ConsiderN(unknown) values of a variableX(discrete or continuous) independently sampled from a distribution with densityf(x, θ), where θ is an unknown vector parameter. Suppose that the values ofXbelonging to a certain regionRare not observable. This article deals with the problem of estimatingNand θ in such situations which arise, for instance, in life testing and capture-recapture census. Asymptotic theory for maximum likelihood estimation ofNand θ is presented here and is shown to yield as corollaries both some existing results and a new result pertaining to the truncated negative binomial distribution.