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Thread: I need some help, please

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    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.

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    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
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    Re: I need some help, please

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

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    I need some help, please

    Let X1, X2, ...,X4 be a random sample from a population that has mean 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

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    Re: I need some help, please

    Hi Fahad,
    This is easy one.

    E[ (cT-\sigma^2)^2 ] =\sigma^4 \times E[ (\frac{cT}{ \sigma^2 }-1)^2 ]
    =\sigma^4 \times E[ (c \frac{T}{\sigma^2}-1)^2 ] =\sigma^4 \times E[ (c W-1)^2] where W follows chi-square with n-1 df
    =\sigma^4 \times {c^2 E[W^2] -2c E[W] -1}

    Now you know the moments of the chi-square distribution. substitute it. you will get the answer
    Last edited by vinux; 11-14-2010 at 10:32 AM. Reason: LATEX correction
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    Re: I need some help, please

    who was that for?

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    Re: I need some help, please

    Quote Originally Posted by Smith194 View Post
    Let X1, X2, ...,X4 be a random sample from a population that has mean 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
    In the long run, we're all dead.

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    Re: I need some help, please

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

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    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!

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    Re: I need some help, please

    well this quadratic expansion cannot be wrong

    More hints:

    1) E[X_i] = \mu, Var[X_i] = \sigma^2

    2) Var[X_i] = E[X_i^2] - E[X_i]^2

    3) E[X_iX_j] = E[X_i]E[X_j] if X_i, X_j are independent.

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    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 and variance σ2.
    E[X1] = , E[X1^2] = Var(X1) + (E[X1])^2 =σ2 +2
    E[X1^2]-2E[X1]E[X2]+E[X2^2] = σ2 +2 - 2 + σ2 +2 = 2 σ2

    Edit: BGM was faster than me
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    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 ??

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    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]

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    Re: I need some help, please

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

    Note: T is an unbiased estimator for \sigma^2
    if and only if E[T] = \sigma^2

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

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    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???

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