Robust estimation in truncated discrete distributions with application to capture-recapture experiments

https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2982096

The Analysis of Library Data

Consider N (unknown) values of a variable X (discrete or continuous) independently sampled from a distribution with density f(x, θ), where θ is an unknown vector parameter. Suppose that the values of X belonging to a certain region R are not observable. This article deals with the problem of estimating N and θ in such situations which arise, for instance, in life testing and capture-recapture census. Asymptotic theory for maximum likelihood estimation of N and 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.

Estimating the Size of a Truncated Sample

Estimating the unknown number of classes in a population has numerous important applications. In a Poisson mixture model, the problem is reduced to estimating the odds that a class is undetected in a sample. The discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one-sided. Confidence intervals for the number of classes are also necessarily one-sided. A sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes.

Estimating the number of classes

/pdf/estimating-the-number-of-classes-474fhz862b.pdf

Maximum likelihood estimators and a modified maximum likelihood estimator are developed for estimating the zero class from a truncated Poisson sample when the available sample size itself is a random variable. All the estimators considered here are asymptotically equivalent in the usual sense; hence their asymptotic properties are investigated in some detail theoretically as well as by making use of Monte Carlo experiments. One modified estimator appears to be best with respect to the chosen criteria. An example is given to illustrate the results obtained.

Alan J. Gross

Papers

Estimating the Complete Sample Size from an Incomplete Poisson Sample