A Unified Treatment of Integer Parameter Models

Abstract

It is shown that there is an analog to the score function (the derivative of the log-likelihood with respect to the parameter), here called the difference score, which plays an important theoretical role in integer parameter models. A unified treatment of integer parameter models can be obtained by recognizing that many commonly used models have a linear difference score (LDS). It is established that many of the exact results available for exponential family models also hold for LDS models. In particular we find an information lower bound, show the uniqueness of the maximum likelihood estimator, establish monotone likelihood ratio, and establish optimality properties in two-sided tests. It is also shown that if one has a multiparameter likelihood in which there is a conditional likelihood depending on N alone, then these exact results apply there as well.

It is also established when and why the usual likelihood methodology will apply to a class of nonstandard models in which the natural approximations require the integer parameter to become infinite. In the process a variety of asymptotically equivalent confidence methods are derived.

Examples possessing the LDS property include: exponential family restricted to a lattice; hypergeometric with unknown population size; binomial with unknown sample size; the inspection problem with an unknown number of defects in a sample in which the items are tested with error; the zero-truncated Poisson with unknown sample size; the Jelinski–Moranda model; and mark–recapture and capture–removal models.

All but the first three models have additional parameters θ—which leads to a variety of approaches to inferential problems. For instance, the mark–recapture model has a conditional likelihood free from θ and so exact conditional methods apply. In this case exact, conditional likelihood and profile likelihood methods are all available to construct confidence intervals. On the other hand, conditional methods are not available in the capture–removal model.

Key Words:

• Capture–recapture model
• Difference score
• Hypergeometric
• Jelinski–Moranda model
• Linear difference score
• Mark–recapture model