Fundamental limits of universal variable-to-fixed length coding of parametric sources

Universal variable-to-fixed (V-F) length coding of d-dimensional exponential family of distributions is considered. An achievable scheme is proposed, which consists of a dictionary to parse the source output stream. The previously-introduced notion of quantized type is employed for the dictionary construction. The quantized type class of a sequence is based on partitioning the space of minimal sufficient statistics into cuboids. The proposed dictionary consists of sequences in the boundaries of transition from low to high quantized type class size. Asymptotics of the ϵ-coding rate of the proposed coding scheme for large enough dictionaries is derived. In particular, we show that the third-order coding rate of the proposed scheme is H d/2 log log M/log M, where H is the entropy of the source and M is the dictionary size. We further provide a converse, showing that this rate is optimal up to the third-order term.