This is a scatter plot showing the Tisserand invariants and MOIDs with Jupiter for the 332 Jupiter-family comets listed on my other page (as of June 8 2007). Notice how there are no JFCs known with Tisserand between 2.6 and 3.0 and MOID greater than 1.5 AU (dark grey rectangle). Furthermore there are only 7 objects with such Tisserand and MOID greater than 1.0 AU (light grey rectangle). And really of those 7, just 4 are well within that parameter space. MOID is a somewhat poor criterion for orbital comparisons since comets tend to move their own orbits by nongravitational forces, but this plot suggests that one has a higher probability of finding dormant or extinct comets among the asteroid population by avoiding asteroids in the grey rectangles.
For what it's worth, the 7 comets are: 159P/LONEOS, 182P/LONEOS, P/2002 LZ11 (LINEAR), P/2005 JY126 (Catalina), P/2005 W3 (Kowalski), P/2005 Y2 (McNaught), and C/2006 F2 (Christensen). The four well-embedded ones are 159P/LONEOS, P/2002 LZ11 (LINEAR), P/2005 JY126 (Catalina), and P/2005 Y2 (McNaught).
Comments are welcome!