Let Xi, , xn be a randon sannple fron f,(z0)-e-(z-0),0 є (-00,00), z > θ a....
Let Xi, , xn be a sample from fx (x10) = e-(z-8),T 〉 θ, θ e(-00,00) 1-Identify a sufficient statistic for θ if it exists. 2.Identify a Minimal sufficient statistic for θ it it exists. O, 0O
Let Xi, , Xn be a sample from U(0,0), θ 0. a. Find the PDF of X(n). b. Use Factorization theorem to show that X(n) is sufficient for θ. C. Use the definition of complete statistic to verify that X(n) is complete for θ.
3. Let X1 , X2, . . . , Xn be a randon sample from the distribution with pdf f(r;0) = (1/2)e-z-8,-X < < oo,-oc < θ < oo. Find the maximum likelihood estimator of θ.
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
5. Let Xi, . . . , Xn be a random sample from f(x:0) = -| for z > 0. (a) Assume that θ 0.2 Using the Inversion Method of Sampling, write a R function to generate data from f(x; 0). (b) Use your function in (a) to draw a sample of size 100 from f(0 0.2 (c) Find the method of moments estimate of θ using the data in (b). (d) Find the maximum likelihood estimate of θ using...
5. (10 points) Let Xi, . . . , xn be 1.1.d. from pdf: f(z:0)-D"-9 rS θ, θ > 0, (a) Show that the order statistics (b) Find MLE of θ (1) and X(n) are jointly sufficient for θ.
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let Xi, Θ C R. Suppose that {f(x,10) : θ E Θ} has monotone likelihood ratio (MLR) in T(Xn). Consider test function if T(%) > c Xn if T(%) < c, where γ E [0, 1) and c 〉 0 are constants. Prove that the power function of φ(Xn) is non-decreasing in θ , xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let...
please answer with full soultion. with explantion. (4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
Please answer the following question and show every step. Thank you. Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...