October 10, 2006 --- Class 13 --- Biased Estimators Using Mathematica

Activities:

	Biased Esimators Using Mathematica

	We considered the problem posed in section H of the handout.
	We started by teaching Mathematica to do multiple integrals
	analytically and then having it get a numerical result as described
	in the handout.

	We trained Mathematica to do multiple integrals analytically 
	for the problem we are considering.  Without
	waiting too long, we were able to get up to a sample size of 6;
	however, this result still shows significant bias.  On Nations in
	~sg/oct19_2004.nb you will find a mathematica notebook showing how to
	time the command for doing a four-dimensional integral and the analytic
	and numerical results.  On the Swain Apple cluster in 
	October_10_2006.nb you will find a notebook with the commands I
	issued during class.

	We found that for small samples sizes, the integrals, which
	represent the average value for 1/(mean value) we get numbers
	larger than 2, even though for an infinite sample we expect 2.

	Bias reduction

	Recognizing that a plot of the result vs. 1/n shows the error is
	linear in 1/n, where n is the sample size:

	S_n = A +e/n

	where A is the limit of n -> infinity and e determines the size of
	the error.  Considering samples sizes n and n-1, it is easy to
	solve for A.

	A = n S_n - (n-1) S_{n-1}

	In the next class, we will apply this idea to bias reduction by
	looking at different sample sizes.  The jackknife method let's us
	resample our n values to also get samples of size (n-1).