Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915629731151872 |
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| author | Fernández, Andrés Aradillas Blanchet, José Olea, José Luis Montiel Qiu, Chen Stoye, Jörg Tan, Lezhi |
| author_facet | Fernández, Andrés Aradillas Blanchet, José Olea, José Luis Montiel Qiu, Chen Stoye, Jörg Tan, Lezhi |
| contents | We consider a class of hypothesis testing problems where the null hypothesis postulates $M$ distributions for the observed data, and there is only one possible distribution under the alternative. We show that one can use a stochastic mirror descent routine for convex optimization to provably obtain - after finitely many iterations - both an approximate least-favorable distribution and a nearly optimal test, in a sense we make precise. Our theoretical results yield concrete recommendations about the algorithm's implementation, including its initial condition, its step size, and the number of iterations. Importantly, our suggested algorithm can be viewed as a slight variation of the algorithm suggested by Elliott, Müller, and Watson (2015), whose theoretical performance guarantees are unknown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent Fernández, Andrés Aradillas Blanchet, José Olea, José Luis Montiel Qiu, Chen Stoye, Jörg Tan, Lezhi Econometrics Statistics Theory We consider a class of hypothesis testing problems where the null hypothesis postulates $M$ distributions for the observed data, and there is only one possible distribution under the alternative. We show that one can use a stochastic mirror descent routine for convex optimization to provably obtain - after finitely many iterations - both an approximate least-favorable distribution and a nearly optimal test, in a sense we make precise. Our theoretical results yield concrete recommendations about the algorithm's implementation, including its initial condition, its step size, and the number of iterations. Importantly, our suggested algorithm can be viewed as a slight variation of the algorithm suggested by Elliott, Müller, and Watson (2015), whose theoretical performance guarantees are unknown. |
| title | Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent |
| topic | Econometrics Statistics Theory |
| url | https://arxiv.org/abs/2511.16925 |