- #1
MaxManus
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Homework Statement
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
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?