The optimal sub-Gaussian normalisation for randomised monotone functions

2023/12/03 by Thomas Anton, Anton, Thomas, Rabee Tourky +2
Economics, Econometrics and Finance · Mathematics · #60E15 #60H07 #62G10 #Advanced Causal Inference Techniques #FOS: Computer and information sciences #FOS: Mathematics #Health Systems, Economic Evaluations, Quality of Life #Methodology (stat.ME) #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2312.01265

openalex publication_date 2023/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M denote the class of randomised monotone functions on ℝ with values in [0,1], and let U_M\colon ℝ₊→ ℝ₊ be the minimal function for which ℙ\ √(η_f) sup_t∈ℝ | f_Z(t) - \Exff_Z(t) | ≥ ε√U_M(η_f) \ ≤ 2\e^-2ε2 holds for every member f_Z of M with finite effective sample size η_f and every positive ε. We prove that for every x> 1, | √U_M(x) - √(log₄ x) | ≤ 2 min \ 1, (2 ln(\e + ln x))/(√(ln x)) \ . The optimal adjustment √U_M(x) matches (1)/(√(2ln 2))√(ln x) for all x>1, with residuals bounded as above.

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