Abstract
In this paper, we address an open problem raised by Levy (2009) regarding the design of a binary minimax test without the symmetry assumption on the nominal conditional probability densities of observations. In the binary minimax test, the nominal likelihood ratio is a monotonically increasing function and the probability densities of the observations are located in neighborhoods characterized by placing a bound on the relative entropy between the actual and nominal densities. The general minimax testing problem at hand is an infinite-dimensional optimization problem, which is quite difficult to solve. In this paper, we prove that the complicated minimax testing problem can be substantially reduced to solve a nonlinear system of two equations having only two unknown variables, which provides an efficient numerical solution.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 61473197, 61671411 and 61273074), Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT 16R53), and Program for Thousand Talents (Grant Nos. 2082204194120 and 0082204151008).
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Song, E., Shi, Q., Zhu, Y. et al. Robust hypothesis testing for asymmetric nominal densities under a relative entropy tolerance. Sci. China Math. 61, 1851–1880 (2018). https://doi.org/10.1007/s11425-016-9021-6
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DOI: https://doi.org/10.1007/s11425-016-9021-6
Keywords
- Kullback-Leibler divergence
- robust hypothesis testing
- min-max problem
- least-favorable densities
- saddle point