It's very hard to compare PD w.r.t to half rate or full rate, since both of them can implement these functions.

In terms of speed, Bang-bang PD has clear advantage since it's only limited by how fast a flip-flop can be built, whereas in linear PD, processing the very narrow pulse will limit the speed in the end, which can be very aggressive.

Of course, linear PD and Bang-bang PD have their own other problems also. Further, this paper may have some thing you want:

[1] R. C. Walker, “Designing Bang-Bang PLLs for Clock and Data Recovery in Serial Data Transmission Systems,” pp. 1–12.