Approximations for the Conditional Case

Abstract

There is, by now, a voluminous literature on approximations. We will highlight those approximations that are interesting from a methodological point of view and could be considered optimal relative to the effort involved. There are two basic approaches used. The methods of moments and product limit approximations are based on spacings, and the other methods are based on cell occupancy numbers. The approximations based on spacings are described in Sections 10.1–10.4, and those on cell occupancy numbers in Sections 10.5–10.7. In order of computational effort they might be ordered as:

• 1. Simple one parameter exponential approximation.

• 2. Crude approximation based on cell occupancy probabilities.

• 3.

  • (a) Markov-type approximation based on cell occupancy numbers.

  • (b) Product limit approximations.

  • (c) Method of moments: two-parameter approximations.

• 4. Average of upper and lower bounds based on three-way intersections.