# Challenging Estimation Question - Help needed

• Nov 2nd 2008, 11:09 AM
math beginner
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.
• Nov 2nd 2008, 11:30 PM
CaptainBlack
Quote:

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
• Nov 3rd 2008, 05:52 AM
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?
• Nov 3rd 2008, 09:00 PM
CaptainBlack
Quote:

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