Neural Generative Modeling of Order Statistics

Order statistics (OS) and induced order statistics (IOS) arise whenever one sorts the components of a random vector, ubiquitous in portfolio construction and impact investing where assets are ranked by an impact factor and paired with returns. Standard generators ignore the combinatorial constraints of sorted data. We analyze neural OS generators through Schucany's and Sukhatme's probabilistic representations, which serve as theoretical devices for deriving approximation guarantees. We prove that ReLU networks with (\mathcal{O}(\varepsilon^{-2}\ln(\varepsilon^{-1}))) parameters approximate the OS of (N) i.i.d. uniforms within (\mathcal{O}(\varepsilon\ln N)) in expected (L_1). On synthetic OS benchmarks, a simple MLP generator with global mean-scale normalization and an OS-aware penalization term improves sortedness while trading off distributional fidelity, outperforming vanilla baselines on ordering metrics.