# Thread: Challenging Estimation Question - Help needed

1. ## Challenging Estimation Question - Help needed

To show that an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: to estimate the mean of a population with the finite variance s2, we first take a random sample of size n. Then we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2,3,…, or n, we use as our estimator the mean of the random sample; otherwise, we use the estimate n2. Show that this estimation procedure is
a) consistent;
b) neither unbiased nor asymptotically unbiased.
Find the variance of this estimator.

2. Originally Posted by math beginner
To show that an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: to estimate the mean of a population with the finite variance s2, we first take a random sample of size n. Then we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2,3,…, or n, we use as our estimator the mean of the random sample; otherwise, we use the estimate n2. Show that this estimation procedure is
a) consistent;
b) neither unbiased nor asymptotically unbiased.
Find the variance of this estimator.
What have you done? Or what is the specific difficulty that you have?

You are given a sequence of estimators for the mean, for this to be a consistent sequence of estimators you need to show that it converges in probability to the population mean. You can use the consistency of sample means as a tool to do this.

For the second part for a given sample size compute the expectation of the estimator, and look at the limit of this expectation as n becomes arbitarily large.

CB

3. i have trouble with the part that says, "otherwise, we use the estimate n2 (squared)". I don't know how to factor that in the calculations. any tips?

4. Originally Posted by math beginner
i have trouble with the part that says, "otherwise, we use the estimate n2 (squared)". I don't know how to factor that in the calculations. any tips?
The estimator $\displaystyle t_n$ is $\displaystyle \bar{x}$ with probability $\displaystyle (n-1)/n$ and $\displaystyle n^2$ with probability $\displaystyle 1/n$.

Hence the expectation of the estimator is:

$\displaystyle E(t_n)=\frac{n-1}{n} \mu + \frac{1}{n} n^2$

To be consistent requires that for all $\displaystyle \varepsilon >0$ :

$\displaystyle \lim_{n \to \infty} Pr(|t_n-\mu|<\varepsilon )=1$

CB