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Thread: I have questions ??

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    I have questions ??




    Hi

    I have 3 questions.

    Let x1, . . . , xn be independent observations of a chi-square(k) random variable, with p.d.f.
    f(x)= 1/(2^k/Γ(k/2)) x^(k/2-1) exp(-x/2)

    I have found E(X)=k , Var(X)= 2k, the method of moments estimator for k which is (k*) = Xbar
    and k* is unbiased of k because E(k*)=E(Xbar)=1/n [E(X1)+E(X2)+...+E(Xn)]= 1/n nk= k

    My questions are:
    1. How can I find Var(k*) ?
    2. Show that the Cram´er-Rao lower bound for the variance of unbiased estimators of k is 4[n(d^2/d ��^2)logΓ(��) (k/2)]^-1
    3. show that k* is approximately fully efficient.

    Many thanks in advance.

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    Re: I have questions ??


    Just by definition
    Var[\bar{X}] = Var\left[\frac {1} {n} \sum_{i=1}^n X_i \right]
= \frac {1} {n^2} \sum_{i=1}^n Var[X_i]
= \frac {1} {n^2} n Var[X_1] = \frac {2k} {n}

    For part b, I do not know what is your problem. Maybe you try
    to begin with the definition and work with that first.

    http://en.wikipedia.org/wiki/Cram%C3...80%93Rao_bound

    I guess part c) is just asking you to compare the answer you obtained
    in part a) and b). If they are close, then the estimator is approximately
    efficient.

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