntevls e shote thn the lengths of the coespondng cedble ntevls.
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