1. Sampling Distributions
Tobias
Econ 472

2. The Design
Consider taking 10 draws from a Uniform distribution on (0,1).
[That is, randomly drawing a number
between 0 and 1].
One can show that if a random variable X has such a distribution, then E(X) =.5 and Var(X) =1/12.
Tobias
Econ 472

3. The Design, Continued
Let x1, x2,  , x10 denote this collection of 10 draws.
Consider two different rules for using this data to estimate E(X) = x:
and
Tobias
Econ 472

4. The Design, Continued Both of these estimators are unbiased, since
(The fact that our first estimator is unbiased was proven in class).
Tobias
Econ 472

5. The Design, Continued
To obtain the sampling distribution of both estimators, we first obtain 5,000 different sets of 10 draws from this uniform distribution.
For each set of 10 draws, we calculate both estimates.
We summarize the 5,000 estimates from each estimator in the following histograms:
Tobias
Econ 472

6. Tobias
Econ 472

7. Results for Sample Mean
As you can see, the sample mean is an unbiased estimator of the population mean x, as the sampling distribution is centered around E(X) = .5.
Tobias
Econ 472

8. Tobias
Econ 472

9. Results for Second Estimator
Again, this is an unbiased estimator of x.
However, it is slightly less efficient than the sample mean, (i.e., it has a larger variance).
This becomes a little more clear when plotting the sampling distributions on the same graph:
Tobias
Econ 472

10. Tobias
Econ 472