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A Question about an unbiased estimator

  1. Mar 6, 2013 #1
    1. The problem statement, all variables and given/known data

    The random sample [itex]X_1, ... , X_n[/itex] has a [itex]N(0, \theta)[/itex] distribution. So now I have to solve for c such that [itex]Y= c \sum^n_{i=1}[/itex] is an unbiased estimator for [itex]\sqrt{\theta}[/itex].



    2. Relevant equations



    3. The attempt at a solution

    [itex]E(c \sum^n_{i=1} |X_i|) = c \sum^n_{i=1} E(|X_i|) = c \sum^n_{i=1} \int \frac{|X_i|}{\sqrt{2(\pi)(\theta)}}e^{-X_i/(2\theta)}[/itex]

    So now I have to solve...

    [itex] c \sum^n_{i=1} \int \frac{|X_i|}{\sqrt{2(\pi)(\theta)}} e^{-X_i/(2\theta)} = \sqrt(\theta)[/itex], right? But how can I integrate the absolute value of [itex]X_i[/itex]?

    Thanks in advance
     
    Last edited: Mar 6, 2013
  2. jcsd
  3. Mar 6, 2013 #2

    haruspex

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    Split the range of integration into x < 0, x > 0.
    Do you mean, X1,...,Xn are independent samples from a N(0,θ) distribution? If so, why the subscript on θi?
     
  4. Mar 6, 2013 #3
    Oh sorry, it's supposed to be just [itex]\theta[/itex]
     
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