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Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:

Let $\xi$ be a random variable with density function $f$ unknow. Given a sample $\widehat{\xi}_{1},\ldots,\widehat{\xi}_{N}$ of $\xi$ we have that the kernel density estimator (KDE) of $f$ is $$\widehat{f}_{h}(x):=\frac{1}{Nh}\sum_{i=1}^{N}\mathcal{K}\left(\frac{x-\widehat{\xi}_{i}}{h}\right).$$ where $h> 0$ and $\mathcal{K}$ is a probability density function such that $\int x \mathcal{K}(x)dx=0$ and $\int x^{2} \mathcal{K}(x)dx=1$.

In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE where

\begin{equation} \mathrm{MISE}(h):=\mathbb{E}_{\mathbb{P}^{N}}\left[ \int \left( \widehat{f}_{h}(x)- f(x)\right)^{2}dx \right]. \end{equation} the randomness is in the vector $\left(\widehat{\xi}_{1},\ldots,\widehat{\xi}_{N}\right)$, that has distribution $\mathbb{P}^{N}=\mathbb{P}\times\cdots \times\mathbb{P}$, where $\mathbb{P}$ where $\mathbb{P}(A):=\int_{A}f(x)dx.$ Therefore, $\mathbb{P}$ is unknow. There are many techniques to estimate MISE.

The question: If $h_{MISE}$ minimizes $\mathrm{MISE}$ then that $h_{MISE}$ is used to determine $\widehat{f}_{h_{MISE}}$ for any sample of size $N$, the same is always used. Is not it better to find an $h$ for each sample?

Given a sample $\widehat{\xi}_{1},\ldots,\widehat{\xi}_{N}$, we consider the expression $$\mathrm{ISE}(h):= \int \left( \widehat{f}_{h}(x)- f(x)\right)^{2}dx. $$ If $h_{ISE}$ minimizes $\mathrm{ISE}$. Is not $\widehat{f}_{h_{ISE}}$ a better estimator than $\widehat{f}_{h_{MISE}}$?

Why usually minimize MISE instead of minimizing ISE?

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  • $\begingroup$ see Härdle (2004) Nonparametric and semiparametric models, page 53-57 for a discussion on an introductory level of the question you pose. They conclude that no single method is better. $\endgroup$ Feb 19, 2019 at 13:52
  • $\begingroup$ Also see page 17 of Li and Racine (2007) Non-parametric econometrics: Theory and practice. $\endgroup$ Feb 19, 2019 at 19:02

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