# Distribution of the rank statistic for an iid sample?

For an iid sample $X_1, \dots, X_n$, its rank statistic is $(r_1, \dots, r_n)$ where $r_k$ is the rank of $X_k$ sorted in nondecreasing order. When there are ties, the ranks of the ties are the same as the average of their random ranks. For example, in $1, 2, 2, 3$, the ranks of the two $2$'s are $(2+3)/2 = 2.5$.

The pmf of the rank statistic of an iid sample of size $n$ and with a continuous distribution, is always $0$ if there is tie, and $1/n!$ if there is no tie. The marginal distribution of each $r_k$ is uniform over $\{1,\dots, n\}$.

I was wondering how to decide its joint pmf and its marginal pmf when the distribution is discrete or other non-continuous case?

I think there might already been some references covering my question. So could you recommend some?

Thanks and regards!

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What precisely do you mean by rank statistic? Do you mean the quotient $\frac{f(x_1,x_2,\dots,x_n)}{g(\tilde x_1,\tilde x_2,dots,\tilde x_n)}$ where $f$ is the pdf of the $iid$ random variables, $g$ the pdf of the order statistics? (where $\tilde x_1\leq \tilde x_2 \leq \dots \leq \tilde x_n$ are the values $x_1, \dots,x_n$ in ascending order? – Tim May 5 '13 at 18:57
@Tim: For an iid sample $X_1, \dots, X_n$, its rank statistic is $(r_1, \dots, r_n)$ where $r_k$ is the rank of $X_k$ sorted in nondecreasing order. When there are ties, the ranks of the ties are the same as the average of their random ranks. For example, in $1, 2, 2, 3$, the ranks of the two $2$'s are $(2+3)/2 = 2.5$. – Tim May 5 '13 at 19:11
Ok thanks. I don't think there's a good answer for that unless $X$ something really simple like a uniform distribution on $(1,2,\dots,n)$. – Tim May 5 '13 at 19:20