Xiaohong Chen, Wayne Yuan Gao, Likang Wen
arXiv 24 Nov 2025 · Econometrics
arXiv:2511.19121 · PDF · Extracted main text
We propose a new formulation of the maximum score estimator that uses compositions of rectified linear unit (ReLU) functions, instead of indicator functions as in Manski (1975,1985), to encode the sign alignment restrictions. Since the ReLU function is Lipschitz, our new ReLU-based maximum score criterion function is substantially easier to optimize using standard gradient-based optimization pacakges. We also show that our ReLU-based maximum score (RMS) estimator can be generalized to an umbrella framework defined by multi-index single-crossing (MISC) conditions, while the original maximum score estimator cannot be applied. We establish the $n^{-s/(2s+1)}$ convergence rate and asymptotic normality for the RMS estimator under order-$s$ Holder smoothness. In addition, we propose an alternative estimator using a further reformulation of RMS as a special layer in a deep neural network (DNN) architecture, which allows the estimation procedure to be implemented via state-of-the-art software and hardware for DNN.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | –- (1985) Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator | 1.000 | 6 | 3 | 100% |
| 2 | –- and Gao, W. Y (2025) Semiparametric learning of integral functionals on submanifolds | 0.941 | 6 | 3 | 83% |
| 3 | Horowitz, J. L (1992) A smoothed maximum score estimator for the binary response model | 0.874 | 13 | 2 | 100% |
| 4 | Kim, J. and Pollard, D (1990) Cube root asymptotics | 0.874 | 9 | 2 | 100% |
| 5 | Gao, W. Y. and Li, M (2024) Identification of semiparametric panel multinomial choice models with infinite-dimensional fixed effects self | 0.874 | 7 | 2 | 100% |
| 6 | Manski, C. F (1975) Maximum score estimation of the stochastic utility model of choice | 0.811 | 4 | 2 | 100% |
| 7 | –-, –- and Xu, S (2023) Logical differencing in dyadic network formation models with nontransferable utilities | 0.737 | 3 | 2 | 100% |
| 8 | Van Der Vaart, A. W. and Wellner, J. A (1996) Weak Convergence and Empirical Processes | 0.644 | 4 | 2 | 50% |
| 9 | Delsol, L. and Van Keilegom, I (2020) Semiparametric m-estimation with non-smooth criterion functions | 0.644 | 3 | 2 | 67% |
| 10 | Chen, X (2007) Large sample sieve estimation of semi-nonparametric models self | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 36 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Nonparametric Uniform Inference in Binary Classification and Policy Values | 0.405 | 1 | 1 |
| 2 | Gaussian Approximation for Maximum Score and Non-Smooth M-Estimators with Multiway Dependence | 0.405 | 1 | 1 |
| 3 | Root-$n$ Asymptotically Normal Maximum Score Estimation | 0.405 | 1 | 1 |