1. (a) Carefully state the Lehmann–Scheff´e theorem. (b) Let X1, ...,Xn be i.i.d. random variables from Binomial(r, ?) distribution where 0

1. (a) Carefully state the Lehmann–Scheff´e theorem. (b) Let X1, ...,Xn be i.i.d. random variables from Binomial(r, ?) distribution where 0 <>< 1="" and="" r="1" is="" an="" integer.="" justify="" thatni="1" xi="" is="" a="" complete="" and="" sufficient="" statistic="" for="" .="" (c)="" derive="" the="" umvue="" of="" =="" p?(x="1)," where="" x="" is="" distributed="" as="" binomial(r,="" ).="" 2.="" let="" x1,="" ....,xn="" be="" a="" random="" sample="" from="" a="" uniform="" distribution="" on="" 0="" to="" .="" (a)="" write="" down="" the="" likelihood="" function="" for="" .="" (b)="" find="" a="" one="" dimensional="" complete="" and="" sufficient="" statistic="" for="" .="" justify="" your="" answer.="" (c)="" find="" the="" mle="" for="">


Binder1.pdf MA 781/782 Qualifier 2011 1. (a) Carefully state the Lehmann–Scheffé theorem. (b) Let X1, ..., Xn be i.i.d. random variables from Binomial(r, θ) distribution where 0 < θ="">< 1="" and="" r="" ≥="" 1="" is="" an="" integer.="" justify="" that="" ∑n="" i="1" xi="" is="" a="" complete="" and="" sufficient="" statistic="" for="" θ.="" (c)="" derive="" the="" umvue="" of="" λ="Pθ(X" ≤="" 1),="" where="" x="" is="" distributed="" as="" binomial(r,="" θ).="" 2.="" let="" x1,="" ....,="" xn="" be="" a="" random="" sample="" from="" a="" uniform="" distribution="" on="" 0="" to="" θ.="" (a)="" write="" down="" the="" likelihood="" function="" for="" θ.="" (b)="" find="" a="" one="" dimensional="" complete="" and="" sufficient="" statistic="" for="" θ.="" justify="" your="" answer.="" (c)="" find="" the="" mle="" for="" θ.="" justify="" your="" answer.="" (d)="" the="" pareto(a,="" b)="" distribution="" has="" pdf="" f(y;="" a,="" b)="aba" ya+1="" ,="" y="" ≥="" b="" (and="" 0="" otherwise).="" if="" a="" prior="" for="" θ="" is="" pareto(a,="" b),="" then="" what="" is="" the="" posterior?="" (e)="" use="" your="" answer="" to="" (d)="" to="" find="" an="" estimator="" for="" θ.="" (f)="" based="" on="" your="" answer="" to="" (d),="" describe="" how="" you="" could="" find="" an="" interval="" estimator="" for="" θ.="" (g)="" describe="" how="" your="" choice="" of="" b="" influences="" the="" posterior.="" 3.="" let="" {x1,="" ...,="" xn}="" be="" a="" random="" sample="" from="" a="" population="" with="" density="" f(x,="" θ)="{θ2" +="" 2θ(1="" −="" θ)}x(1="" −="" θ)2(1−x),="" x="{0," 1},="" 0="">< θ="">< 1="" thus="" the="" random="" variable="" x="" can="" assume="" only="" two="" possible="" values,="" 0="" and="" 1.="" (a)="" derive="" the="" moment="" estimator="" of="" θ.="" (b)="" derive="" the="" mle="" of="" θ="" by="" using="" the="" em-algorithm.="" you="" need="" to="" explicitly="" write="" down="" the="" update="" algorithm.="" (c)="" which="" estimator="" you="" would="" prefer="" for="" estimating="" θ?="" 4.="" let="" x1,="" ...,="" xn="" be="" i.i.d.="" n(θ,="" θ2),="" θ=""> 0. For this model X and cS are unbiased estimators of θ, where X is the sample mean,S is the sample standard deviation, and c is a constant given by c = √ n−1Γ((n−1)/2)√ 2Γ(n/2) . (a) Consider an estimator of the form T1 = a1X + a2(cS), where a1 + a2 = 1. Show that the estimator T1 is an unbiased estimator of θ for any choices of {(a1, a2) : a1 + a2 = 1} and find the best estimator in this class that minimizes the variance. Let’s call this estimator T ∗1 . (b) Now consider another estimator of θ of the form T2 = a1X+a2(cS), where we do NOT assume that a1 + a2 = 1. Find the best estimator in this class that minimizes the MSE. Let’s call this estimator T ∗2 . (c) Between MSE(T ∗1 ) and MSE(T ∗2 ), which one is smaller? Justify your answer. –2– MA 781/782 Qualifier 2011 5. Let X be a sample of size 1 from a p.d.f fθ. Find MP test of size α = 0.05 for H0 : θ = θ0 vs H1 : θ = θ1 when (a) fθ(x) = 2[θx + (1 − θ)(1 − x)], 0 ≤ x ≤ 1, 0 ≤ θ0 < θ1="">< 1;="" (b)="" fθ0="" is="" the="" p.d.f.="" of="" n(0,="" 1)="" and="" fθ1="" is="" the="" p.d.f.="" of="" cauchy="" distribution="" c(0,="" 1).="" 6.="" let="" x1,="" x2,="" ...,="" xn="" be="" independent="" random="" samples="" from="" f(x|θ)="1" θ="" xθ−1="" 0="" ≤="" x="" ≤="" 1,="" 0="" ≤="" θ0="">< θ1="">< 1;="" (a)="" find="" a="" ump="" test="" of="" size="" α="" for="" testing="" h0="" :="" θ="" ≤="" θ0="" versus="" h1="" :="" θ=""> θ1. (b) Can you find a UMPU test of size α for testing H0 : θ = θ0 versus H1 : θ �= θ1? If yes find the UMP test. If not give provide reasoning and find the next best test in the class of most powerful tests. 7. Let (Xj,1, Xj,2), j = 1, 2, . . . , n be independent pairs of poisson random variables with E(Xj,1) = λ1 and E(Xj,2) = λ2. for j = 1, 2, . . . , n (a) Construct the likelihood ratio test to test H0 : λ1 = λ2 vs HA : λ1 = λ2 . Provide a size α critical value for the above hypothesis based on the LRT. (b) Using appropriate reparameterization construct the Wald Test and the Score Test for the same hypothesis i.e. H0 : λ1 = λ2 vs HA : λ1 = λ2. Provide size α critical values for each of Wald Test and the Score Test. 8. Let Xi = θ2z 4 i + �i, i = 1, . . . , n, where zi’s are constants and �i ∼ N(0, σ2) , σ known (a) Obtain the UMA confidence interval for θ with confidence coeffecient (1 − α). (b) Find the unbiased estimator of θ and use it to construct another (1 − α) confidence interval for θ. (c) Compare the length of the intervals in (a) and (b) and comment on which one is more accurate. –3– GEORGiY Rectangle 2 3 4 GEORGiY Rectangle 7 8 GEORGiY Rectangle 9 10 11 12 13 14 GEORGiY Rectangle 15 16 17 18 20 19 21 22 23 24 25 26 27 28