This is a scatter plot showing the Tisserand invariants and MOIDs with Jupiter for the 517 Jupiter-family comets listed on my other page (as of Nov 3 2015).
Most JFCs have MOIDs less than 1 AU, and very few have MOIDs as large as 1.5 AU. Specifically, only 9.1% (47 out of 517) of the comets have MOIDs larger than 1 AU, and only 4.4% (23 out of 517) have larger than 1.5 AU. This includes the "Main-Belt Comets," which are grouped together in the upper-right part of the plot with very high Tisserand and very high MOID. If we exclude those, the percentages are 7.3% and 2.8%, respectively.
Notice also that there are zero JFCs with Tisserand between 2.6 and 3.1 and MOID greater than 1.5 AU (dark grey rectangle). There are 13 comets in that range of Tisserand with MOID between 1.0 and 1.5 AU (light grey rectangle). That is 3.4% (13 out of 384). MOID is a somewhat poor criterion for orbital comparisons since comets tend to move their own orbits by nongravitational forces, but this plot does suggest that dormant or extinct comets within the asteroid population won't be residing within the dark grey rectangle, and are unlikely to be residing within the light grey rectangle. In other words, asteroids with Tisserands between 2.6 and 3.1 and MOIDs larger than 1 to 1.5 AU are evidently not likely to be dormant or extinct comets.
Comments are welcome!