I have a question on how to define mixture distributions for continuous random variables. In short, what I'm confused about is whether they can be equivalently written using the cdf (cumulative distribution function) or the pdf (probability density function).
Let me explain my doubt with an example.
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.
(A2) Assume that $Y$ is a continuous random variable, with well defined pdf.
Consider now the pdf $f(\cdot)$ of $Y$ evaluated at $y\in \mathcal{Y}$:
$$f(y)=\sum_{x\in \mathcal{X},w\in \mathcal{W}} f(y,x,w)=\overbrace{\sum_{x\in \mathcal{X},w\in \mathcal{W}} \overbrace{p(x,w)}^{\text{Mixing weight}}\times \overbrace{f(y| X=x, W=w)}^{\text{Mixing density}}}^{\text{Finite mixture}}
$$
where $p(\cdot,\cdot)$ is the probability mass function of $(X,W)$.
It seems to me that the pdf of $Y$ can be written as a finite mixture (assuming that $f(y|X=x, W=w)$ is well-defined at each $x\in \mathcal{X}, w\in \mathcal{W}$).
I don't think though that the same relation holds for the cdf of $Y$. In other words, let $F(\cdot)$ denote the cdf of $Y$; I think that
$$
F(y)\neq \sum_{x\in \mathcal{X},w\in \mathcal{W}} p(x,w)\times F(y| X=x, W=w)
$$
Therefore, I would like you help to understand the following:
-
Can we write $F(\cdot)$ as a mixture?
-
If yes, using which weights and what is the relation of those weights with the weights $\{p(x,w)\}_{x,w}$ used above?
-
If not, why do I have the impression that in the stat literature mixtures are flexibly defined using cdf or pdf, as in this Wikipedia article?
Best Answer
For any Borel set $B$ we can write:$$P(Y\in B)=\sum_{x\in \mathcal{X},w\in \mathcal{W}}P(X=x,W=w)P(Y\in B\mid X=x,W=w)=\sum_{x\in \mathcal{X},w\in \mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-\infty,y]$ we find something like:
$$F_Y(y)=\sum_{x\in \mathcal{X},w\in \mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.