5. (10 points) Let Xi, . . . , xn be 1.1.d. from pdf: f(z:0)-D"-9 rS...
Let Xi, , xn be a randon sannple fron f,(z0)-e-(z-0),0 є (-00,00), z > θ a. Show that X(1) is a complete statistic for θ. Hint: First find the PDF of XI) b. Show that the sample variance S is an ancllar statistic,and use this result to show that Xa) and S2 are independent.
3. Let X1,... ,Xn be a random sample from a population with pdf 0, otherwise, where θ > 0. (a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ. (c) Find the pdf of θ in (b).
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 sample from the PDF 25 points 2r a. Show that X(n) is a complete-sufficient statistic for θ. b. Show that (3/2)X is unbiased for 0 c. Find the UMVUE for based on Xi, x2, x". Note: your final answer should be written as a computable formnla based on an olbserved sample. Con- sider using order statistics and their conditional distributions
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
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
Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of the form 0-1) otherwise, where θ > 0. (a) Find θ the method of moments (MOM) estimator for θ For what values of θ does θ exist? Why? (b) Find θ, the maximum likelihood estimator (MLE) for θ. (c) Show explicitly that the MLE depends on the sufficient statistic for this Pareto family but that the MOM estimator does not