Is my cat missing the litter box on purpose?

 It depends on what do you call "purpose." Usually owners consider missing the litter box like a desire to take revenge something. Actually that is wrong. Our pets never do that! 

Revenge is an action planned in advance and performed after an event caused it, sometimes, much later. However, a lot of experiments show that animals (including apes) can't make long-term plans. Thus, missing the litter box is usually a sign of anxiety that an animal feels. Another question is why your pet feeling anxiety. There are a lot of reasons. To answer that question an owner needs to seek professional help from a pet expert.