Global navigation satellite system (GNSS) is a key positioning system for intelligent transport systems (ITS). The GNSS receiver should be able to autonomously perform integrity monitoring, including detection of faulty observations. In this contribution, new models for fault detection in the position domain are presented that are tailored for ITS. The fault detection tests are parameterised for the track frame of the vehicle, and in a combined single test form. Another new form is presented where the detection testing is parameterised in the direction of the maximum possible error. The tests are formed where position errors are assumed to have a zero-mean Gaussian distribution, which is a working hypothesis in the open environment. The case of positioning in the urban environment is also addressed using two approaches. The first is by using a logistic distribution that is found to empirically better fit a very large sample of position errors compared to the normal distribution in this environment. The second approach is to use of an overbounding Gaussian distribution. The protection levels (PLs) in the track frame are presented, and the advantage of expressing PL along the maximum direction is shown. The presented methods are experimentally demonstrated in practice through a kinematic test.