(This summarizes what you basically need to know about joint distributions in this course.)

Transcription

1 HG Ot. ECON 430 H Extra exerses for o-semar week 4 (Solutos wll be put o the et at the ed of the week) Itroduto: Revew of multdmesoal dstrbutos (Ths summarzes what you basally eed to kow about jot dstrbutos ths ourse.) I the letures I have oly talked about jot two-dmesoal dstrbutos, but everythg metoed about them geeralzes straghtforward to hgher dmesos. For the sake of ompleteess I revew the bass here (Re s a bt vague o ths): If X, X,, X are rv s, ther jot df s defed by Fx (, x,, x) = PX ( x X x X x) The orrespodg jot pdf or pmf s defed by PX ( = x X = x) dsrete ase f( x, x,, x) = F( x, x,, x) otuous ase x x x I the otuous ase we have: [Note that everythg below hold for dsrete dstrbutos as well, replag tegrals by sums ad pdf s by pmf s.] The df s determed by the pdf by x x x x F( x, x, x ) f ( u, u,, u ) du du du du = alulated by startg from the ermost tegral ad workg out step by step outwards eah sgle tegral. The margal jot pdf for ay sub-olleto of rv s s obtaed by tegratg away all the other varables. To smplfy otato osder the four rv s, XYZUwth,,, pdf f( xyzu,,, ). For example, the margal pdf s of Y ad ( X, ZU, ) are respetvely f ( y) = f ( x, y, z, u) dxdzdu ad f ( x, z, u) = f ( x, y, z, u) dy

2 The expetato of ay futo, g( XY,, ZU, ), of XYZU,,, a be foud as before [ ] = E g( X, Y, Z, U ) g( x, y, z, u) f ( x, y, z, u) dxdydzdu wheever the tegral exsts. The odtoal dstrbuto of Y keepg ( X, ZU, ) fxed to the umbers ( xzu,, ) s determed by the odtoal pdf defed (just as the two-dmesoal ase) by () f( y xzu,, ) = f( xyzu,,, ) f( xzu,, ) Note that ths desrbes a oe-dmesoal dstrbuto of Y where xzu,, appear as parameters. The odtoal expeted value of Y (), sometmes alled the regresso futo of Y wth respet to ( X, ZU, ), ad the odtoal varae of Y () are futos of ( xzu,, ) µ σ ( xzu,, ) = EY ( xzu,, ), ( xzu,, ) = var( Y xzu,, ) The law of total expetato (also alled the law of double expetato ) holds geeral (same proof as the two-dmesoal ase): [ ] [ µ ] EY ( ) = E EY ( X, ZU, ) = E ( X, ZU, ) [ ] [ ] σ [ µ ] var( Y) = E var( Y X, ZU, ) + var EY ( X, ZU, ) = E ( X, ZU, ) + var ( X, ZU, ) If X, X,, X are depedet, the jot pdf (as well as the jot df) a be fatorzed to a produt of the margal pdf s (df s): () f( x, x,, x ) = f ( x ) f ( x ) f ( x ) ad F( x, x,, x ) = F ( x ) F ( x ) F ( x ) X X X X X X Smlarly, the expetato of the produt XX X fatorzes uder depedee: (3) = EXX ( X) EX ( ) EX ( ) EX ( ) If X, X,, X are depedet, the mgf (f t exsts) of the sum, S = X+ X + + X, a also be fatorzed to the produt of the dvdual mgf s: (4) ts MS() t = E e = M X () t M () () X t M X t whh follows dretly from (3) as show the leture for the ase =.

3 3 A remark o modellg: To model a jot pdf (or pmf) lke (e.g.) f( xyzu,,, ) dretly s ofte dffult beause of our ommo lak of tuto o the omplete jot behavour. Se t s usually easer to model oe-dmesoal dstrbutos tha multdmesoal oes, the task s ofte aomplshed by deomposg the jot pdf (pmf) to a produt of oedmesoal pdf s (pmf s) whh s always possble due to (). For example, the - dmesoal ase, usg (), we have f( x, y) = f( y x) f( x) ad the 4-dmesoal ase, usg () several tmes, f( xyzu,,, ) = f( y xzu,, ) f( xzu,, ) = f( y xzu,, ) f( x zu, ) f( zu, ) = = f( y x, z, u) f( x z, u) f( z u) f( u) A e example of ths prple you a fd the exerse (semar week 40) o the ROSCA Narob oerg the dstrbuto of ( V, X ). There the margal dstrbuto of V s modelled as dsrete uform over,,,, ad the odtoal dstrbuto of X, gve V = v, s modelled as bomal ( v, p). From ths, f eeded, we get the full jot pmf of ( V, X ): v = = x f(, vx) = for x= 0 ad v= 0 otherwse x v x p ( p) for x 0,,, v ad v,3,, We would eed ths, e.g., to derve the margal dstrbuto of X, whh s slghtly omplated ad ot bomal(!). Lukly, ths ase, we do t eed the margal dstrbuto of X to derve the expeted value, EX ( ). Muh smpler s to use the law of total expetato as the exerse. So, ths ase we do t eed to bother about the (omplated) jot dstrbuto at all to aswer our questos of terest. Exerse Let X, X,, X be d ad eah expoetally dstrbuted, X ~ exp( λ ). Use (3) above to prove that for ay, the mea, X X = parameters ths dstrbuto expressed by =, s exatly gamma-dstrbuted. Idetfy the λ ad. I the followg expressos I have used the futo symbol, f, geerally the sese that the varous f s represet dfferet futos. It s the struture of the argumets that determes whh futo we are talkg about. Ths artefat s sometmes used mathematal texts to smplfy otato.

4 4 [Ht. Idetfy frst the gamma-dstrbuto for dstrbuto of X = S.] S = X usg (3). The fd the = Exerse Let X, X,, X be d ad eah ormally dstrbuted, X ~ N ( µσ, ), where µ = EX ( ), σ = var( X). I ths exerse we wll ompare three estmators of σ : ˆ σ = S = ( X X) = ˆ σ = S = ( X X) = S = ˆ σ = S = ( X X) = S = a. Put V = ( X X) =. Usg the learty of the - operator, show that ( ) = = V = X X = X X [Ht: Remember that sums (lke tegrals) have learty propertes lke, for example, ( a + bx + y ) = a + b x + y = = = where a, b, are umbers ot depedg o. ] V b. Use supplemetary exerse 4 ad the fat that σ s χ - dstrbuted to show that ˆ σ = S s ubased (.e., ES ( ) = σ ), ad has varae 4 σ var( ˆ σ ) = var( S ) =. Show that both ˆ σ k, k =,, are based dowwards,.e., E( ˆ σk) = kσ, k =,, where both k <. Fd, ad expla why both estmators satsfy, E( ˆ σ ) σ whe k

5 5. (Ths property s usually expressed by sayg that both estmators are asymptotally ubased). d. Gross varae. There are may dfferet ways to ompare estmators. I the bas ourse you leared to ompare ubased estmators by the muh used rtero: For two dfferet ubased estmators of the same parameter, hoose the oe wth smallest varae. Ths rtero s useless our stuato where some of the estmators are based. However, t s easy to geeralze t by usg (stead of varae) the mea squared estmato error. Let ˆ θ be a estmator of some ukow parameter, θ. The the mea squared estmato error (also alled gross varae or brutto-varas Norwega) s defed by ˆ ˆ GVar( θ) = E ( θ θ) () Expla why GVar( ˆ θ) = Var( ˆ θ) wheever ˆ θ s ubased. () Show geeral that, whe the varaes exst, we have (4) ˆ ˆ ˆ ( ˆ ) GVar( θ) = E ( θ θ) = Var( θ) + E( θ) θ [Ht: Add ad subtrat exeute the squarg. ] E( ˆ θ ) sde ˆ ˆ ˆ ˆ ( θ θ) = θ E( θ) + E( θ) θ, ad Our exteded rtero s ow (5) For two dfferet estmators (ot eessarly ubased) for the same parameter, hoose the oe wth smallest mea squared error. e. Show, usg (4), that GVar( ˆ σ ˆ ) < GVar( σ ). Hee ˆ σ s preferable to ˆ σ aordg to rtero (5). [Note that ˆ σ s the same as the so alled maxmum lkelhood estmator, as you wll see later the ourse.] f. Defg a estmator, σ = S, for ay ostat > 0, we obta a whole lass of potetal estmators of σ. Show that mmzg GVar( σ ) s gve by

6 6 Hee, σ = + = σ s the optmal estmator of ˆ σ ths lass - aordg to rtero (5). [Ht: Mmze the expresso you get wth respet to after usg (4) o GVar( ) σ.] g. Prove the followg result (whh mples that all three estmators for σ are osstet): Let ˆ θ, =,, be a sequee of asymptotally ubased estmators for θ wth varaes that overge to zero whe. The ˆ θ s a osstet estmator for θ (.e., ˆ θ P θ ). [Ht: Show frst, usg Markov s equalty, that, f GVar( ˆ ) 0, the θ ˆ θ P θ. The use (4) to dedue the result asked for. ] [Note. I eoometrs we operate wth several oepts of basedess/ubasedess of a estmator. I ths ourse you lear two oepts,.e., (u)basedess ad ()osstey : If ˆ θ s a estmator of θ, we say that ˆ θ s ubased (based) f E( ˆ θ) = θ ( E( ˆ θ) θ ) ˆ θ s osstet (osstet) f plm( ˆ θ) = θ ( plm( ˆ θ) θ ) These two oepts are ot equvalet. Ubasedess does ot geeral mply osstey, ad osstey does ot geeral mply ubasedess. Ubasedess s a property of ˆ θ for a sgle gve, whle osstey s a property of the whole sequee, ˆ θ, =,, O the other had, osstey has wder applablty tha ubasedess beause of the otuty property (f ˆ θ s osstet for θ, the g( ˆ θ ) s osstet for g( θ ) wheever gx ( ) s a otuous futo), a property ot shared by the ubased-oept (eve f E( ˆ θ ) = θ, the qute ofte Eg( ˆ θ ) g( θ) - exept whe g s lear ( g( x) = a + bx )). ]