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 $P$.