TOPIC:Consistent but biased estimator.
Exercise 2.4: Suppose θ is an estimator for θ with probability function Pr 2(n-1)/n and Pr[θ θ+n]...
2. (a) Define the bias of ˆ θ as an estimator for the parameter θ. [2 marks] (b) For independent random variables X1,X2,...,Xn, assume that E(Xi) = µ and var(Xi) = σ2, i = 1,...,n. (i) Show that ˆ µ1 = {(X1+Xn)/2}is an unbiased estimator for µ and determine its variance. [3 marks] (ii) Find the relative efficiency of ˆ µ1 to the unbiased estimator ˆ µ2 = X, the sample mean. [2 marks] (iii) Is ˆ µ1 a consistent...
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need only C,D, and E Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
please solve 6 4. Let Xi. X2. . Xnbe ap (1 I: 1 Xi ) 1/n is a consistent estimator for θ e . BAN. [Show that n(θ-X(n)) G (1, θ the estimator T0(X) = (n + 2)X(n)/(n + 1) in this class has the least MSE. an 5. In Problem 2, show that TX)Xm) is asymptotically biased for o 6.In Problem 5, consider the class of estimators T(X) cX(n), c 0. Sho 4. Let Xi. X2. . Xnbe ap...
Please answer as neatly as possible. Much thanks in advance! Question 1: 6. In Problem 1, show that θ2 is a consistent estimator for θ. Deduce that Y(n) is a consistent estimator for θ and also asyınpt○tically unbiased estimator for θ. 1. Let Yi, ½, . . . ,y, denote a random sample from an uniform distribution on the interval (0,0). We have seen that (1) and 62 Ym are unbiased estimators for 0. Find the efficiency of 6 relative...
1. Suppose X ~Bin(n, 6). (a) Show that the maximum likelihood estimator (MLE) for θ is θ (b) Show that E(0)-0 and that var(0) 0(1-0)/m X/n.
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
-2.4 (1 point) Suppose we have the following likelihood function, L (9) and there are only 3 possible values for θ : θ 0.505, 0.405, 0.3. Use L(0) to determine which value is the maximum likelihood estimate: T, ผู้โ
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE