Given a uniformly sampled permutation matrix $\Pi\in\{0,1\}^{n\times n}$, what can we say about the matrix $E$ where $$ \Pi = I + E, $$ where $I$ is the identity matrix. More precisely. what can we say about the following:
- What is the statistical distribution of $E$?
- What is the mean of $E$?
- What is the variance of $E$?
Thanks.
Motivation: This is useful in the shuffled linear regression problem where observations are in the form of $$ y=\Pi X\beta=X\beta+EX\beta, $$ where $y$ and $X$ are observed, $\Pi$ and $\beta$ are unknown, and the goal is to estimate $\beta$. I was wondering if we can view $EX\beta$ like some sort of noise, akin the usual linear regression with additive noise.