(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Consistent Estimator

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