# Parameter estimation of a transformation

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.

• 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? – Demetri Pananos Jun 10 at 23:35
• No, you don't observe them in pairs. The samples are independent. – Jochi Toborochi Jun 11 at 1:51

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.

• Thanks for your answer. Can you elaborate a bit more? – Jochi Toborochi Jun 11 at 1:52
• @JochiToborochi I've added more details. – gunes Jun 11 at 8:10
• 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 ? – Pohoua Jun 11 at 10:01
• Well, not even, since here $\phi$ is not monotonous... but I think the likelihood of $X$ is more complex than just $\hat{f}_Y(\phi(x))$. – Pohoua Jun 11 at 10:27
• 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. – user20160 Jun 11 at 22:58