Consistent Estimator of Balanced One Way ANOVA Model

  • Thread starter MaxManus
  • Start date
In summary, the balanced one-way ANOVA model with random effects has the form X_{ij} = s + a_i + e_{ij}, where a's and e's are independent and normally distributed with E(a_i) = 0, var(a_i) = tau^2, E(e_ij) = 0, and var(e_ij) = sigma^2. X-bar is consistent for s if k -> infinity and n is fixed, but not if n -> infinity and k is fixed.
  • #1
MaxManus
277
1

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?
 
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  • #2
For b)Var(X_{ij}) = \frac{1}{k^2 n^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, but not if n is fixed and k goes to infinity.So here X-bar would not be consistent
 

1. What is a consistent estimator of balanced one-way ANOVA model?

A consistent estimator of balanced one-way ANOVA model is a statistical method used to estimate the population mean of a continuous variable by taking into account the differences between multiple groups. It is considered consistent because as the sample size increases, the estimated value approaches the true population mean.

2. How is a consistent estimator of balanced one-way ANOVA model calculated?

A consistent estimator of balanced one-way ANOVA model is calculated by taking the average of the group means and then adjusting for the differences between the groups. This adjustment involves dividing the sum of squares of the differences between the group means by the degrees of freedom, and then subtracting this value from the overall mean.

3. Why is a consistent estimator of balanced one-way ANOVA model important?

A consistent estimator of balanced one-way ANOVA model is important because it allows us to make accurate inferences about the population mean based on a sample of data. This can help us identify any significant differences between groups and make informed decisions based on these results.

4. What are the assumptions for using a consistent estimator of balanced one-way ANOVA model?

The assumptions for using a consistent estimator of balanced one-way ANOVA model include independence of observations, normality of the data within each group, and homogeneity of variances between groups. Violating these assumptions can lead to inaccurate results.

5. How can we test the validity of a consistent estimator of balanced one-way ANOVA model?

We can test the validity of a consistent estimator of balanced one-way ANOVA model by performing a hypothesis test, such as the F-test, to determine if the differences between group means are statistically significant. Additionally, we can check the assumptions of the model and make sure they are not being violated.

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