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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.

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  • $\begingroup$ "Reasonable" implies you have a loss function in mind, for otherwise this question is unanswerable. What is it? And are there any assumptions you can make about $P$ at all? $\endgroup$ – whuber Jun 6 '13 at 16:24
  • $\begingroup$ Good point @whuber. I've tried to clarify what I mean. $\endgroup$ – DavidR Jun 6 '13 at 17:20
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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)

enter image description here

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)

enter image description here

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  • $\begingroup$ Yes, can you explain more what a binomial rate smoother does or post a helpful link? $\endgroup$ – soakley Jun 8 '13 at 21:01
  • $\begingroup$ Is it possible that the CDF estimate won't itself be a valid CDF? In particular, it seems to me that the estimator could occasionally decrease as c increases (at least for a finite sample of c's). $\endgroup$ – DavidR Jun 20 '13 at 19:22
  • $\begingroup$ Great point ! My oversight; indeed we need to enforce a monotonic constraint. Luckily, the library scam which builds upon mgcv allows to incorporate "shape constraints" such as monotonicity, convexity, etc. I have added new code above $\endgroup$ – Markus Loecher Jun 22 '13 at 8:04
  • $\begingroup$ Looks good to me -- thanks! I've been wondering what a nonparametric maximum likelihood estimator would like here, similar in spirit to the Kaplan-Meier estimator for right censored data. $\endgroup$ – DavidR Jun 28 '13 at 20:43

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