Abstract
As mentioned in the previous chapter, a minimax robust test can be designed based on a suitable choice of a distance between probability measures, which is most likely decided for, depending on the application. Instead of searching for a distance and performing a tedious design procedure, it is probably most convenient to choose a simple parameter, which accounts for the distance. In this way, the designer has the flexibility to choose both the degree of robustness as well as the type of the distance with only setting a few parameters.
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Notes
- 1.
Notice that \(\alpha \)-divergence is preferred against the Rényi’s \(\alpha \)-divergence because Rényi’s \(\alpha \)-divergence is convex only for \(\alpha \in [0,1]\) [CiA10, p. 1540].
- 2.
In general \(\mathrm {arg}\sup \) may not always be achieved since \(\mathscr {G}_0\) and \(\mathscr {G}_1\) are non-compact sets in the topologies induced by the \(\alpha \)-divergence distance. In this book, existence of \(\hat{g}_0\) and \(\hat{g}_1\) is due to the KKT solution of the minimax optimization problem, which is introduced in Sect. 4.3.3.
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Gül, G. (2017). Robust Hypothesis Testing with Multiple Distances. In: Robust and Distributed Hypothesis Testing. Lecture Notes in Electrical Engineering, vol 414. Springer, Cham. https://doi.org/10.1007/978-3-319-49286-5_4
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