  We study a new parametric family of distributions on the ordered simplex
$\nabla^{d-1} = \{y \in \mathbb{R}^d: y_1 \geq \dots \geq y_d \geq 0,
\sum_{k=1}^d y_k = 1\}$, which we call Generalized Rank Dirichlet (GRD)
distributions. Their density is proportional to $\prod_{k=1}^d y_k^{a_k-1}$ for
a parameter $a = (a_1,\dots,a_d) \in \mathbb{R}^d$ satisfying $a_k + a_{k+1} +
\dots + a_d > 0$ for $k=2,\dots,d$. The density is similar to the Dirichlet
distribution, but is defined on $\nabla^{d-1}$, leading to different
properties. In particular, certain components $a_k$ can be negative. Random
variables $Y = (Y_1,\dots,Y_d)$ with GRD distributions have previously been
used to model capital distribution in financial markets and more generally can
be used to model ranked order statistics of weight vectors. We obtain for any
dimension $d$ explicit expressions for moments of order $M \in \mathbb{N}$ for
the $Y_k$'s and moments of all orders for the log gaps $Z_k = \log Y_{k-1} -
\log Y_k$ when $a_1 + \dots + a_d = -M$. Additionally, we propose an algorithm
to exactly simulate random variates in this case. In the general case $a_1 +
\dots + a_d \in \mathbb{R}$ we obtain series representations for these
quantities and provide an approximate simulation algorithm.
