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Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in {\zeta(s)}. For example, integrating {|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto {2T}, we obtain an asymptotic formula when the shift α is roughly bigger than {\frac{1}{\log T}}and {k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as {\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function.
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Extended fiducial inference for individual treatment effects via deep neural networks
Abstract Individual treatment effect estimation has gained significant attention in recent data science literature. This work introduces the Double Neural Network (Double-NN) method to address this problem within the framework of extended fiducial inference (EFI). In the proposed method, deep neural networks are used to model the treatment and control effect functions, while an additional neural network is employed to estimate their parameters. The universal approximation capability of deep neural networks ensures the broad applicability of this method. Numerical results highlight the superior performance of the proposed Double-NN method compared to the conformal quantile regression (CQR) method in individual treatment effect estimation. From the perspective of statistical inference, this work advances the theory and methodology for statistical inference of large models. Specifically, it is theoretically proven that the proposed method permits the model size to increase with the sample sizenat a rate of$$O(n^{\zeta })$$ for some$$0 \le \zeta <1$$ , while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range$$0\le \zeta < \frac{1}{2}$$ required by the classical central limit theorem. Furthermore, this work provides a rigorous framework for quantifying the uncertainty of deep neural networks under the neural scaling law, representing a substantial contribution to the statistical understanding of large-scale neural network models.
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- PAR ID:
- 10591105
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Statistics and Computing
- Volume:
- 35
- Issue:
- 4
- ISSN:
- 0960-3174
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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