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.


This is the analogue when you have to sum over a continuum.

If you were to do the sum over all of the integers, it makes sense to write a sum over the integers. You can write the infinite sum as a limit of a sequence of partial sums. Fourier series do this.


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: $\langle f(x), g(x)\rangle={\displaystyle \int } f(x)g(x)dx$ over some interval, perhaps $[0,1]$ or $[-\pi, \pi]$.

  • $\begingroup$ 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! $\endgroup$
    – CLL
    Feb 25 '20 at 9:08

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