![. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show that theestimator θ t914(1-t)a, is also an unbiased estimator of θ for any value of the constant t. Suppose V[6]:ơİ and v[62] of. Ifa, anda,are independent, find an expression for V[d] in terms of t, σ and σ Find the value of t that produces an estimator of the form 6 ะเอิ,+(1 that has the smallest possible variance. (Your final answer will be in terms of of and e).](http://img.homeworklib.com/questions/a01685d0-7289-11ea-9cf1-c3a0fc19a528.png?x-oss-process=image/resize,w_560)
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. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show...
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and...
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and 02 are independent. Consider the following estimator: (a) Show that θ3 is unbiased for θ (b) Find the value of a that minimizes the variance of θ3 (c) which estimator would you use? θ, θ2, or 6, when using the value of a found in part (b)
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0,...
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0, and θ2 are independent. Consider the following estimator: 6, - a+(1 -@ (a) Show that @g is unbiased for θ (b) Find the value of a that minimizes the variance of 03 (c) Which estimator would you use? θί,02, or th when using the value of a found in part (b)
(3) Suppose that E (0,) θ, Ε(92) θ,V(4) of, and V(92)-σ . Assume that 0, and...
(3) Suppose that E (0,) θ, Ε(92) θ,V(4) of, and V(92)-σ . Assume that 0, and θ2 are independent. Consider the following estimator: (a) Show that a, is unbiased for θ (b) Find the value of a that minimizes the variance of 83 (c) Which estimator would you use? 01.02, or 얘 when using the value of a found in part (b)
6. In Problem 1, show that θ2 is a consistent estimator for θ. Deduce that Y(n) is a consistent e...
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...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these po...
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these point estimates? (4pts) b) Find a point estimate with mean square error less than or equal to 0.01. (2pts) c) Only use ël and Ộ2, find the unbiased estimator with the smallest variance possible. What is that estimator? What is the smallest variance? (6pts)
Suppose we...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Estimators' Properties A researcher knows a random variable X has E[X] = and V[X] = o...
Estimators' Properties A researcher knows a random variable X has E[X] = and V[X] = o both finite. Here the researcher knowns the true value for (so they don't have to estimate it) but does not know the value for o? and wants to estimate it. a) Consider the following estimator: Ż (X; – w.)? a) Use the weak law of large numbers to construct the probability limit of õ?. Is õ? a consistent estimator for o?? b) Is 72...
A) Find the variance of each unbiased estimator. b) Use the Central Limit Theorem to create an ap...
a) Find the variance of each unbiased estimator.
b) Use the Central Limit Theorem to create an approximate 95%
confidence interval for theta.
c) Use the pivotal quantity Beta(alpha=13, beta=13) to create an
approximate 95% confidence interval for theta.
d) Use the pivotal quantity Beta(alpha=25, beta=1) to create an
approximate 95% confidence interval for theta.
Suppose that Xi, , x25 are i.i.d. Unifom(0,0), where θ is unknown. Consider three unbiased estimators of 6 25 26 25 25 26 max (X...,...
Suppose that E h ˆθ1 i = E h ˆθ2 i = θ, Var h ˆθ1...
Suppose that E h ˆθ1 i = E h ˆθ2 i = θ, Var h ˆθ1 i = σ 2 1 , Var h ˆθ2 i = σ 2 2 , and Cov h ˆθ1, ˆθ2 i = σ12. Consider the unbiased estimator ˆθ3 = aˆθ1 + (1 − a) ˆθ2. What value should be chosen for the constant a in order to minimize the variance and thus mean squared error of ˆθ3 as an estimator of θ?
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and 02 are independent. Consider the following estimator: (a) Show that θ3 is unbiased for θ (b) Find the value of a that minimizes the variance of θ3 (c) which estimator would you use? θ, θ2, or 6, when using the value of a found in part (b)
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0, and θ2 are independent. Consider the following estimator: 6, - a+(1 -@ (a) Show that @g is unbiased for θ (b) Find the value of a that minimizes the variance of 03 (c) Which estimator would you use? θί,02, or th when using the value of a found in part (b)
(3) Suppose that E (0,) θ, Ε(92) θ,V(4) of, and V(92)-σ . Assume that 0, and θ2 are independent. Consider the following estimator: (a) Show that a, is unbiased for θ (b) Find the value of a that minimizes the variance of 83 (c) Which estimator would you use? 01.02, or 얘 when using the value of a found in part (b)
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...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these point estimates? (4pts) b) Find a point estimate with mean square error less than or equal to 0.01. (2pts) c) Only use ël and Ộ2, find the unbiased estimator with the smallest variance possible. What is that estimator? What is the smallest variance? (6pts)
Suppose we...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Estimators' Properties A researcher knows a random variable X has E[X] = and V[X] = o both finite. Here the researcher knowns the true value for (so they don't have to estimate it) but does not know the value for o? and wants to estimate it. a) Consider the following estimator: Ż (X; – w.)? a) Use the weak law of large numbers to construct the probability limit of õ?. Is õ? a consistent estimator for o?? b) Is 72...
a) Find the variance of each unbiased estimator.
b) Use the Central Limit Theorem to create an approximate 95%
confidence interval for theta.
c) Use the pivotal quantity Beta(alpha=13, beta=13) to create an
approximate 95% confidence interval for theta.
d) Use the pivotal quantity Beta(alpha=25, beta=1) to create an
approximate 95% confidence interval for theta.
Suppose that Xi, , x25 are i.i.d. Unifom(0,0), where θ is unknown. Consider three unbiased estimators of 6 25 26 25 25 26 max (X...,...
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