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A gentle introduction to the Cramér-Rao lower bound (with a cat)

It is summer, Edgard and Sofia are comfortably sitting on the terrace, watching the beautiful light of the end of the day. Edgard starts:

“Let’s play a game to see who is the better statistician! Immanuel my cat will give each of us a secret number strictly greater than zero. The other person will have to guess it.”

“How are we going to guess?”

“Let’s say that the secret numbers are the means of some Poisson variable. We generate samples at random. The one who gets the closest estimate by dinner time wins.”

“That sounds easy! Will Immanuel give us the same number?”

“What is the fun in that? Let’s ask him to give two different numbers. You know what to do. Just give me your first sample whenever you are ready and I will try to guess your secret number.”

Immanuel whispers something in the ear of Sofia and then does the same with Edgard. Sofia opens her laptop and after a few keystrokes she says “The first number I have for you is 1.”

“OK, I give up. You win.”

Sofia is puzzled at first, but then she notices how Immanuel is rolling...






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