8. Let X, X,, ... , X, be a rs from a distribution with mean u and variance o. Which of п these unbiased estimators has...
7. Let X,X,,...,X, be a rs from a distribution with mean u and variance o?. Which of the following are unbiased estimators of ju? If the estimator is biased, compute the bias. ☺ x a) 4X, b) 4X,-37 c) 4X, -27 d) e) x, f) - n-1
7. Let X, X,,..., X be a rs from a distribution with mean u and variance o”. Which of the following are unbiased estimators of u? If the estimator is biased, compute the bias.
(a) Are they unbiased estimators for µ? (b) Compute the MSE for all the 4 estimators. (c) Which one is the best estimator for µ? Why. PLEASE answer all parts, thanks Let X1, X2, ..., X, be and i.i.d. sample from some distribution with mean y and variance o? Let us construct several estimators for . Let îi = X, iz = X1, A3 = (X1 + X2)/2, W = X1 + X2 (a) Are they unbiased estimators for ?...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...
1) LetX,, ,X, be i.i.d. Uniform (0 , ) random variables for some > 0 (unknown). Which of the following estimators of0 are unbiased and which ones are biased? For each of the biased estimators ofO, find the MSE. (a)2X, (b) the smallest order statistic, (e) the largest order statistic, (d) x, /2 ) For each of the unbiased estimators of 0 in the above problem, find the variance. Which unbiased estimator has the smallest variance? Find the relative efficiency...
Let x and x, be independent random variables with Mean u and variance o2. Suppose that we have two estimators Of u : A @= X1 + X2 2 and ©2 = X, +3X2 2 (a) Are both estimators unbiased estimators of u? (b) What is the variance of each estimator?
Let X1,..., X10 be a random sample from a population with mean u and variance o2. Consider the following estimators for pe: X1 + ... + X10 ê 3X1 - 2X5 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
Let X1,..., Xn be a random sample from a distribution. Suppose Ti (X),T2(X) and U(X) respectively are sufficient, minimal sufficient, and unbiased estimators for the parameter θ of the distribution. Let U1(X) = E U(X) T, (X), U2(X) = EU㈤ T2(X)] a. Show that U1(X) and U(X) are unbiased for θ. b. Show that U2(x)-E[Uj(X)ITLX] c. Show that U2 has a smaller variance than U
Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for ji: X1 +...+ X10 3X1 - 2X3 + 3X10 Ô1 = @2 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...