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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.

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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.

$$\sum_{\mathbb{Z}}$$

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

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  • $\begingroup$ 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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