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We consider the model of nonparametric regression:

$$Y_{i}=f(X_{i})+\xi_{i}$$

with $f:[0,1]\rightarrow \mathbb{R}$, $\xi_{i}$ are i.i.d with density $p_{\xi} $ with respect to the Lebesgue measure on $\mathbb{R}$ and $X_{i}$ are deterministic.

Somebody can explain me why $P_{j}$(the distribution of $Y_{1},\ldots,Y_{n}$, for $f=f_{nj}$) admits $p_{j}(u_{1},\ldots,u_{n})=\Pi^{n}_{i=1}(p_{\xi}(u_{i}-f_{nj}(X_{i})))$, $j=0,1$ as density with respect to the Lebesgue measure on $\mathbb{R}^{n}$.

m0nhawk
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user44677
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    It seems that you ask a simple question, but it is couched in highly technical language. That language would suggest your question ought to be read with great care and caution to make sure every aspect is correctly interpreted, in order to appreciate any unexpected subtleties. In particular, exactly what is "$f_{nj}$," how does it differ from "$f$," and why are you using pairs of subscripts on it, including one ($n$) which seems to be fixed? Why are you subscripting $p$ sometimes with "$j$" and sometimes with "$\xi$"? Why is the domain of $f$ merely $[0,1]$ and not $\mathbb{R}$? – whuber Apr 29 '14 at 19:04
  • I want to apply the technique (in the book "Introduction to Nonparametric Estimation" based on two hypotheses to obtain lower bounds in the nonparametric regression model.$f_{0n}$ and $f_{1n}$ are two hypotheses – user44677 Apr 29 '14 at 19:45

1 Answers1

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Without foregoing any of the generality or rigor of the original notation, we may more simply write that

$$Y_i = x_i + \xi_i$$

for constants $x_i = f(X_i)$ and independent random variables $\xi_i$ having distributions $p_{\xi_i}$. Independence implies the joint density of $(Y_1, Y_2, \ldots, Y_n)$ is the product of their densities $p_{Y_i}$.

Upon noting that $\xi_i = Y_i - x_i$ we obtain the densities for the random variables $Y_i$ as

$$p_{Y_i}(y) = p_{\xi_i}(y-x_i)$$

whence (abusing the "$p$" notation for densities) the joint density is the product

$$p_{\xi_1,\xi_2,\ldots,\xi_n}(y_1, y_2, \ldots, y_n) = p_{\xi_1}(y_1-x_1) p_{\xi_2}(y_2-x_2) \cdots p_{\xi_n}(y_n-x_n).$$

This reduces to the expression given in the question when the $\xi_i$ have a common density $p_{\xi_i}=p_\xi$ and we apply it separately to hypotheses $j=0$ and $j=1$.

whuber
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