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Consistent Estimator

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    A balanced one way anova model with random effects is on the form:
    [itex]X_{ij} = s + a_i + e_{ij}[/itex]
    i = 1,...,k
    j = 1,...,n

    a's and e's are independent and normal distributed
    E(a_i) = 0
    var(a_i) = tau^2
    E(e_ij) = 0
    Var(e_ij) = sigma^2

    [itex]\bar{X} = \frac{1}{k n} \sum_{i=1}^k \sum_{j =1}^n X_{ij}[/itex]

    a) Prove that X-bar is consistent for s if k -> infinity and n is fixed
    and b) that X bar is not consistent if J is fixed and n -> infinity

    3. The attempt at a solution

    For a)
    A sufficient condition for X-bar to be a consistent estimator is that E(X-bar) goes to t and that var(X_bar) -> 0 as k-> infinity and n is fixed

    E(X_ij) = t for all i and j so that one is OK

    [itex] Var(X_{ij}) = \frac{1}{n^2 k^2} \sum_{i=1}^k \sum_{j =1}^n var(s + a_i + e_{ij})[/itex]
    [itex] \frac{1}{n^2 k^2} \sum_{i=1}^k \sum_{j =1}^n \tau^2 + \sigma^2 [/itex]

    [itex] \frac{nk (\tau^2 + \sigma^2}{n^2 k^2} [/itex]

    = [itex] \frac{(\tau^2 + \sigma^2}{n k} [/itex]

    which goes to zero both if n goes to infinity and k is fixed and if n is fixed and k goes to infinity.

    So what do I do wrong?
  2. jcsd
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