With respect to mid-air collisions, how is a near miss calculated, detected, and reported? Does ATC have radar detection, or is it strictly up to pilots to report a close encounter with another aircraft?
First I'd just like to comment that "near miss" is a bad term (as falstro also noticed). It seems to say that the planes "nearly missed", implying they didn't miss. The FAA uses the terms "Near Midair Collision" (NMAC) for serious incidences, and "loss of separation" for less serious ones. As such, terms like "near miss", "close call", and "narrowly escaped disaster" are not well defined.
ATC does usually have radar. The data is recorded and can be analyzed later. Of course, not all areas have radar coverage. In these cases it is up to pilots to report if they feel safety was compromised. They can use the Aviation Safety Reporting System, which is part of the Aviation Safety Information Analysis and Sharing system.
Especially with commercial flights, Traffic Collision Avoidance System (TCAS) is also used. If TCAS issues a Resolution Advisory (RA), which commands pilots to take action to avoid a collision, the pilots are usually required to report the incident.
A "loss of separation" would be anything less than the required separation.
According the the FAA:
That and more information about reporting incidents can be found in Section 6 of the AIM.