This chapter provides further insight into the constant false alarm rate (CFAR) property of some of the sliding window detection processes examined thus far. A substantial development to ensure that the CFAR property is acquired at the detector design stage can be found in the article of E. Conte and others, who examine multidimensional signal detection problems. Invariance in a radar detection context was examined in A. De Maio and S. Iommelli, with the concept of generalised CFAR and existence of a uniformly most powerful invariant detector introduced in De Maio. The concept of a uniformly most powerful CFAR is introduced in the article of A. Ghobadzadeh and others, which is defined as the optimal CFAR test among all such decision rules for a specific problem space. It is then shown that the uniformly most powerful CFAR test for a minimally invariant hypothesis testing problem is given by the likelihood ratio of the maximal invariant under the minimally invariant group.