<p>A number of geometric problems, including affine hyperbolic spheres, Hilbert metrics and Minkowski type problems, are reduced to a singular Monge-Ampère equation which can be written locally as a class of Monge-Ampère equations with singularity at the boundary. We estimate the boundary derivatives of all orders for the solutions to the class of singular equations as good as possible, and prove that the solution to the equation is globally analytic if the dimension of the space is even. As a corollary, we obtain the global analyticity of affine hyperbolic spheres in the even dimensional space.</p>