It is not difficult... why don't you start solving it.
MSS is E[ (cT-σ^2)^2 ] ... Now rest part is only algebra
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.
It is not difficult... why don't you start solving it.
MSS is E[ (cT-σ^2)^2 ] ... Now rest part is only algebra
In the long run, we're all dead.
Actually, I started, but I could not find the final solution
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
who was that for?
ive tried and tried to expand the terms, i get something really dodge. i simply dont understand it.
when i expanded the terms i got E[X1^2-2X1X2+X2^2], i know this is wrong. please guide me!
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 )
E[X1] =µ , E[X1^2] = Var(X1) + (E[X1])^2 =σ2 +µ2Let X1, X2, ...,X4 be a random sample from a population that has mean µ and variance σ2.
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
In the long run, we're all dead.
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 ??
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]
k[2σ2+2σ2]=σ2
k[4σ2]=σ2
k=1/4
is this correct or have i completely lost it???
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