Universal coding with point type classes

With a customized characterization of types, every universal one-to-one coding algorithm can be described as follows: assign sequences to binary strings based on their type class sizes from smallest to largest. With this view, the universal coding problem is to optimally characterize types. In this paper, this Type Size approach is studied for universal source coding of an exponential family of distributions, using the most natural type class definition: two sequences are in the same type class if and only if they are indistinguishable in the sense that they have the same probability for every distribution in the family. This characterization is called the point type class. Exact third-order coding rate is derived for the resulting compression algorithm, revealing that the point type approach, while natural, is sub-optimal compared to the quantized type method, which was previously proposed by the authors.