Suppose I have a population of observations I want to model as being drawn from some distributional family, which I believe adequately represents the true distribution. My goal is to estimate the parameters of this distribution.

Suppose further that these observations are randomly censored, and the probability of censorship increases monotonically with the value of the draw. Censored observations appear as a visible blank – think of them as refusals to answer a survey questionnaire. 

Finally, suppose that I have some ancillary data source that lets me estimate the parameters and distributional form of the probability of censorship, but does not provide true values for the censored observations.

Can the uncensored  observations from the population, together with the observations of censored values and the knowledge of the distribution of censorship, suffice to allow me to correctly estimate the  parameters of the partially censored distribution? If so, how? 

Note: Because the true functional form of the distribution is not known with certainty, I have a mild preference for GMM or GEE estimators over ML estimators. But I'll take whatever I can get.