. The lower the MSE, the better the estimator.

Transcription

1 Estmator Qualty Bas s ( ˆ) [ ˆ θ θ E θ θ] θ A estmator s uased f ( ˆ θ θ ) 0 A estmator s asymptotcally uased f lm ( ˆ θ θ ) 0 A estmator s cosstet f for allδ > 0, lm P( ˆ θ θ < δ) Mea square error s MSE ( ˆ) [( ˆ ) ] var( ˆ) [ ( ˆ θ θ E θ θ θ + θ θ)] The lower the MSE, the etter the estmator Emprcal Dstruto for Complete Data f ( ) The hazard rate fucto h s defed y h ( ) S ( ) The cumulatve hazard fucto H s defed y H( ) l S( ) h( u) du A ogve s a cumulatve dstruto fucto, F ( ), ased o a set of data pots, each gve proalty / where we learly terpolate etwee the data pots A hstogram s the dervatve of a ogve, f ( ), ad s costat etwee data pots j f ( ) c ( ) j cj 3 Kapla-Meer ad Nelso-Åale Estmators The Kapla-Meer estmator s defed y j s S () t r, yj t < yj where s s the umer of deaths ad r s the set of all dvduals at rsk at tme y 0 The Nelso-Åale estmator s defed y j ˆ s () Ht, yj t yj r < ˆ St ˆ( ) e H () t

2 4 Varace of Kapla-Meer ad Nelso-Åale Estmators Greewood s appromato, the varace of the Kapla-Meer estmator, s defed y j s var[ S( yj)] S( yj) r ( r s ) The varace of Nelso-Åale estmator s defed y j var[ ˆ s H( yj )] r The cofdece terval for S ( t) s / U ( S ( t), S ( t ) ) where U U z ep S()l t S() t α / ˆ σ S The cofdece terval for Ht ˆ () s Ht ˆ (), ˆ HtU ( ) U 5 Kerel Smoothg where U z H ep Ht ˆ () α / ˆ σ ˆ The uform kerel desty fucto s gve y k y 0 < y- or > y+ ( ) y- y+ The uform kerel dstruto fucto s gve y 0 < y- ( y ) Ky ( ) y- y+ > y+ The ftted desty s gve y ˆ( ) ( ) ( ) [ ( ) f p y ky F + F ( )] The ftted dstruto s gve y Fˆ ( ) p ( y ) Ky ( )

3 The tragular desty ad dstruto fuctos are gve y: If the oservato, y, s more tha uts to the left of the estmato pot,, K s If the oservato, y, s wth uts to the left of the estmato pot,, the tragular kerel desty fucto ca e show y: ky ( ) y y y + ( ) where y ( ) y+ k ad the tragular kerel dstruto fucto ca e show y the area of the tragle uder the desty fucto or: ( y+ ) K y( ) ( ase heght) [( y + ) ] [ ky( )] If the oservato, y, s wth uts to the rght of the estmato pot,, the tragular kerel desty fucto ca e show y: ky ( ) y y y + ( ) where y ( ) y k ad the tragular kerel dstruto fucto ca e show y the area of the tragle uder the desty fucto or: ( y ) K y( ) ( ase heght) [ ( y )] [ ky( )] If the oservato, y, s more tha uts to the rght of the estmato pot,, K s 0 3

4 6 Appromatos For Large Data Sets c j the rght edpot of the j-th terval d j the umer of left trucated oservatos the terval j j [ c, c + ) Ths ca also e see as the umer of ew etrats or the umer of polces wth a deductle u j the umer of rght cesored oservatos the terval[ cj, c j + ) Ths ca also e see as the umer of wthdrawals or the umer of polces wth a upper lmt j the umer of evets the terval[ cj, c j + ) Ths ca also e see as the umer of deaths or the umer of clams r j the rsk set avalale the terval[ cj, c j + ) The populato at tme c j s gve y j j P d u If we assumeα of the ew etres occur efore tme c j ad β wthdrawals occur efore ths tme, the rj Pj + αdj βuj Uder the UDD assumpto, rj Pj + 05( dj uj) 7 Method Of Momets For a dstruto wth k parameters, calculate the frst k momets of the sample Equate these to the correspodg theoretcal momets ad solve the resultg smultaeous equatos [Note: If matchg the varace, use 8 Percetle Matchg ( ) stead of ( ) ] A smoothed emprcal percetle s otaed y addg oe to the sample sze ad multplyg y the percet If that s a teger, use that order statstc If ot, ut t s etwee ad, terpolate etwee the order statstcs It s udefed f the product s less tha or greater tha 4

5 9 Mamum Lkelhood Estmators Take the log of the desty fucto f ( ), l f ( ) Take the dervatve(s) of l f ( ) wth respect to each parameter l f ( ) Sum up the for each parameter over all s ad set ths result equal to zero to θ estmate the parameter(s) The method of momets uses data,,, for logormal parameters, whle MLEs are calculated usg l,l,,l for logormal parameters The MLE for a uform dstruto o [ a, θ ] s gve yθ ma{,,, } If the data s cesored, or s grouped wth cesored data, the the MLE s the cesorg pot tmes the umer of oservatos dvded y the umer of oservatos elow the cesorg pot For grouped data wth o cesorg, the MLE s the lower of the upper oud of the hghest terval ad the result of the calculato as f the lower oud of the hghest terval s a cesorg pot For epoetal dstrutos, Posso dstrutos, ad gamma dstrutos wth a fed α, the MLE s equal to the sample mea For ormal dstrutos, the MLE for µ s the sample mea ad the MLE forσ s the square root of the emprcal varace For the epoetal dstruto, whe trucated ad cesored, the MLE s gve y the sum of the oservatos less the trucato pot d over the umer of ucesored oservatos τ τ d For a Weull dstruto wth a fedτ, the MLE sθ τ where s the umer of ucesored oservatos ad cludes all data Trucato ad Cesorg For grouped data, the lkelhood of a oservato s Fc ( j) Fc ( j ) for a oservato the terval ( cj, cj ) f ( ) For data left trucated at d, each oservato has a codtoal lkelhood of Sd ( ) For data rght cesored at u, each oservato of u has a lkelhood of Su ( ) Su For data that are oth left trucated ad rght cesored, we have lkelhood ( ) Sd ( ) Fc ( j ) Fd ( ) Grouped data etwee d ad c j wth trucato at d has lkelhood Fd ( ) 5

6 0 Varace Of Mamum Lkelhood Estmators The Delta Method For ay fucto ( ) For two varales, [ ] g, for oe varale, the varace s gve y [ g X ] dg var ( ) var( X) d g g g g var g( X, Y) var( X) + cov( X, Y) + var( Y) y y I geeral, var( g ( X)) ( g) Σ( g ) where g g g,, ad σ σ Σ σ σ The asymptotc covarace matr of the MLE s the verse of the formato matr: l f ( ) l f( ) l f( ) I( θrs ) E E θr θs θr θs The formato matr ca e appromated y the oserved formato: [Note: l( θ ) l( θ) l( θ) I( θrs ) E E θr θs θr θs a d Σ I ] c d ad c c a Whe the MLE for a sgle-parameter dstruto s the same as the method of momets estmator, the varace of the sample mea sσ Dscrete Dstrutos Bomal - Posso - Negatve Bomal - Co Proportoal Hazards Model > s s < s The Co model s gve y β ( ) ( ) z h h e 0 where h 0 ( ) s the asele hazard fucto 0 H( ) l S( ) h( ) ad βz c e 6

7 Relatve Rsk A e A B R e where R s the relatve rsk ad A ad B are two sets of B βz wth e dfferet covarates ( z, z,, z ) r l R A B f e e h ( ) ch 0 0 0( ) ( ) [ ] [ ] l( θ, c) l L( θ, c) l f( ) Solve c terms ofθ, the solve forθ usg Partal Lkelhood l c ad l θ If the k memers of the rsk set at tme y have proportoalty costats c, c,, ck ad memer j des, the the partal lkelhood of the evet s c β j j e k k β c e If the k memers of the rsk set at tme y have proportoalty costats c, c,, ck ad d memers j,, jd de, the the Breslow partal lkelhood s d d β j c j e k d k d β c e ( ) ( ) The Nelso-Åale estmato for the asele hazard rate s ˆ s () j H0 t y j t c R( y j ) The accelerated falure tme model s gve y S( z, β ) S ( e ) 0 T β z where for ay set of factors Z, the survval fucto ca e epressed y multplyg the age y a costat, the costat ot varyg over tme or age 7

8 3 Hypothess Tests If oserved data s trucated at d F F( ) F( d) ( ) Fd ( ) ad f f ( ) ( ) Fd ( ) A p-p plot ca e used y plottg the emprcal dstruto o the -as agast the j j ftted dstruto o the y-as creatg a set of pots( F ( ), F ( )), where F j ( j) + ad the oservatos are sorted If the slope of the emprcal dstruto s less tha 45, the ftted dstruto s puttg too lttle weght that rego; f the slope of the emprcal dstruto s greater tha 45, the ftted dstruto s puttg too much weght that rego Kolmogorov-Smrov Tests The Kolmogorov-Smrov statstc D s the asolute mamum dfferece etwee the emprcal ad ftted dstrutos, ma F ( ) ( ; ˆ F θ ), d u where d s the lower trucato pot ad u s the upper cesorg pot To evaluate the statstc, usg sorted data pots, at each oservato pot j, take the mamum of F ( j ) ad j j F ( j ) If ay two pots j ad + are equal, take the mamum of j j+ F ( j ) ad j F ( j ) D s the mamum of these mama over all j Ch-Square Tests F( ) F( ) F( ) F( ) For each group of data, let p j e the proalty that X s the j-th group uder the hypothess, let e the total umer of oservatos, ad let j e the umer of oservatos group j Let Ej pj The ch-square statstc s k ( j Ej ) Q E j j The epected umer of oservatos should e at least 5 each terval 8

9 Degrees of Freedom To determe the umer of degrees of freedom, fgure out how may of the oservatos could e chose at radom ad are ot derved Whe you dvde clams to k groups ad estmate r parameters, the umer of degrees of freedom s k r For data grouped to k groups, the hypothess that the parametrc ft s good s accepted f the ch-square statstc Q s less tha the crtcal value at the approprate umer of degrees of freedom If a dstruto wth parameters s gve, there are k degrees of freedom If they say the parameters were ftted y some techque such as mamum lkelhood, the there are k rdegrees of freedom Kolmogorov-Smrov Aderso-Darlg Ch-square Should e used oly for dvdual data Should e used oly for dvdual data May e used for dvdual or grouped data Oly for cotuous fts Cotuous or dscrete fts Should lower crtcal value fu < Should lower crtcal value fu < No adjustmet of crtcal value s eeded for u < Crtcal value should e lowered f parameters are ftted Crtcal value should e lowered f parameters are ftted Crtcal value s automatcally adjusted f parameters are ftted Crtcal value decles wth larger sample sze Crtcal value depedet of sample sze Crtcal value depedet of sample sze No dscreto No dscreto Dscreto groupg of Uform weght o all parts of dstruto Lkelhood Rato Algorthm Hgher weght o tals of dstruto data Hgher weght o tervals wth low ftted proalty To determe the model wth the approprate umer of parameters to use, we pck the model where twce the crease the loglkelhood from to + parameters eceeds the ( α)th percetle of ch-square wth the approprate degrees of freedom X [ l( θ,, θ) l( θ,, θ + )] Whe a restrcto, θ cθ, s placed o two models wth separate parameters θ adθ, frst ota ther mamum loglkelhoods separately ad sum them, ad the add the restrcto ad ota the mamum loglkelhood of the comed model 9

10 Schwarz Bayesa Crtero The lkelhood rato algorthm thresholds wll e easer to meet as grows, so the Schwarz Bayesa Crtero compesates for ths y applyg a pealty of ( r )lto each loglkelhood model efore comparg them, where r s the umer of parameters Now choose the ew mamum loglkelhood 4 Iterpolato ad Smoothg f ( ) f ( ) y where ( 0)( ) ( ) f ( ) ( )( ) ( ) 0 For ay + pots, there s a uque polyomal terpolatg the pots The terpolatg polyomal may oscllate, ad therefore ot e sutale for terpolato The terpolatg polyomal s ot recommeded for etrapolato due to oscllato Polyomals may e used for smoothg as well as terpolato Whe used for smoothg, a polyomal of degree lower tha s used for + pots The polyomal selected s the oe mmzg the least squares dfferece from the pots Cuc Sples 3 ( ) ( ) ( ) ( ) f a + + c + d for 0,,, The followg codtos must e met: a) f must go through (, y) ad ( +, y+ ) f( ) y ad f( + ) y+ for 0,,, (Ths codto results costrats) ) The dervatves of the f must e cotuous at the kots f ( + ) f + ( + ) for 0,,, (Ths codto results costrats) c) The secod dervatves of the f must e cotuous at the kots f ( + ) f + ( + ) for 0,,, (Ths codto results costrats) There are the + ( ) + ( ) 4 costrats 0

11 Let m f ( ) f ( ) for 0,,, h + for 0,,, g ( h + h) ( + ) for,, y+ y y+ y s h for 0,,, u s s + for,, The h m + g m + hm 6u for,, + Ad so the coeffcets ecome: a y 3 y + a c h d h h m c [Note: Do t memorze, just derve t from a, c, ad d ] d m + 6h m The formulas for a clamped sple are gve y 3 m m0 s 0 f 0 ( 0) h m0 s ad m [ f ( ) s ] h h tmes the slope you re gog to less the slope you re gog from, the mus m s 3 m tmes the slope you re gog to less the slope you re gog from the mus h Type Restrctos Natural Sple m0 m 0 Curvature Adjusted Sple m0 ad m are specfed Paraolc Ru-Out Sple m0 m ad m m m m0 m m m m m m Cuc Ru-Out Sple ad h h h h 0 Clamped Sple f0 ( 0) ad f ( ) are specfed Curvature The square orm measure of curvature s defed y S [ f ( )] d hj For cuc sples ths ecomes S ( mj + mjmj+ + mj+ ) j 0 3 a m m

12 Etrapolato To etrapolate eyod the terval[ 0, ]: f ( ) f( ) + f ( )( ) for > f ( ) f( 0) + f ( 0)( 0) for < 0 Smoothg Sples Cuc sples may e used for smoothg as well as terpolato Ft s measured y least squares dfferece etwee the curve ad the kots The dfferece s weghted y the recprocals of the varaces Smoothess s measured y the square orm measure of curvature A weght p etwee 0 ad s assged to ft If p 0, the method reduces to a weghted least squares regresso le To costruct the sple, the pots the curve goes through are determed y solvg a lear system The sple s the terpolated etwee those kots 5 Lmted Fluctuato Credlty Let F e the amout of uts ecessary for full credlty CV f e the coeffcet of varato of the frequecy dstruto CV e the coeffcet of varato of the severty dstruto s y ad 0 k, where ( + P) y Φ The horzotal as of the tale flls the lak You wat to e wth k of epected P of the tme ad the vertcal as of the tale flls the lak How may are eeded for full credlty? Eperece Epressed I: Eposure Uts Numer Of Clams Aggregate Losses Numer of Clams CV 0 f Credlty For: Clam Sze (Severty) 0 s CV µ 0µ f CV f CV 0 s 0 s fcvf µµ µ CV f 0 s s Aggregate Losses/ Pure Premum CV s 0 CVf + µ f 0 µ f CV 0 µµ f s f CVf CV + µ s f CV + µ s f

13 Partal Credlty Whe there s adequate eperece for full credlty, we must determe Z, the credlty factor Z wll e used to determe the credlty premum P C : P ZX + ( Z) M M + Z( X M) C where M s the maual premum, the premum tally assumed f there s o credlty The square root rule states that the credlty factor for epected clams s: Z F where F s the umer of epected clams eeded for full credlty 6 Bayesa Estmato Dscrete I the frst row, eter the pror proalty that the rsk s each class (e, the proaltes of eg a good or ad drver) I the secod row, eter the lkelhood of the eperece gve the class (e, the proaltes of havg o crashes oe year ad oe crash the et year) 3 The thrd row s the product of the frst two rows Sum up the etres the thrd row 4 The fourth row s the quotet of the thrd row over ts sum These are the posteror proaltes of eg each class gve the eperece 5 I the ffth row, eter the epected value, gve that the rsk s the class These are the hypothetcal meas (e, ES ( ) ENEX ( ) ( ) ) 6 I the sth row, eter the product of the fourth ad ffth rows Sum up the etres the sth row Ths sum s the epected sze of the et loss for ths rsk, gve the eperece Ths s kow as the Bayesa premum Two shortcuts are avalale: If the proaltes of eg each of the classes are equal, you ca skp the frst le ad treat the secod le as f t s the thrd le If you are ot terested the posteror proaltes ut oly the predctve epected value, you ca skp le 4 ad stead, wegh the hypothetcal meas y the jot proaltes le 3 3

14 Cotuous The pror desty, π ( θ ), s the tal desty fucto for the parameter whch descres the model The model desty, f( θ ), s the desty fucto descrg the tem eg modeled The ucodtoal desty, f ( ), s the desty fucto for the tem eg modeled f a perso s pcked at radom f ( ) f( θ ) πθ ( ) dθ The posteror desty, π ( θ,, ), s the revsed desty fucto for the parameter ased o data,, The predctve desty, f ( +,, ), s the revsed ucodtoal f ( ) ased o the oservatos,, f ( +,, ) f( θ ) πθ (,, ) dθ The posteror desty s the: f(,, θ ) πθ ( ) f(,, θπθ ) ( ) πθ ( ) f( ) f (,, θ ) πθ ( ) dθ For the loss fucto mmzg mea square error, l( ˆ θ, θ) ( ˆ θ θ), the Bayesa pot estmate s the mea of the posteror dstruto For the loss fucto mmzg asolute value of the error, l( ˆ θ, θ) ˆ θ θ, the Bayesa pot estmate s the meda of the posteror dstruto For the zero-oe loss fucto, whch s 0 f ˆ θ θ ad s otherwse a costat, the Bayesa pot estmate s the mode of the posteror dstruto 7 Bayesa Credlty Cojugate Prors a) Posso/Gamma Suppose clam frequecy s Posso wth parameter λ ad λ vares accordg to a gamma dstruto wth parametersα adγ θ There are eposures ad clams The the posteror dstruto of λ s a gamma dstruto wth parameters α α + ad γ γ + P C α α + γ γ + wherez γ + 4

15 ) Normal/Normal Suppose clam sze has a ormal dstruto wth meaθ ad varaceν ad θ vares accordg to a ormal dstruto wth mea µ ad varace a The the posteror dstruto s gve y: v( µ ) a( ) PC µ + va ad a a where Z v+ a v + a v+ a + v/ a c) Bomal/Beta Suppose clam frequecy s a Beroull dstruto wth parameter p ad p vares accordg to a eta dstruto wth parameters a ad There are m trals wth k successes The the posteror dstruto s a eta dstruto wth parameters a a+ k ad + m k a a k PC a + m where Z + a + + m a + + m Recogze these dfferet forms of eta fuctos: ( a ) ( a, ) 6 ( ) ( a, ) 5 ( a 5, ) d) Epoetal/Iverse Gamma Suppose clam sze has a epoetal dstruto wth parameterθ adθ vares accordg to a verse gamma dstruto wth parametersα ad β There are clams,, oserved The the posteror dstruto s a verse gamma dstruto wth parametersα α + ad β β + β β + PC where Z α α + α + 8 Bühlma Credlty Let Θ represet the hypothess as to the rsk class to whch a eposure elogs µ ( Θ ) (the hypothetcal mea) v( Θ ) (the process varace) [ ( )] [ ] [ ] µ EΘ µ Θ (the overall mea) v EΘ v( Θ ) (the mea of the process varace) a var µ ( Θ ) (the varace of the hypothetcal mea) Θ 5

16 Bühlma s credlty factor s gve y: a Z where k + k a+ v v ad s the umer of oservatos a Whe there are oly two possle values, X ad X, oe wth proalty q, the Beroull shortcut gves the varace etwee those values to e ( X X ) q( q) Whe provded wth hypothetcal meas ad process varaces stead of separated y rsk class, µ ad v rema the same, ut a must cota the varace etwee the hypothetcal meas as well as the varato of the hypothetcal meas wth each group Ths s gve y: a var( N) E var( N I) + var E( N I) [ ] [ ] Suppose m j s the eposure for class j, ad we oserve a average loss per eposure of X j The the credlty factor ecomes: am Z + am where m s the sum of the recprocals of the varaces for all the eposures, v j For the Posso/gamma par, the Bühlma k s θ For the ormal/ormal par, the Bühlma k s va For the omal/eta par, the Bühlma k s a Bühlma As Least Squares Estmate Of Bayes + Let Y e the -th Bayesa estmate Y ˆ e the -th least squares Bühlma estmate of the Bayesa estmate X e the -th oservato p e the proalty of oservg var( X ) px ( X) a+ v cov( X, Y) pxy XY a X The: P ( X ) ( Z) E( X) + ZX C a cov( X, Y) where Z ad a+ v var( X) E( X) EY ( ) EY ( ˆ) 6

17 9 No-Parametrc Emprcal Bayesa Estmato Let m e the umer of eposures group year j m j m e the umer of eposures group over all years j j m m e the total umer of eposures all groups over all years Zˆ ma ˆ e the credlty factor for group maˆ+ vˆ Uform Eposures No-Uform Eposures r r ˆ µ j j r ˆ µ j m r v ˆ ( j ) r j r vˆ aˆ ( ) r vˆ aˆ r j r r j m ( ) j j ( ) m ( ) vˆ ( r ) m m r m Whe asked to use the method preservg total losses, use the credlty weghted mea gve y: Zˆ ˆ ˆ µ Z 0 Sem-Parametrc Emprcal Bayesa Estmato If a model s assumed to have a Posso dstruto, µ ad v are equal The overall sample varace s gve y: ( ) s vˆ+ aˆ ( ) ˆ a vˆ 7

18 Smulato Iverso Method Assumg the dstruto fucto of X, FX ( X ), s cotuous ad mootocally creasg, f u s a uform radom umer of [0,] the: P [ F ( u) ] P [ F( u) F( )] F( ) ad so we set F ( u) If F does ot assume the valueu ecause t jumps, so that Fc ( ) aad Fc (), the map all radom uform umers a u to c If F( ) a a rage[, ), the map the uform radom umer a to To smulate s o a uform terval [ a,, ] use a+ ( a ) u To smulate s for a ormal dstruto, fd the stadard ut ormal value, Z, such that Φ ( Z) u The o-ut ormal value s gve y, X µ + Zσ Ad the logormal value s gve y e X e µ + Zσ To smulate s for a Posso process o[0, t] wth mea λ, smulate s for a epoetal dstruto wth meaθ λ to ota the tmes utl each evet occurs Accumulate these terarrval tmes ad cout the umer of evets efore tme t The smulate the quatty questo per evet wth the same process Whe smulatg s for a med dstruto, dstruto A wth proalty p ad dstruto B wth proalty p, treat the dstrutos as a cumulatve fucto y usg the frst radom umer to smulate the dstruto to use ad the et to smulate For a dscrete dstruto, the dstruto fucto s: F( ) p P( X ) The a uform radom umer the terval[ p, p ) j j gets mapped to 8

19 Numer Of Data Values To Geerate The mmum umer of smulatos to ru, N, s gve y: s y N N0CV k ( ) where s ad s aˆ Bootstrap Appromato Let θ ( F) e a parameter of the dstruto fucto F, ad let g (,, ) e a estmator ased o a sample of tems The estmate s the mea square error of g appled to the emprcal dstruto from θ ( F e ), where F e s the emprcal dstruto Sce g s a fucto of tems, each of them havg a emprcal dstruto wth possle values, dfferet possltes must e cosdered If a estmator uses oly k of the pots, the umer of ootstrap samples s k, sce varyg the remag k pots has o effect o the estmate The ootstrap appromato of the mea square error of the sample mea as a estmator for the true mea s equal to the emprcal dstruto varace: s ( ) ( ) Mscellaeous E[( X X) ] E[ X ] 3 E[ X] E[ X ] + E[ X] Skewess γ 3 3 σ σ E[( X X) ] E[ X ] 4 E[ X] E[ X ] + 6 E[ X] E[ X ] 3 E[ X] Kurtoss γ 4 4 σ σ 9