4.35 Estimating Features of a Distribution from Binomial Data

A statistical problem that arises in several fields is that of
estimating the features of an unknown distribution, which may be
conditioned on covariates, using a sample of binomial observations on
whether draws from this distribution exceed threshold levels set by
experimental design.  One application is destructive duration
analysis, where the process is censored at an observation test
time. Another is referendum contingent valuation in ource economics,
where one is interested in features of the distribution of values
placed by consumers on a public good such as an endangered species.
Sampled consumers are asked whether they would vote for a referendum
that would provide the good at a cost specified by experimental
design.  This paper provides practical estimators for moments and
quantiles of the unknown distribution in this problem.  Under mild
regularity conditions and a randomized design for thresholds, we show
that the moments estimators are root-N consistent and asymptotically
normal, despite the limited information in binomial response, while
quantile estimators converge at a lower rate equal to the optimal rate
for nonparametric regression estimation of the distribution of
responses.