# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 51

To prove that bias of $\, \hat \theta_J \,$ is of the order of $\, n^{-2} \,$, we need to show $\, \Exp(\hat \theta_J) \,$ contains one term equals to $\, \theta \,$ and remaining terms contains $\, n^{2} \,$ or higher powers in denominator.

Given:

Proof:

Last step is reduced by removing summation and division by $\, p \,$ because every terms occurs $\, p \,$ times in the sum.

Thus all the terms contains $\, n^2 \,$ or higher powers in the denominator.

$$\tag*{\blacksquare}$$