MLE vs MOM

Last week I had a severe case of a swollen throat.  Normally, life would continue, but being a teacher, I decided to take the week off from classes.  It was such a memorable time, as I received such wonderful attention.  First of all, Kyungbok, God bless her, bombarded me with tea, medicine, juice, fruits, and porridge.  Then my Headmistress came herself and gave me three chunks of kingfish.  Then a number of my students came to greet me.  They are all such wonderful people.  I was touched.

With a sudden influx of free time, I had this sudden impulse to solve a question that I wanted to answer while I was in college. The question at hand stems from an exercise in the Mathematical Statistics textbook.  It asked something along the lines of:

There are n numbers, from 1 to n, and a student chooses 50….

And then the question continues, but the question involved using the given 50 numbers to estimate the n.  There was a minor debate between me and the professor in regards to the method of estimation.  The professor suggested the most likelihood estimator to find n, which was basically the largest number in the sample of 50, while I suggested the method of moments, which was to find the average of the 50 numbers, and then multiplying by two.  It seemed that both were decent methods, but the quest for accuracy remained in my mind, until today.  The question, therefore, is

In order to find n, is it better to use the MLE or the MOM?

Wow, this post is extremely nerdy.  It turns out the MLE is better!  But seriously, who cares!  So with that, I end this post.