  Universal variable-to-fixed (V-F) length coding of $d$-dimensional
exponential family of distributions is considered. We propose an achievable
scheme consisting of a dictionary, used to parse the source output stream,
making use of the previously-introduced notion of quantized types. The
quantized type class of a sequence is based on partitioning the space of
minimal sufficient statistics into cuboids. Our proposed dictionary consists of
sequences in the boundaries of transition from low to high quantized type class
size. We derive the asymptotics of the $\epsilon$-coding rate of our coding
scheme for large enough dictionaries. In particular, we show that the
third-order coding rate of our scheme is $H\frac{d}{2}\frac{\log\log M}{\log
M}$, 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.
