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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 discover a rich relationship between the two. We discover the score function and so much more exciting.

Ideas

Video

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Food For Thought

- 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.

**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?

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