Exam Statistics 6 th September 2017 Solution
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1 Exam Statstcs 6 th September 17 Soluto Maura Mezzett Exercse 1 Let (X 1,..., X be a raom sample of... raom varables. Let f θ (x be the esty fucto. Let ˆθ be the MLE of θ, θ be the true parameter, L(θ be the lkelhoo fucto, l(θ be the loglkelhoo fucto, u(θ the score fucto, a I(θ be the Fsher formato matrx 1. A statstc T = T (X s a sucet statstc for θ a T = T (X s always a ubase estmator for θ, b T = T (X s a cosstet estmator for θ, c T = T (X s the Metho of Momet estmator for θ, oe of the prevous s correct.. Whch of the followg s ot true? a b c u(ˆθ = [ ] l(θ x E = ( ( logl(θ x logl(θ x V ar = E [ ( logl(θ x ] ( logl(θ x E = E 3. To verfy the hypothess H : µ = µ versus H 1 : µ µ, whch of the followg statemets s NOT true: a Wal test requres oly the urestrcte moel to be estmate. 1
2 b c Score test requres oly the restrcte moel to be estmate. LR test requres both the restrcte a urestrcte moels to be estmate The three statstcal tests prove the same values the observe sample 4. Whch of the followg statemets s TRUE? a b c A very low sgcace level creases the chaces of a Type I error. Type I error occurs whe the ull hypothess s rejecte a the ull hypothess s actually true. Type I error occurs whe the ull hypothess s accepte a the ull hypothess s actually true. Type II error occurs whe the ull hypothess s rejecte a the ull hypothess s actually false. Exercse We coser two cotuous epeet raom varables U a W ormally strbute wth N(, σ. The varable X ee by X = U + V has a Raylegh strbuto wth a parameter σ f(x; θ = x σ exp ( x, x raom varables s- Let (X 1,..., X be a raom sample of... trbute as X 1. Apply the metho of the momets to the estmator ˆσ MOM of the parameter σ. Soluto x ( x x xf(x; θx = σ exp = 1 x ( σ σ exp x σ π 1 x = exp σ πσ x ( x x
3 Y N(, σ y exp ( y πσ y = E(Y = V ar(y + E(Y = σ xf(x; θx = π 1 σ σ = σ π π E(X = σ σ MOM ˆ = x π. F ˆσ MLE maxmum lkelhoo estmator (MLE for σ a scuss propertes of ths 1estmator. Soluto L(σ x = x ( σ exp x logl(σ x = log(x logσ logl(σ x σ = σ + x σ 4 x x logl(σ x σ = f σ = logl(σ x σ = + σ 4 logl(σ x σ ˆ ˆ σ MLE = x σ = + ṋ σ 4 ṋ σ 4 < x 3. Compute the score fucto a the Fsher formato. Soluto Score(σ = logl(σ x σ = σ + logl(σ x σ = σ 4 x σ 4 x 3
4 x f(x; θx = x 3 ( σ exp x x Itegrato by parts f g = f g x x ( σ exp x x = ( x exp E(X = = f g ( x x σ exp ( x x ( x exp ( x x ( logl(σ x I(θ = E σ ( = E σ 4 x I(θ = σ 4 σ I(θ = σ 4 4. Specfy asymptotc strbuto of ˆθ MLE. Soluto ( σ ˆ MLE N σ, σ4 5. Coser the test H : σ = σ versus H 1 : σ = σ 1 wth σ 1 > σ. Determe the UMP test (Uformly Most Powerful of sze α. Exercse 3 Let (X 1,..., X be a raom sample of... raom varables strbute as follows: f(x; θ = 1 x 1 θ θ x < c, θ > θc 1 θ 1. F ˆθ MLE maxmum lkelhoo estmator (MLE for θ a scuss propertes of ths estmator. 4
5 Soluto L(θ x = 1 θ c θ x 1 θ θ log L(θ x = log (θ θ log(c + 1 θ θ log L(θ x log L(θ x = θ + θ log(c 1 θ log(x = θ θ 3 log(c + 1 θ 3 log(x log(x log L(θ x log L(θ x = f ˆθ = log(c log(x ˆθ = θ < ˆθ MLE = log(c log(x. Compute the score fucto a the Fsher formato. (Amt that E(log(X = log(c θ Soluto log L(θ x u(θ = ( log L(θ x I(θ = E = θ + θ log(c 1 θ log(x = θ + θ 3 log(c 1 θ 3 E(log(x I(θ = θ + θ 3 log(c 1 θ 3 (log(c θ I(θ = θ 3. Specfy asymptotc strbuto of ˆθ MLE. Soluto ˆ θ MLE N (θ, θ 5
6 4. Assume that log(c = 4: Coser a sample of sze N = 1 for whch the realsato of the ML estmator ˆθ =, a we wat to test H : θ = 1.5 versus H 1 : θ 1.5 Compute ONE of the followg test statstc, specfy the correspog strbuto a take your cocluso base o the sample: (a The Lkelhoo Rato test statstc. (b The Wal test statstc. ˆ θ MLE N θ MLE ˆ /1 = 3.33 p value <.1 (1.5, Reject ull hypothess (c The Score test statstc. u(θ = log L(θ x = θ + θ log(c 1 θ x Exercse 4 Let (X 1,..., X be epeet etcally strbute raom varables wth p..f. f(x = θ x exp( θx x > Is T (X 1,..., X = 1/X 1 a ubase estmator of θ? Exercse 5 Prove correct statemet for Cramer-Rao equalty 6
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