Rank Distributions for Independent Normals with a Single Outlier

Abstract

Thurstone's latent-normal model, introduced a century ago to describe human preferences in psychometrics (1927), remains a cornerstone for modeling random rankings. Yet when the underlying normals differ in distribution, the joint law of ranks $R_i:={\sum }_{j=1}^n{1}_{X_j\le X_i}$ is virtually unexplored. We study the simplest non-identically-distributed case: $n+1$ independent normals with $X_0\sim \mathcal{N}\left({\mu }_0,{\sigma }_0^2\right)$ and $X_i\sim \mathcal{N}\left(\mu ,{\sigma }^2\right)$ for $1\le i\le n$. Here, $R_0\mid X_0\sim 1+\mathrm{Binomial}(n,\Phi ((X_0-\mu )/\sigma \left)\right)$, and the success probability $\Phi \left(\right(X_0-\mu )/\sigma )$ is accurately modeled by a beta distribution. Exploiting beta-binomial conjugacy, we observe that $R_0-1$ follows a beta-binomial law, which then yields a precise approximation for the joint distribution of $\left(R_0,R_{i_1},\dots ,R_{i_m}\right)$. We derive closed-form expressions for $\mathbb{E}{R}_i$, $\mathrm{Cov}\left(R_i,R_j\right)$, and the limiting distributions of $\left(R_0,R_{i_1},\dots ,R_{i_m}\right)$ as key parameters grow large or small.

47 pages, 9 figures