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When is the following unbiased?

  1. Apr 6, 2015 #1
    1. The problem statement, all variables and given/known data
    Let $$S_{1}^2, S_{2}^2, S_{3}^2$$ be sample variances of sample size $$n_{1}, n_{2}. n_{3}$$ respectively. The populations have means $$\mu_{1}, \mu_{2}, \mu_{3}$$ respectively with common variance $$\sigma^2.$$

    When is $$\pi_{1}S_{1}^2 + \pi_{2}S_{2}^2 + \pi_{3}S_{3}^2$$
    unbiased?

    2. Relevant equations


    3. The attempt at a solution

    I was under the impression that the sample variance was always unbiased. Wouldn't the expected value of the above expression always yield the same expression?
     
  2. jcsd
  3. Apr 6, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    How are the S defined? They might be biased estimators for σ2, then you have to account for that.
     
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