M2S1 - EXERCISES 8: SOLUTIONS

Transcription

1 MS - EXERCISES 8: SOLUTIONS. As X,..., X P ossoλ, a gve that T ˉX, the usg elemetary propertes of expectatos, we have E ft [T E fx [X λ λ, so that T s a ubase estmator of λ. T X X X Furthermore X X X From propertes of expectatos, varaces, a the Posso strbuto, X X X. E fx [ X V ar fx [X + {E fx [X} λ + λ λ λ +. X Now, from propertes of epeet Posso raom varables Y X P ossoλ so therefore, takg expectatos the above, E ft [T E fx [ X E f Y [ Y λ λ + λ λ + λ λ + λ λ + [λ λ + λ λ + λ.. From the theorem lectures, a by propertes of the Gamma strbuto, we ca wrte V s σ where X χ Gamma, χ Gamma, V X,, so that, usg the Gamma expectato a varace results, E fx [X / / μ, Var f X [X / / σ, say. Hece, by the Cetral Lmt Theorem, V μ σ V Z N,. Hece, substtutg the efto for V, s σ s σ σ Z N,, a fally, by a locato/scale trasformato to Z σ + σ Z, we have the result that approxmately s Z N σ σ 4,. MS SOLUTIONS 8: page of 7

2 3. X,..., X Gammaα, β so that E fx [X α/β a Var fx [X α/β a therefore [ { } E fx X α VarfX [X + E fx [X β + α α α + β β. Hece for the metho of momets estmators ˆα MM a ˆβ MM, ee to smultaeously solve the followg: FIRST MOMENT SECOND MOMENT Elemetary algebra gves 4. For > x x α β x x + S α α + β, where S x x ˆα MM x S, ˆβMM x S. x x. STEP L f X x ; x STEP log L log + log x x {log L} + log x ˆML / log x Hece For > {log L} < for all ESTIMATE : ˆ ML, log x ESTIMATOR:. log X STEP L f X x ; + x + STEP log L log + + log x + + x {log L} + log x ˆML / log x Hece {log L} < for all + ESTIMATE : ˆ ML, ESTIMATOR: log x. log X MS SOLUTIONS 8: page of 7

3 For > STEP L f X x ; x exp { x } STEP log L log + log x x { } x exp x {log L} x ˆML x Hece {log L} < ESTIMATE : ˆ ML for all, ESTIMATOR:. x X v Because of the costrat the pf that x x 3 x 3, x,..., x, STEP L f X x ;, otherwse. STEP log L log + log 3 log x At ths pot we ote that the lkelhoo s mootocally creasg, a hece the lkelhoo s maxmze whe s as large as possble but so that the costrat x,..., x s stll satsfe, hece ESTIMATE : ˆ ML m {x,..., x }, ESTIMATOR: m {X,..., X }. v Notg the costrat the pf that x, we have STEP L, f X x ; x + + x, STEP log L, log + log + log x x,..., x {log L, } + log log x ˆ ML {log L, } log x log ˆ ML The seco of the partal ervatve equatos cates aga that the maxmum of the lkelhoo occurs MS SOLUTIONS 8: page 3 of 7

4 whe s as large as possble, that s, whe ˆ ML m {x,..., x }. Hece ESTIMATES: ˆML [ log x log {m {x,..., x }} ˆML m {x,..., x } ESTIMATORS: [ log x log {m {X,..., X }} m {X,..., X } 5. Follow the four step proceure that ca be summarze as follows: for a observe raom sample x,..., x from a strbuto represete by mass/esty fucto f X x; STEP : Form the lkelhoo fucto L STEP L Take atural log to obta log L f X x ; : F the value of at whch log L a hece L s maxmze wth the parameter space Θ by fferetato : Check the maxmum value has bee fou. Formally, we efe the maxmum lkelhoo estmate of, ˆ ML, as Hece, for the P ossoλ case ˆML arg max L. STEP Lλ f X x ; λ e λ λ x e λ λ x! x! STEP log Lλ log x! λ + x log λ {log Lλ} + x λ λ ˆλML x x {log Lλ} λ x λ < for all λ Therefore ESTIMATE : ˆλ ML x, ESTIMATOR: X, a from questo o ths sheet, we kow that f T X X the E ft [T λ a T s ubase. Now, f τ τλ e λ so that λ log τ, x MS SOLUTIONS 8: page 4 of 7

5 we ca reformulate the lkelhoo terms of τ, gvg log Lτ log x! + log τ + x log log τ a {log Lτ} λ x τ log τ + τ ˆτ ML exp whch ca be show to be the value that maxmzes the lkelhoo, so that ˆτ ML λ τ ˆλML. { } x exp { x}, 6. X,..., X Expoetal/ so that E fx [X a hece, usg staar mgf techques, we have X X Gamma, E fx [X, so that f T X X the E ft [T, a hece T s a ubase estmator of. Now f Y m {X,..., X }, the prevous orer statstcs results gve that { F Y y { F X y} e y/} e y/, y >, so that Y Expoetal. Hece f T Y the a hece T s a ubase estmator of. E fy [Y, E f T [T, Straghtforwar calculatos show that the MLE s gve by ˆ /T, where T s gve by T X a s strbute as Ga, [use momet geeratg fuctos f ecessary. We have that the pf of T s f T t; t e t /Γ. Note that sce s a teger Γ Γ. We have E/T t e t t Γ t e t t Γ, sce the tegra s the pf of Ga,. Hece ˆ has expectato /, so the MLE s base, but ˆ/ s ubase. Trasformato gves that X s strbute as Ga, /.e. χ, so t s pvotal. Let c a c be the α/ a α/ quatles of χ. We the have P c < T < c α. Pvot to obta P c /T < < c /T α, MS SOLUTIONS 8: page 5 of 7

6 so that c /T, c /T s a α% cofece terval. To test H : accept H ff s the cofece terval costructe from the ata sample. 7. We have f X x, x +, F Xx x x +, x +. E fx [X by tegrato, or by otg that the pf s costat a hece symmetrc about a hece, usg staar expectato techques, we have that f T X E ft [T E fx [X, a hece T s a ubase estmator of. Now f Y m {X,..., X } a Y max {X,..., X }, the prevous orer statstcs results gve that f Y y f X y { F X y} a f Y y f X y {F X y} For the expectatos, E fy [Y + y + { { y { } + y y } y } { } + y, y +, { } + y, y +. + t t t, settg t + y /y + t t t t t a E fy [Y + +, + y { } + y y t t t, settg t + y /y t t t, + t t so that f M Y + Y / the by propertes of expectatos E fm [M E f Y [Y + [ [ E f Y [Y , MS SOLUTIONS 8: page 6 of 7

7 a hece M s a ubase estmator for. [Alteratvely, ote that, by symmetry, we have that Y has the same strbuto as Y. Hece, EY E Y. Rearrage, usg propertes of expectato, to get EY + Y. 8. The lkelhoo fucto s L Ix < Imax{x,..., x } <. Coserg ths as a fucto of, t s maxmse at max{x,..., x }. So, the MLE s Y max{x,..., X }. Drectly, we have P ˆ/ y y. The we ca check that the gve terval satsfes the ecessary property: P ˆ ˆ/α / P ˆ/ α / P ˆ/ α / α. 9. The key here s that X a X are IID N,, so that X +X s strbute as χ. The, rectly, we calculate: P {, S} {P N,.36} { [ Φ.36} { ++.95}.95, usg the ht. Also, P {, C} P χ , o lookg up Tables of the ch-square strbuto. [Note that χ s actually expoetal, wth mea, so we ca calculate the strbuto fucto etc. rectly, wthout the ee for tables. The sesble crtero to scrmate betwee S a C s area more geerally, the volume of a cofece set: we wat the raom set to be small, whle cotag the true ukow parameter value wth the specfe probablty, here 95%. Note that here S a C have fxe o-raom areas. The area of S s , whle the area of C s π , so C s preferable accorg to the crtero.. Suppose X,..., X are IID Nμ, σ. Whe σ s ukow, we costruct the cofece terval usg the pvotal quatty ˉX μ S/ t, Stuet s t strbuto, o egrees of freeom. The a α% cofece terval for μ s ˉX t α/ S/, ˉX + t α/ S/, terms of the α/ quatle of t. Note that S X ˉX { X ˉX }. Here, 5, α.5, so t α/.776. Also, s.79. Hece the wth the the cofece terval s.776 s From lectures, the wth of the cofece terval whe σ s kow s z α/ σ , sce σ. The probablty questo s P S > z α/ σ/t α/ P S >.96/.776. But we kow that S χ 4 sce σ, so the requre probablty s P χ 4 > 4.96/.776 P χ 4 > MS SOLUTIONS 8: page 7 of 7