TITLE: What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order?
QUESTION [17 upvotes]: What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from the distribution of this N-dimensional vector:

Create an N-1 dimensional vector of uniformly distributed random numbers between 0 and 1, inclusive. For example, for N=3, [0.564, 0.243].
Append the numbers 0 and 1 to the vector, then sort it in ascending order. For example, [0, 0.243, 0.564, 1].
Compute the 1st-order differences of the vector, yielding an N-dimensional vector of uniformly distributed random numbers that sum to 1. For example, [0.243, 0.321, 0.436].
Sort this vector into descending order. For example, [0.436, 0.321, 0.243].

It is easy to demonstrate by graphical analogy that for N=2, the expected value of the vector is exactly [3/4, 1/4].
For N=3, by sampling I have determined that the expected value is approximately [0.611, 0.277, 0.111].
For N=4, by sampling I have determined that the expected value is approximately [0.520, 0.271, 0.146, 0.063].
What is the analytical form of the expected value for any value of N?

REPLY [15 votes]: Liviu has given an excellent answer with a nice geometric flavor to it. This answer is meant to serve as a complement with a slightly more probabilistic bent. In the process, some of the computations get buried in favor of bringing out the connections to other areas of elementary probability theory.
A well-known alternative mechanism for generating points uniformly on the simplex is as follows: Let $Z_1,\ldots,Z_n$ be iid exponential random variables. Then $(Z_1/Z,\ldots,Z_n/Z)$ is uniformly distributed on the simplex where $Z = Z_1 + \cdots + Z_n$. 
This is a special case of the Dirichlet distribution. In your question, you are effectively interested in computing $\mathbb E \frac{Z_{(k)}}{Z}$ for each $k$ where $Z_{(k)}$ is the $k$th order statistic from your iid exponential sample.
By Rényi representation, we can construct $Z_{(k)}$ directly as follows: Let $Z_{(0)} = 0$, and let $(\widetilde Z_i)_{i \geq 1}$ be a sequence of iid exponentials. Then, form 
$$
Z_{(k)} = Z_{(k-1)} + \frac{\widetilde Z_k}{n-k+1}\>.
$$
Note that $Z = \sum_{k=1}^n Z_k = \sum_{k=1}^n Z_{(k)} = \sum_{k=1}^n \widetilde Z_k$.
Hence
$$
\mathbb E \frac{Z_{(k)}}{Z} = \sum_{i=1}^k \frac{1}{n-i+1}\mathbb E \frac{\widetilde Z_i}{Z} = \frac{1}{n} \sum_{i=1}^k \frac{1}{n-i+1} \>,
$$
since the $\widetilde Z_i/Z$ have the same $\text{Beta}(1,n-1)$ distribution for each $i$.
References

A. Rényi (1953), On the theory of order statistics, Acta Mathematica Hungarica, vol. 4(3–4), pp. 191–231. 
L. Devroye (1986), Non-Uniform Random Variate Generation, Springer-Verlag. (Chapter 5: Uniform and Exponential Spacings).

Postscriptum: Note that the algorithms given in this answer for generating a point on the standard simplex and on its sorted variant are both $O(n)$ compared to the $O(n \log n)$ approach in the question, so if $n$ is large and computing $\log x$ for $0 < x < 1$ is cheap, this may lead to meaningful gains in sampling efficiency.
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