# On the properties of a finite mixture

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 $$x\in \mathcal{X}$$ 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 $$y\in \mathcal{Y}$$. 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 $$\epsilon_x\sim \mathcal{N}(\alpha_x,\sigma^2_x)$$, then $$Y|X=x, W=w\sim N(h(x,w)+\alpha_x,\sigma^2_x)$$.

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 $$y\in \mathcal{Y}$$ as $$F(y)=\sum_{x\in \mathcal{X}, w\in \mathcal{W}} p(x,w) G(y-\mu_{x,w})$$ 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 $$H_{x,w}: t\in \mathbb{R}\mapsto G(t-\mu_{x,w})$$ are characterised by equivalent central moments.

$$F\left(y\mid X=x_{i},W=w_{j}\right)=P\left(h\left(X,W\right)+\epsilon_{X}\leq y\mid X=x_{i},W=w_{j}\right)=$$$$P\left(h\left(x_{i},w_{j}\right)+\epsilon_{x_{i}}\leq y\mid X=x_{i},W=w_{j}\right)=P\left(h\left(x_{i},w_{j}\right)+\epsilon_{x_{i}}\leq y\right)=G_{i}\left(y-h\left(x_{i},w_{j}\right)\right)$$ where $$G_{i}$$ denotes the CDF of $$\epsilon_{x_{i}}$$.
Only if the distribution of $$\epsilon_{x_{i}}$$ does not depend on $$i$$ then you can write $$F\left(y\mid X=x_{i},W=w_{j}\right)=G\left(y-h\left(x_{i},w_{j}\right)\right)==G\left(y-h\left(x_{i},w_{j}\right)\right)$$and:$$F(y)=\sum_{x\in \mathcal{X}, w\in \mathcal{W}} p(x,w) G(y-h(x,w))$$ where $$G$$ denotes the common CDF of the $$\epsilon_{x_{i}}$$.
• Thanks. Just to clarify "Only if the distribution of $\epsilon_{x_i}$ does not depend on $i$": is this equivalent to say "Only if $\epsilon_{x_1}, \epsilon_{x_2}$ are identically distributed"? – STF Dec 28 '18 at 16:50
• Also, maybe there is a typo in your answer: what do you mean by $G(y-h(x_i, w_j))==G(y-h(x_i, w_j))$? – STF Dec 28 '18 at 17:16