# Usefulness of MISE

I'm currently in a class on nonparametric smoothing, and, while talking about density estimation in general, the professor introduced the notion of MISE (mean integrated square error):

$$\text{MISE}\left( \hat{f}_X(x) \right) = E\left[ \int \left( \hat{f}_x(x) - f_X(x)\right)^2dx \right]$$

Why is considering MISE useful? Why not just consider MSE? I've tried to find answers or work out differences myself but am not arriving at anything insightful so far.

$$\sum_{\mathbb{Z}}$$
However, it doesn’t make sense to talk about a sequence of partial sums for $$\sum_{\mathbb{R}}$$.
But you can integrate over $$\mathbb{R}$$, so that’s what is done. This is the same idea as integrating to find the inner product between two functions: $$\langle f(x), g(x)\rangle=\int } f(x)g(x)dx$$ over some interval, perhaps $$[0,1]$$ or $$[-\pi, \pi]$$.
• This is exactly what I was looking for. It makes perfect sense - we would like some way to measure the difference between the estimator of the function and the actual function across all $x$, and integration is the way to do that. I may wait a little while to see if anyone else submits an answer before accepting, but thank you for your answer! – CLL Feb 25 '20 at 9:08