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imagine you have a sample $X_1, X_2, \ldots, X_n$ from a random variable $X$, and another sample $Y_1, Y_2, \ldots, Y_m$ from a random variable $Y$. You know that $Y = \phi(X)$. For concreteness, say $Y = a_0 + a_1 X + a_2 X^2$. How can you estimate $a_0$, $a_1$ and $a_2$ from the samples?

You don't how $X$ or $Y$ are distributed, and your samples do not come in pairs. In fact, $n \neq m$.

I am stuck trying to solve this. Perhaps it is a well-known problem in the statistics community, but I am unable to find anything about it.

Thanks.

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  • $\begingroup$ So to be clear, I don't observe pairs of $X$ and $Y$? That is to say $Y_1$ is NOT the transformation of $X_1$? $X_1$ and $Y_1$ are unrelated? $\endgroup$ – Demetri Pananos Jun 10 at 23:35
  • $\begingroup$ No, you don't observe them in pairs. The samples are independent. $\endgroup$ – Jochi Toborochi Jun 11 at 1:51
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One method would be constructing a KDE for $Y$ and calculating the likelihood of $\hat Y_i=a_0+a_1X_i+a_2X_i^2$, and then getting maximum likelihood estimate for the parameters $a_0,a_1,a_2$.

More Explanation: Since we don't have corresponding pairs in both datasets, one thing we can do is to estimate the PDF of $Y$, by using some approach. KDE (Kernel Density Estimation) is a useful, non-parametric method for doing this. KDE will gives us an estimate of $f_Y(y)$, i.e. $\hat f_Y(y)$.

Then, we can calculate the likelihood of the $X$ data using: $$L(X|a)=\prod_{i=1}^n \hat f_Y(a_0+a_1x_i+a_2x_i^2)$$

And, find $a_k$ that maximizes this expression. This is mostly a numerical approach.

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  • $\begingroup$ Thanks for your answer. Can you elaborate a bit more? $\endgroup$ – Jochi Toborochi Jun 11 at 1:52
  • $\begingroup$ @JochiToborochi I've added more details. $\endgroup$ – gunes Jun 11 at 8:10
  • $\begingroup$ I think the likelihood function of $X$ is $\hat{f}_Y(\phi(x)) / \phi'(x)$, so here $ \hat{f}_Y(\phi(a_0 + a_1 x + a_2 x^2)) / (a_1 + 2 a_2 x)$, no ? $\endgroup$ – Pohoua Jun 11 at 10:01
  • $\begingroup$ Well, not even, since here $\phi$ is not monotonous... but I think the likelihood of $X$ is more complex than just $\hat{f}_Y(\phi(x))$. $\endgroup$ – Pohoua Jun 11 at 10:27
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    $\begingroup$ Hmm. One point of concern here is that the proposed objective function is trivially maximized by setting $a_1$ and $a_2$ to zero, and $a_0$ to the mode of $\hat{f}_Y$. This would mean that the estimated $a_0$ doesn't depend on $X$, and $a_1$ and $a_2$ don't depend on the data at all. $\endgroup$ – user20160 Jun 11 at 22:58

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