Ytukyg

Let $Y,X,W$ be real-valued random variables, respectively with supports denoted by $mathcal{Y},mathcal{X},mathcal{W}$.

(A1) Assume that $mathcal{X},mathcal{W}$ are finite. Without loss of generality, assume that $mathcal{X}equiv {x_1,x_2}$ and $mathcal{W}equiv {w_1,w_2}$.

(A2) For each realisation $xin mathcal{X}$ of $X$, let $epsilon_x$ be another random variable. Assume that, $forall x in mathcal{X}$, $epsilon_x$ is stochastically independent of $X,W$.

(A3) Assume that the following relation holds
$$
Y=h(X,W)+epsilon_{X}
$$

Consider now the cdf $F(cdot)$ of $Y$ evaluated at $yin mathcal{Y}$. Following here, we can write

$$F(y)=p(x_1,w_1)times F(y| X=x_1, W=w_1)
$$
$$
+p(x_2,w_1)times F(y| X=x_2, W=w_1)
$$
$$
+p(x_1,w_2)times F(y| X=x_1, W=w_2)$$
$$
+p(x_2,w_2)times F(y| X=x_2, W=w_2) $$

where $p(x,w)$ is the probability mass function of $(X,W)$ evaluated at $(x,w)$ and $F(cdot| X=x, W=w)$ is the cdf of $Y$ conditional on $X=x,W=w$.

The lines above highlight that $F(cdot)$ can be expressed as a finite mixture.

Let's focus on the relation between $F(cdot| X=x, W=w)$, (A3), and the cdf of $epsilon_x$.

For any $(x,w)$, $F(cdot| X=x, W=w)$ is determined by (A3) and the cdf of $epsilon_x$.

For example, if $epsilon_xsim mathcal{N}(alpha_x,sigma^2_x)$, then $Y|X=x, W=wsim N(h(x,w)+alpha_x,sigma^2_x)$.

In my exercise, I want to remain non-parametric about the distribution of $epsilon_x$ and I'm looking for non-parametric features of $F(cdot |X=x,W=w)$ that are compatible with (A3). Specifically, this is my question:

Question: is (A3) compatible with writing $F(cdot)$ at any $yin mathcal{Y}$ as
$$
F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) G(y-mu_{x,w})
$$
where $G: mathbb{R}rightarrow [0,1]$ is a cdf symmetric around zero [i.e., $G(y)=1-G(-y)$] and ${mu_{x,w}}_{x,w}$ are real numbers all different between each other?

In other words, I'm wondering whether the differences across
$$
F(y| X=x_1, W=w_1), F(y| X=x_1, W=w_2),F(y| X=x_2, W=w_1) ,F(y| X=x_2, W=w_2)
$$
could be captured by a location shift $mu_{x,w}$ differing across $(x,w)$.

Further thoughts: notice that, as explained here, the cdf's ${H_{x,w}}_{x,w}$ with
$$
H_{x,w}: tin mathbb{R}mapsto G(t-mu_{x,w})
$$
are characterised by equivalent central moments.

$$Fleft(ymid X=x_{i},W=w_{j}right)=Pleft(hleft(X,Wright)+epsilon_{X}leq ymid X=x_{i},W=w_{j}right)=$$$$Pleft(hleft(x_{i},w_{j}right)+epsilon_{x_{i}}leq ymid X=x_{i},W=w_{j}right)=Pleft(hleft(x_{i},w_{j}right)+epsilon_{x_{i}}leq yright)=G_{i}left(y-hleft(x_{i},w_{j}right)right)$$
where $G_{i}$ denotes the CDF of $epsilon_{x_{i}}$.

The third equality is based on independence.

Only if the distribution
of $epsilon_{x_{i}}$ does not depend on $i$ then you can write
$$Fleft(ymid X=x_{i},W=w_{j}right)=Gleft(y-hleft(x_{i},w_{j}right)right)==Gleft(y-hleft(x_{i},w_{j}right)right)$$and:$$
F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) G(y-h(x,w))
$$
where $G$ denotes the common CDF of the $epsilon_{x_{i}}$.