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Let $(x_1,Y_1), \dotsc, (x_n,Y_n)$ be a random sample from the regression model $$Y_t=m(x_t)+\epsilon_t.$$ When authors want to develop the asymptotic properties of the local linear estimator of $m$ when the design points are fixed (for example, $x_t=t/n$), they usually assume things that I don't really undersand. Fan and Gijbels Assumed the design density $f$ satisfies $$x_j=G^{-1}(j/n) \text{ with } G(x)=\int_{-\infty}^x f(y)dy.$$

Fernandez Assumed that the regular design density satisfies $$\int_0^{x_t}f(y)dy=\frac{t-1}{n-1}, 1\leq t\leq n. $$

Does anyone understand what is the intuition behind it? Why the design points are defined in terms of cumulative distribution functions?

*I will print the relevant parts of their papers:

Fan

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Fernandez

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kjetil b halvorsen
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Celine Harumi
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    It's interesting, when pondering this question, to consider what alternative approaches there might be. Since you're studying asymptotics, you have to supply some way to produce a sequence of $x$ values for each $n.$ Clearly whatever your procedure might be, it has to allow the sequences to vary (because their lengths vary!). What would you propose? – whuber Mar 02 '20 at 21:21
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    @whuber Nice comment. For Fernandez procedure, I see that $P(x_{t-1,n}\leq X\leq x_{t,n} )=1/(n-1)$. So, given the density $f_X$, the distance between $x_{t-1,n}$ and $x_{t,n}$ must be $O(1/n)$. I think this is a good way to start thinking the model. That is, when $n$ grows, the regular design points become "dense". And this is what the asymptotics requires, in this case. – Celine Harumi Mar 03 '20 at 12:39

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