# Consistent Estimator

1. Nov 6, 2011

### MaxManus

1. The problem statement, all variables and given/known data

A balanced one way anova model with random effects is on the form:
$X_{ij} = s + a_i + e_{ij}$
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

$\bar{X} = \frac{1}{k n} \sum_{i=1}^k \sum_{j =1}^n X_{ij}$

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

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

=
$\frac{nk (\tau^2 + \sigma^2}{n^2 k^2}$

= $\frac{(\tau^2 + \sigma^2}{n k}$

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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