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Math Help - Estimators

  1. #1
    Junior Member
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    Oct 2009
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    Estimators

    Please help me to solve this:
    Let x1,...xn be a sample random from pdf
    f(x\t)=tx^(t-1) , 0<x<1 , t>0
    1) to show that the pdf belongs to the exponential family
    2)To find the lower limit of variance of two estimators unbiased to t
    3)Find a enough statistics to t and its distribution
    4)Suggest a estimator unbiased to t that be from unbiased function and check if it is efficient

    Thank you
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  2. #2
    MHF Contributor harish21's Avatar
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    1)

    the function belongs to a exponential family if its pdf can be written as:

    f(x|t) = \bigg\{exp\bigg(c(t) \; T(x) \; + b(t)+S(x)\bigg)\bigg\}


     t.x^{t-1} = \mbox{exp}(log(t)).\mbox{exp}((t-1)log(x)) = \mbox{exp}\bigg(logt+(t-1)log(x)\bigg) =...
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