Kant thought the third type (analytic a posteriori) is self-contradictory, so he discusses only three types as components of his epistemological framework.
However, Stephen Palmquist treats the analytic a posteriori not only as a valid epistemological classification but also as the most important of the four for philosophy.
In “Knowledge and Experience – An Examination of the Four Reflective ‘Perspectives’ in Kant’s Critical Philosophy”, Kant-Studien 78:2 (1987), pp.170-200, Stephen Palmquist shows how Kant’s own discussion of the role of hypotheses (and the “as if” approach) in philosophy can be understood only as an example of analytic aposteriority.
See also the revised version of this article, reprinted as Chapter IV of Stephen Palmquist, Kant’s System of Perspectives: An architectonic interpretation of the Critical philosophy (Lanham: University Press of America, 1993).
In “A Priori Knowledge in Perspective: (II) Naming, Necessity and the Analytic A Posteriori”, The Review of Metaphysics 41:2 (December 1987), pp.255-282, Palmquist argues that Saul Kripke uses Kant’s terms incorrectly when he analyzes naming as contingent a priori; when Kripke’s use of the key terms is translated to make it consistent with Kant’s usage, Kripke’s position can be understood as defending the analytic a posteriority of naming.