being sufficient statistics, types are not useful in the representation of continuous-alphabet sources. As Csiszár [@Csiszar1998] writes: “extensions of the type concept to continuous alphabets are not known.” In the case that $\mathcal{X} = \mathbb{R}$, we make extensive use of the basic distribution theory of order statistics. When the sequence of random variables $X_1,\ldots,X_n$ is arranged in ascending order as $X_{(1:n)} \le \cdots \le X_{(n:n)}$, $X_{(r:n)}$ is called the $r$th order statistic. It can be shown that the order statistics for exchangeable variates are complete, minimal sufficient statistics [@DavidN2003]. For alphabets of vectors of real numbers, there is no simple canonical form for expressing the minimal sufficient statistic since there is no natural ordering of vectors [@Barnett1976].

Lossless Coding {#sec:entropyrate}
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Consider the lossless coding of multisets of $n$ letters drawn from the discrete alphabet $\mathcal{X}$. Since there are $n!$ permutations of a sequence of length $n$, it would seem that a rate savings of $\log(n!)$ relative to sequence coding might be possible. Specifically, the upper bound $H(J) \leq \log(n!)$ combined with (\[eq:Hdecomp\]) gives the lower bound $$H(\{X_i\}_{i=1}^n) \ge H((X_i)_{i=1}^n) - \log n! \mbox{.}
\label{eq:orderlowerbound}$$ Since this lower bound can be negative, there must be more to the story.

The 