Nonparametric Identification from Order Statistics

Question

Suppose a vector of random variables $(X_1,...,X_n,Y_1,...,Y_m)$ is such that $X\sim F(\cdot)$ and $Y\sim G(\cdot)$. So $X$ are distributed independently and identically as $F(\cdot)$ and $Y$ as $G(\cdot)$. We only observe $n+m$ ordered variables $(Z_1<Z_2<.....<Z_{n+m})$. The question is, can we recover the two CDFs $F(\cdot)$ and $G(\cdot)$ from the $Z's$?

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Edit:

When I said recover I meant to identify, which is a different problem from estimate but related. the idea being that if I have "infinitely" many iid observations from $F$, I can identify and then estimate $F$ (using ECDF and invoke some consistency). Here I only know $Z_1<\ldots <Z_{n+m}$ and I know that there are two CDFs that generate $Z$'s. I am sorry if my statement was confusing.

Answer 1

While I do not provide a whole solution, I will at least try to provide a starting point for formalizing and answering your question. I hope you or someone else can fill in the details.

So you have distribution functions $F$ and $G$ and a random vector (for $n \in \mathbb N$):

$$ (X,Y) \sim F^n\otimes G^n$$

And you are asking whether $F$ and $G$ are identifiable from the distribution of the order statistics of $(X,Y)$ (after "unpacking" it).

I think that to answer this question it is more instructive to look at the case $n=1$.

$$ (X,Y) \sim F\otimes G$$

Now you are asking: Are $(F,G)$ identifiable from the distribution of $(\min(X,Y), \max(X,Y))$?

since

$$\Pr[\min(X,Y) \leq t] = F(t) + G(t) - F(t)G(t)$$

and

$$\Pr[\max(X,Y) \leq t] = F(t)G(t)$$

you see that you can identify some of the information about $F$ and $G$. You also have available information about the covariance of $(\min(X,Y), \max(X,Y))$ (more generally: their copula) which I have not analyzed here.

But is this enough to identify $F$ and $G$? Obviously it is not because you definitely cannot distinguish between $F\otimes G$ and $G\otimes F$. Is it identifiable up to "naming"? I think it might be with some further conditions, such as strict monotonicity of $F$ and $G$, but I have not thought through them.

Answer 2

The answer is yes and moreover, you can identify $F$ and $G$ only using two order statistics: $Z_1$ and $Z_{m+n}$.

As in air's answer, let's consider the case $m=n=1$. Let $H(t)=\Pr[Z_1\le t]$ and $I(t)=\Pr[Z_2\le t]$ be the known cdfs of $Z_1=\min(X,Y)$ and $Z_2=\max(X,Y)$. Again, as in air's answer, we have that $H$ and $I$ are related to $F$ and $G$ by the system of equations:

\begin{align*} H=&1-(1-F)(1-G)=F+G-F\,G\\ I=& F\,G \end{align*}

Thus, since $F=I/G$ by the second equation, we obtain from equation one that $H = \frac{I}{G}+G-I$, or $$ G^2- (H+I) G + I =0$$.

Solving the quadratic equation give us: $G=\frac{(H+I)+\sqrt{(H+I)^2-4I}}{2}$ (the negative root is not consistent with $1\ge H\ge I \ge 0$) and so $F=\frac{2I}{(H+I)+\sqrt{(H+I)^2-4I}}$. That is, we identified $F$ and $G$.

Now consider the general case, $H(t)=\Pr[Z_1\le t]$ and $I(t)=\Pr[Z_{m+n}\le t]$ and so:

\begin{align*} H=&1-(1-F)^n(1-G)^m\\ I=& F^n\,G^m \end{align*} Although, we might not be able able to explicitly solve for $F$ and $G$, it is still possible to solve it numerically for $F(t)$ and $G(t)$ given the values of $H(t)$ and $I(t)$. Thus, you can identify $F$ and $G$ only using two order statistics: $Z_1$ and $Z_{m+n}$.