Question

Suppose X1 is from a population with mean µ and variance 1 and X2 is from...

Suppose X1 is from a population with mean µ and variance 1 and X2 is from a population with mean µ and variance 4 (X1, X2 are independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show that ?̂ is unbiased for ?. Find the most efficient estimator in this class, that is, find the value of a such that the estimator has the smallest variance.

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

TOPIC:Unbiased and efficient estimators.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose X1 is from a population with mean µ and variance 1 and X2 is from...
Suppose X1 is from a population with mean µ and variance 1 and X2 is from a population with mean µ and variance 4 (X1, X2 are independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show that ?̂ is unbiased for ?. Find the most efficient estimator in this class, that is, find the value of a such that the estimator has the smallest variance.
Let X1, X2 be two normal random variables each with population mean µ and population variance...
Let X1, X2 be two normal random variables each with population mean µ and population variance σ2. Let σ12 denote the covariance between X1 and X2 and let ¯ X denote the sample mean of X1 and X2. (a) List the condition that needs to be satisfied in order for ¯ X to be an unbiased estimate of µ. (b) [3] As carefully as you can, without skipping steps, show that both X1 and ¯ X are unbiased estimators of...
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ: 1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased estimator for the mean? What is the variance of the estimator? Can you find a more efficient estimator?) ( 10 Marks)
Suppose that X is Poisson, with unknown mean λ, and let X1, ..., Xn be a...
Suppose that X is Poisson, with unknown mean λ, and let X1, ..., Xn be a random sample from X. a. Find the CRLB for the variance of estimators based on X1, ..., Xn. ̂̅̂ b. Verify that ? = X is an unbiased estimator for λ, and then show that ? is an efficient estimator for λ.
Suppose that X and Y are random samples of observations from a population with mean μ...
Suppose that X and Y are random samples of observations from a population with mean μ and variance σ2. Consider the following two unbiased point estimators of μ. A = (2/3)X + (1/3)Y B = (5/4)X - (1/4)Y Find variance of A. Var(A) and Var(B) Efficient and unbiased point estimator for μ is = ?
A random sample X1, X2, . . . , Xn is drawn from a population with...
A random sample X1, X2, . . . , Xn is drawn from a population with pdf. f(x; β) = (3x^2)/(β^3) , 0 ≤ x ≤ β 0, otherwise (a) [6] Find the pdf of Yn, the nth order statistic of the sample. (b) [4] Find E[Yn]. (c) [4] Find Var[Yn]. (d)[3] Find the mean squared error of Yn when Yn is used as a point estimator for β (e) [2] Find an unbiased estimator for β.
Let X1, X2, . . . , Xn be iid exponential random variables with unknown mean...
Let X1, X2, . . . , Xn be iid exponential random variables with unknown mean β. (1) Find the maximum likelihood estimator of β. (2) Determine whether the maximum likelihood estimator is unbiased for β. (3) Find the mean squared error of the maximum likelihood estimator of β. (4) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (5) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (6)...
X1 is normal with mean mu and variance sigma^2. X2 is normal with mean mu and...
X1 is normal with mean mu and variance sigma^2. X2 is normal with mean mu and variance 2sigma^2. Consider the estimator cX1 + (1-c)X2. For what value of c is the variance minimized?
a. If ? ̅1 is the mean of a random sample of size n from a...
a. If ? ̅1 is the mean of a random sample of size n from a normal population with mean ? and variance ?1 2 and ? ̅2 is the mean of a random sample of size n from a normal population with mean ? and variance ?2 2, and the two samples are independent, show that ?? ̅1 + (1 − ?)? ̅2 where 0 ≤ ? ≤ 1 is an unbiased estimator of ?. b. Find the value...
5. Consider a simple case with only four independently and identically distributed (iid) observations, X1, X2,...
5. Consider a simple case with only four independently and identically distributed (iid) observations, X1, X2, X3, X4, on a random variable X. Consider these two estimators: µˆ1 = 1/12 (2X1 + 4X2 + 4X3 + 2X4), µˆ2 = 1/12 (X1 + 5X2 + 5X3 + X4). a Show that each is unbiased, and that one is more efficient than the other. b Show that the usual sample mean is more efficient than either. Explain why the others given above...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT