The Rank of the Odd Normal Out (2401.00952v2)

Abstract: A century ago Thurstone wrote about using $\mathbf{X}\sim\mathcal{N}{n}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$ to model our preferences (1927a,b,c). Today latent normals frequently model random rankings. Despite this, for $R{i}:=#\left{ 1\leq j\leq n:X_{j}\leq X_{i}\right} $, little is known about $\left(R_{i_{0}},\ldots,R_{i_{m}}\right)$'s distribution in non-i.i.d. settings. We consider the simplest of such settings, namely that with $n+1$ independent normals, where $X_{0}\sim\mathcal{N}\left(\mu_{0},\sigma_{0}^{2}\right)$ and $X_{1},X_{2},\ldots,X_{n}\sim\mathcal{N}\left(\mu,\sigma^{2}\right)$. In this setting $\left(\left.R_{0}\right|X_{0}\right)\sim1+\mathrm{Binomial}\left(n,\Phi\left(\left(X_{0}-\mu\right)/\sigma\right)\right)$, and $\Phi\left(\left(X_{0}-\mu\right)/\sigma\right)$ is approximately beta-distributed. The beta distribution's conjugacy for the binomial implies that $R_{0}-1$ is roughly beta-binomial. We approximate the distribution of $\left(R_{i_{0}},\ldots,R_{i_{m}}\right)$, deriving $\mathbb{E}R_{i}$, $\mathrm{Var}\left(R_{i}\right)$, $\mathrm{Cov}\left(R_{i},R_{j}\right)$, and its limiting distributions as key parameters grow large or small.