What expansion factor should be used in binned probability proportional to size sampling?

In probability proportional to size sampling, including prism cruising and other forms of point and line sampling, calculation of an exact expansion factor requires that size be recorded exactly. When sizes are binned or recorded by class, this information is lost. While several alternatives for calculating the expansion factor have been proposed, theoretical attention has been lacking. A decision-theoretic perspective helps distinguish between the alternatives and offers some support to the use of the arithmetic mean size in calculating the expansion factor, a choice which had previously come under some criticism. However, consistency arguments strongly favor estimators based on squared error loss or minimax principles. Some new alternatives are suggested when prior information about the diameter distribution in a stratum is available.