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In your paper https://arxiv.org/pdf/1710.03870.pdf, at definition 4.0.3 page 12, I think there a little error.

$$Z_t(\hat{Y}_0) = \begin{cases} 1, & A:=\bigcup_{k=1}^n\{R_t^{k,a}\ge 1\} \cap \bigcup_{k=1}^n\{R_t^{k,b}\ge 1\} \not= \emptyset,\\ -1, & B:=\bigcup_{k=1}^n\{R_t^{k,a} < 1\} \cap \bigcup_{k=1}^n\{R_t^{k,b} < 1\} \not= \emptyset,\\ 0, & (A \bigcup B)^c \not= \emptyset,\\ \end{cases}$$

$Z_t$ is not well define because it is not a function. It can take more than 2 values for a unique $Ŷ_0$. $A$ and $B$ might be non-empty at the same time. Am I wrong? If not, how to fix that error?

Jeremie
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    It's not my paper. – Arnaud Mortier Aug 02 '18 at 00:49
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    Jokes apart, you should not expect people to read a whole paper to understand what the $R$ variables are, how the RHS actually depends on $\hat{Y}_0$, whatever $\hat{Y}_0$ means, with the risk that in the end, it turns out that you misread a comma. – Arnaud Mortier Aug 02 '18 at 00:53

1 Answers1

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The definition of $\,A,B\,$ implies that $\,A \cap B = \emptyset\,$ since, for example, $\,R_t^{k,a}\ge 1\,$ and $\,R_t^{k,a}\lt 1\,$ are mutually exclusive.

Therefore the definition of $\,Z_t\,$ is of the following form, which is well defined because the three cases are mutually disjoint, and their union covers the applicable universe:

$$Z_t(\hat{Y}_0) = \begin{cases} 1, & \hat{Y}_0 \in A \\ -1, & \hat{Y}_0 \in B \\ 0, & \hat{Y}_0 \not\in A \bigcup B \\ \end{cases}$$

dxiv
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