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Given $N$ data points $x$ and $N$ targets $t$, considering a new point $x$ and the corresponding new target $t$, what would be:

  • The conditional density
  • The conditional mean
  • variance

of the Nadaraya-Watson model

$$p(x,t) = \frac{1}{N}\sum_{n=1}^Nf(x - x_t, t - t_n)$$

Where the kernel is an isotropic Gaussian with mean $(0,0)$ and covariance is $\sigma^2I$, in terms of $k(x, x_n)$?

How do you verify that the sum of the functions = 1, since it is a distribution?

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    $\begingroup$ This can be found in any textbook on nonparametric econometrics; see Li & Racine (2006), for example. $\endgroup$ May 5, 2014 at 5:35
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    $\begingroup$ What is your observation? X? And you're doing an inference about the variable $t$, is it right? Please make this clear. $\endgroup$
    – Jundiaius
    May 5, 2014 at 8:06
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    $\begingroup$ A nice introduction to the Nadaraya-Watson estimator can be found through Bruce Hansen's (University of Wisconsin-Madison) online lecture notes ssc.wisc.edu/~bhansen/718/NonParametrics2.pdf. $\endgroup$ Aug 25, 2016 at 1:14

1 Answer 1

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Here you have well described the conditional density (with $y$ in the place of your $t$): http://en.wikipedia.org/wiki/Kernel_regression#Derivation

The conditional density is the $f(y|x) = \frac{f(x,y)}{f(x)}$.

The conditional mean would be the mean for that density $E(Y|X)$ and the same thing for variance $E(Y-E(Y|X)|X)$.

It is the integral of the conditional density $f(y|X)$ over all y that shall be equal to 1. That will lead you to an integral of all K functions.

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