Estimator quality

Post #12 in this thread https://www.physicsforums.com/showthread.php?t=616643&highlight=inverse gives three estimators for the variance that you can investigate.

You haven't made your project clear. Is the primary purpose to analyze a particular set of data or is the primary purpose to investigate the statistical properties of various estimators? - i.e. is is this project for a statistics course? - a science project? - a work assignment?

Can you use computer software (or write programs) to do Monte-Carlo simulations?

 

Investigating the properties of estimators for the mean and variance of a normal distributiion is going down a well-trodden path. (Perhaps that is why someone has sent you down it.) Experts evaluating that sort of project would know the answers in advance, although the average science-fair judge might not. If you do a moderate amount of reading about the subject, you will know the answers in advance too.

A less well known path begins with the fact that you don't have samples from a normal distribution. Real measurements can't be taken exactly; the results are only given to a certain precision. The mathematical ideal of sampling from a normal distribution has no such restriction. Your measurements are not samples from a normal distribution; they are samples from a discrete distribution that you might assume is related to some normal distribution. How does using non-exact samples affect estimators of the mean and variance?

That's a line of research that I find personally interesting. I don't know what your advisors would think.

 

You'll have to describe the scenario better. For example:

Scenario 1:
Dr. Stats Teacher: "Mark, why don't you do your science project on comparing the performance of diferent estimators for the mean and variance of a normal distribution?"

Scenario 2:
Mark: "I've read about estimators. I wonder what the good ones are to use for the mean and variance of a normal distribution."
Mr. Glubbston: "Gosh, that's a good question, Mark. I've forgotten all statistics that I ever took. Maybe you should do a science project on that."

Scenario 3: The judges at Wednesday's science fair will be
Dr. Myron Glevash, chairmain of the statistics department at Oggerland University
Mr. Sydney Dappslie, quality control engineer at Morlin Dynamic Systems

Scenario 4: The judges at Wednesday's science fair will be
Mr. Herman Vipsdale, president of the First National Bank
Dr. Edward Laxtilly, M.D.

 

If (as in scenario 1) an expert in statistics has suggested the path of your investigations then you should take it. The expert is correct that your path of investigation will produce useful results.

These are considerations that I see. 1) The results of such investigations are already well-known to experts, so experts evaluating your work will know the results that you "should" get and will criticize any omissions. 2) You can read enough about the topic to know more than any non-statistical experts who ask you questions 3) Experts won't give you any points for originality unless you also investigate some side-line to the commonly used estimators. (Perhaps your advisor already has such a side-line in mind and will reveal it once you've become familar with more well-known results.)

 

Simulate batches of N samples from a know normal distribution, apply the estimators to each batch, histogram how estimates are distributed. See which estimators tend to cluster most tightly around the true value of the parameter being estimated.

Then you should start reading about the mathematical theory of estimators and study the known theoretical results if you have a sufficient mathematical background.