Estimating a distribution from above/below observations

Question

Let $P$ be an unknown distribution on $(-\infty,\infty)$. Let $X_1,\ldots,X_n$ be an iid sample from $P$. Let $c_1,\ldots,c_n\in(-\infty,\infty)$ be a known set of constants. We observe $Y_1,\ldots,Y_n$, where $Y_i = 1(X_i < c_i)$. (That is, $Y_i=1$ if $X_i<c_i$ and is $0$ otherwise.) I'm looking for some "reasonable" estimators of the distribution $P$.

And what do I mean by "reasonable"? Suppose, for simplicity, that $P$ is supported on [0,1], and the $c_i$ are an iid sample from Uniform[0,1]. $P$ probably has atoms, but we can ignore that if it invites more answers.

Call $\hat{P}_n$ a "reasonable" estimator of $P$ if, with probability $1$ over the selection of the $c_i$'s, $\hat{P}_n$ converges to $P$ in distribution.

NOTE: We can actually consider the $c_i$ as design points that can be chosen by the experimenter. That seems like a separate issue, and I don't want to complicate the question. But if you have an estimator that requires a particular set of $c_i$'s, that's fine too.

Answer 1

You could try to directly estimate the CDF via a binomial rate smoother ? Here is an idealized example for x stemming from a normal distribution:

   ci = seq(from=-3,to=3,length=500)
   X = rnorm(500)
   Y = rep(NA, 500)
   for (i in 1:500) Y[i] = as.numeric(X[i] < ci[i] )
   plot(ci,Y, type="s")
   library(mgcv)
   library(boot)
   fit=gam(Y~s(ci), family=binomial(link="logit"))
   plot(fit, trans=inv.logit, shade = TRUE)

To enforce monotonic behavior, in the above example, change the code to:

library(scam)
fitMonotone=scam(Y~s(ci,bs="mpi"), family=binomial(link="logit"))

InvLogit = function(x, SCALE=TRUE) {
  if (SCALE) x = x -mean(x)
  return(exp(x)/(1+exp(x)))
}

plot(fitMonotone, trans=InvLogit, shade = TRUE)