In this lecture on method of moment, we have:
why is gradient of psi inverse a dxd matrix?
K-th moment $m_k$is defined as $ \mathbb E[X^k] $ and can be estimated by the average using Law of Large Numbers which here is represented by $\hat m_k$
My understanding is that the inverse function $\psi^-1 $ takes the vector of moments of size d, so why isn't the gradient of size d?
I have found the answer. Normally, for a function $\psi$ that goes from $\mathbb R^d$ to $\mathbb R$ then you have to take the derivative of the function in regards to each of the d parameters and you will have a vector of size d. In this case $\psi$ is a function from $\mathbb R^d$ to $\mathbb R^d$ therefore you will end up with a d-dimensional array for each of d elements in the "y" side.