Maximum Likelihood Estimation is an algorithm to find a reasonable estimator. Personally, it really woos my mind - simple and yet so beautiful. Method of Moments is simpler. It doesn't woo me :p. However, still, they have a lot of similarities. Thus, we have set off to explore them. Finally, we ask for a lot of food for thought. After all, we are all explorers at heart.
We ask "Is MLE = MOM? If not, when?"
We discover a rich relationship between the two. We discover the score function and so much more exciting.
Hints, Solution, and More
Find out examples where the estimates of Maximum Likelihood and Method of Moments are same.
Find out examples where the estimates of Maximum Likelihood and Method of Moments are not same.
Prove that Maximum Likelihood Estimation is same as solving \(\sum_{i=1}^{n} \frac{\partial}{\partial \theta} \log f\left(X_{i} \mid \theta\right)=0\).
Prove that Method of Moments Estimation is same as solving \((\frac{1}{n} \sum_{i=1}^{n} X_{i}^{k}-\mu_{k}(\theta)=0)\).
Let's explore the connection in the video.
Don't forget the food for thought.
Enjoy the video
Build your foundations.
Ace your Exams.
Learn. Enjoy. Practice. Repeat.
Above all, prove that \(E(h(X, \theta))=0\) for Maximum Likelihood Estimation and Method of Moments Estimation.
In addition, what is the intuition of the score function? Thus, we ask what is the intuition of the variance of the score function?
Do you think that method of moments and maximum likelihood estimate is equal for the exponential family?
As a result, we explore a one-parameter family. Thus, can you find out the pdf of the distributions for which the two estimates will be the same?
However, can you comment on the sufficient statistic, if the estimates are the same?