  In this paper, we investigate the redundancy of universal coding schemes on
smooth parametric sources in the finite-length regime. We derive an upper bound
on the probability of the event that a sequence of length $n$, chosen using
Jeffreys' prior from the family of parametric sources with $d$ unknown
parameters, is compressed with a redundancy smaller than
$(1-\epsilon)\frac{d}{2}\log n$ for any $\epsilon>0$. Our results also confirm
that for large enough $n$ and $d$, the average minimax redundancy provides a
good estimate for the redundancy of most sources. Our result may be used to
evaluate the performance of universal source coding schemes on finite-length
sequences. Additionally, we precisely characterize the minimax redundancy for
two--stage codes. We demonstrate that the two--stage assumption incurs a
negligible redundancy especially when the number of source parameters is large.
Finally, we show that the redundancy is significant in the compression of small
sequences.
