1. ## I need some help, please

Hi,

I have found some difficulty in solving this exercise.

[Suppose that Y1,…,Yn are observations of independent N(μ,σ^2) random variables and that we are interested in estimating σ^2 using estimators of the form cT, where T=∑(Yi - ̅Y)^2 and c is non-random (does not depend on Y).
Show that the mean squared error of cT is

σ^4 {c^2 (n^2-1)-2c(n-1)+1}

and that this is minimized at c=(n+1)^(-1). This estimator is known as the Pitman estimator for σ^2 in the Gaussian model. Show that the unbiased estimator has c=(n-1)^(-1) and compare its mean squared error to that of the Pitman estimator.]
HINT: Use the fact that T/σ^2~chi-squared (n-1) and the moments of the chi-squared distribution are known.

2. ## Re: I need some help, please

It is not difficult... why don't you start solving it.

MSS is E[ (cT-σ^2)^2 ] ... Now rest part is only algebra

3. ## Re: I need some help, please

Actually, I started, but I could not find the final solution

4. ## I need some help, please

Let X1, X2, ...,X4 be a random sample from a population that has mean µ and variance σ2.

Find E [(X1-X2)^2] and hence the value of k such that
T = k[(X1-X2)^2 + (X3-X4)^2]
is an unbiased estimator of σ2.

i need help in this question, ive read the books but still ive never come across this type of question. please guide me!

Thanks

5. ## Re: I need some help, please

This is easy one.

where W follows chi-square with n-1 df

Now you know the moments of the chi-square distribution. substitute it. you will get the answer

6. ## Re: I need some help, please

who was that for?

7. ## Re: I need some help, please

Originally Posted by Smith194
Let X1, X2, ...,X4 be a random sample from a population that has mean µ and variance σ2.

Find E [(X1-X2)^2] and hence the value of k such that
T = k[(X1-X2)^2 + (X3-X4)^2]
is an unbiased estimator of σ2.

i need help in this question, ive read the books but still ive never come across this type of question. please guide me!

Thanks
Hint: I guess X1, X2,X3,X4 are independent.. Expand the terms... substitute the individual expectations

8. ## Re: I need some help, please

ive tried and tried to expand the terms, i get something really dodge. i simply dont understand it.

9. ## Re: I need some help, please

when i expanded the terms i got E[X1^2-2X1X2+X2^2], i know this is wrong. please guide me!

10. ## Re: I need some help, please

well this quadratic expansion cannot be wrong

More hints:

1)

2)

3) if are independent.

11. ## Re: I need some help, please

E[X1^2-2X1X2+X2^2] = E[X1^2]-2E[X1]E[X2]+E[X2^2]
( E[X1X2] =E[X1] E[X2] if X1 and X2 are independent )

Let X1, X2, ...,X4 be a random sample from a population that has mean µ and variance σ2.
E[X1] =µ , E[X1^2] = Var(X1) + (E[X1])^2 =σ2 +µ2
E[X1^2]-2E[X1]E[X2]+E[X2^2] = σ2 +µ2 - 2 µ µ + σ2 +µ2 = 2 σ2

Edit: BGM was faster than me

12. ## Re: I need some help, please

thanks for the help, i just want to ask that are you sure that its 2sigma^2, because it says unbiased estimator of sigma^2. so dont you think the answer has to be sigma^2 ??

13. ## Re: I need some help, please

if E[(X1-X2)^2] = 2 σ2
than does E[(X3-X4)^2] = 2 σ2 ????
if it does than what should i do for the next step to find K?
Should i just do this

T= k[2 σ2 + 2 σ2]

14. ## Re: I need some help, please

than does E[(X3-X4)^2] = 2 σ2 ????
sure.

Note: is an unbiased estimator for
if and only if

Use this relationship, together with the previous result, solve for

15. ## Re: I need some help, please

k[2σ2+2σ2]=σ2
k[4σ2]=σ2
k=1/4

is this correct or have i completely lost it???