Eksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>.

Transcription

1 Eco 43 Eksame 6 H Utsatt SENSORVEILEDNING Settet består av 9 delspørsmål som alle abefales å telle likt. Svar er gitt i <<.. >>. Problem a. Let the radom variable (rv.) X be expoetially distributed with desity fuctio (pdf) x e for x> gx ( ) = otherwise (i) Fid (i.e., calculate) PX ( < EX ( )) ad PX ( > EX ( )). (ii) Show that the media of X is approximately,3. << Sice EX ( ) = ad the cdf: Gx ( ) = e x, we fid PX ( < EX ( )) = PX ( < ) = PX ( ) = G() = e =, 63, ad PX ( > EX ( )) =,63 =,368 Gm ( ) = / gives m = l() =, 693 >> b. Let by Y = + X where > is a ukow parameter. Show that the pdf of Y is give y ee for < y< f( y; ) = y otherwise << The cdf is for < y < : X F( y) = P( Y y) = P + = P X = e y y derivatio. >> y which gives the pdf by Pritig mistake: Should be,693

2 c. Let Y, Y,, Y be iid with Yi ~ f( yi; ) as i b. (i) Derive the maximum likelihood estimator (mle), ˆ, for, based o Y, Y,, Y. (ii) Show that the Fisher iformatio for oe observatio is I( ) = (iii) Calculate a approximate 95% cofidece iterval (CI) for whe the estimate is ˆ =, 85 ad =. << (ii) l( f( y; )) = l( ) + l( y). Hece y l( f ) =, which gives I( ) =. (i) The log likelihood becomes l( f ) = + y ad l( ) = l( f( yi; )) = (l( ) + ) l( yi) i yi l'( ) = i + y leads to the mle: ˆ = i i Y i (iii) We have by the asymptotic theory for mle s combied with Slutsky s lemma, that ˆ is approximately N(, ) distributed, givig the CI: ˆ ˆ ˆ ±,96 =,85 ±,36 = (, 49,, ). >> d. Let deote the true value of. (i) How may observatios,, is eeded (approximately) to esure that the estimatio error, ˆ, is less tha, with probability at least,95, assumig that the true is =? [Hit: solve P( ˆ,),95 with respect to.] (ii) For ay, how may observatios,, is eeded to esure that the relative estimatio error, ˆ is less tha, with probability at least,95? ˆ << Z = ~ N(,) approximately, givig (i) P( ˆ,) = P( Z (,) / ),95 for (,) / = (, 5),96 or

3 3, 96 = 537,5. ˆ Z (ii) P(,) = P(,) = P( Z (,) ),95 for or, 96 = 384,. >> Problem Poor people i the third world have seldom the opportuity to obtai loas i the bak, or eve a bak accout, ad have great difficulties to save for durable cosumer goods. I the slum area i Nairobi by the ame of Kibera may have orgaized themselves i a maer that actually makes it easier to save. Oe type of orgaizatio is called ROSCA (Rotatig Savig ad Credit Associatio), which is usually orgaized as follows: A group of poor people, mostly wome, agree to establish a mutually bidig lottery that lasts for oe year. Every member commits herself to participate i the lottery for the whole year. The year is divided ito periods. I each period every member cotributes a amout, s/, to a commo pool (also called pot) which, at the ed of each period, thus, cotais the total amout, s. At the ed of the period the group comes together ad chooses by lottery oe perso amog those that have ot already bee chose i earlier periods. I the lottery everyoe eligible have the same chace to be chose. The chose perso the gets the total amout, s, of the pot for disposal. We assume here, at least iitially, that o member fails her commitmet ad drops out of the lottery immediately after havig bee paid the amout s. Hece every member cotributes a total of s distributed over paymets durig the year. I this way each participat gets hold of the sum s at a radom poit i time durig the year without havig to save by hidig moey i the mattress or other places where it is likely that the husbad or others may fid the moey ad disappear with it. a. Imagie that you are oe of the participats of a ROSCA, ad suppose that you wi the pot after V periods. The lottery implies that V is a discrete radom variable with a uiform distributio over the itegers,,,,, i.e., PV ( = v) = for v=,,, Show that + ( + ) EV ( ) = ad var( V) =. [Hit: The followig formulas may be useful: Pritig mistake for var(v). Should be ( )

4 = ( + ) ( + )(+ ) = ] 6 << ( + ) + E( V ) = vp( V = v) = v v = = v= ( + )(+ ) ( + )(+ ) EV ( ) = vpv ( = v) = v v = = 6 6 v= ( + )(+ ) ( + ) ( + )(+ ) 3( + ) var( V ) = = = >> 6 4 b. Let Y be the waitig time, measured i years, util you wi the pot. Explai why + EY ( ) = Explai why, uder the preset coditios, it may be a advatage with may participats i a ROSCA. << V + EY ( ) = E = = + decreases with >> c. We will ow remove the somewhat urealistic assumptio that all participats keep their promises give at the start of a ROSCA. Assume istead that there is a positive probability, p, that someoe who wis the pot immediately afterwards drops out of the ROSCA ad therefore does ot pay the cotributio s/ o ay of the remaiig meetigs. Suppose you wi the pot i period v. Let X be the umber of participats who have dropped out before you wo the pot. For example, if you wi the pot i the first period, X =. If you wi i the secod period, X = or X =. If you wi i the third period, X =, or, ad so o. Assume (i this sectio oly) that p is the same for every participat ad period.

5 5 (i) Explai why X is biomially distributed with E( X V = v) = ( v ) p whe V = v is give, ad the participats take their decisios idepedetly of each other. (ii) Show that the margial expectatio of X is EX ( ) = p (iii) Fid a expressio for the margial variace, var(x), expressed by ad p. << (i) ---- (ii) EX ( ) EEXV ( ( )) EpV ( ( )) p p = = = = (iii) var( X) = E(var( X V)) + var( E( X V)) = E(( V ) p( p)) + var( p( V )) = = p( p) + p >> d. Suppose that the sactio agaist someoe who drops out maily cosists of exclusio from future participatio i the ROSCA, ad exclusio from other ROSCA s as well (the ROSCA s are familiar with each other ad with the records of potetial participats). Let τ deote the welfare loss per year as a result of exclusio, expressed as a amout of moey uits. By itegratig the discouted welfare loss up to the horizo T, we obtai the total discouted welfare loss caused by exclusio, W(T), as T t τ T W ( T ) = τe dt = ( e ) where is the discout rate. Suppose further that a participat believes that her remaiig lifetime, T (measured i years) i the slum is gamma distributed with shape parameter α > ad scale parameter λ >. The parameters αλ, may deped o the perso i questio but ot o. Show that the expected total discouted welfare loss, ω, is τ λ ω = EWT ( ( )) = λ+ α α λ << The mgf for T is Mt () = λ t ad is well defied for t < λ. Obviously T Ee ( ) = M( ) which gives ω. >> e. Whether participats drop out or ot after havig wo the pot will aturally deped o the

6 6 size of. Hece the probability p should deped o as well. With fewer participats the social relatioships that help keepig the participats i the ROSCA are stroger. Let us assume that it is oly by the threat of exclusio that this cotrol works. As a ratioal but heartless aget you cosider droppig out whe you wi the pot i period v accordig to the followig rule: If the remaiig amout, Q, to be paid to the ROSCA util the ed of the year after the wiig period v, exceeds the expected total discouted welfare loss by exclusio, ω, you decide to drop out. What does this rule mea i terms of the correspodig probability p of droppig out? (Hit: Set up a expressio for Q. You ca igore ay discoutig i Q sice the paymet period is so short.) Suppose that the loss per year caused by exclusio, τ, depeds o i such a way that it decreases whe icreases. This assumptio appears reasoable sice, if the ROSCA s were geerally larger, it would be easier to seak back ito a ROSCA at a later year, ad the exclusio would therefore be less efficiet as a sactio. Suppose that all participats are ratioal ad heartless i the sese of adoptig the decisio rule for droppig out give above. Discuss how p varies (i.e., icreases or decreases) as a fuctio of v ad. It is clear that i order to be able to start a ROSCA at all, the idividual p s should be close to zero, eve for the oe that wis after first period. Does this explai why the ROSCA s i actual practice usually are rather small? (Aroud = 3 is a commo size.) ( vs ) v << The remaiig amout to be paid is Q= = s, where v is the time of the year that the wiig occurs. The heartless has p = if Q v = s > ω ad p = if Q ω. Note that ω depeds o oly through τ ad, hece, decreases whe icreases. For give, the p is i geeral largest for the first periods, i.e., v = or. Cosider two scearios: (i) If we fix v to oe of the earlier periods ( to v = say), ad let icrease, the Q icreases ad ω decreases. Hece p icreases with ad the ROSCA is feasible oly whe is sufficietly small. (ii) If we fix the time of the year, ( v), whe a participat wis the pot, ad let icrease, the Q is fixed whileω decreases, causig p to icrease, thus agai leadig to a smaller. O the other had, give these coditios for a ROSCA to appear, there is also a tedecy to choose as large as possible i order to give shortest possible expected waitig time to wiig the pot. >>