Let $X \sim Binomial (n, p)$ with both $n$ and $p$ known. Suppose for some non-increasing function $G:[0,1] \rightarrow [0,1]$, and some fixed $c_0 \in [0,1]$, we have that \begin{align} \label{ineq} P(G(X/n) \leq c_0) \leq \alpha \end{align} where $\alpha$ can be chosen as 0.1. Let $\mathcal{G}$ be set that contains all the functions satisfy the above inequality. I want to find the smallest such function $G_0$ such that $G_0 \in \mathcal{G}$ and $G_0(x) \leq G(x), \quad \forall x \in [0,1]$, where $G$ is any other function in $\mathcal{G}$. Can anyone give some hint on looking for such a function.
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1$\begingroup$ It seems this is a discrete problem, because it involves only the values of $G$ at $0, 1/n, 2/n, \ldots, 1$: that might help you work through some instructive examples as well as help you reformulate the question in a simpler manner. Note that in general a smallest such function does not exist. $\endgroup$– whuber ♦Sep 7, 2012 at 18:46
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1$\begingroup$ I am curious about what is the underlying motivation for the question. It looks very abstract but i have a hunch that you have some specific application in mind. $\endgroup$– Michael R. ChernickSep 7, 2012 at 18:52
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$\begingroup$ I agree $G_0$ might not exist in general. I was wondering whether in this particular case the problem is solvable. Thanks for your comment Whuber. $\endgroup$– user13154Sep 7, 2012 at 20:16
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$\begingroup$ Is $c_0$ the same for all functions in the $\mathscr{G}$-class? $\endgroup$– ZenSep 7, 2012 at 21:02
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$\begingroup$ $c_0$ is the fixed for all functions in $\mathcal{G}$. This question is not related to any statistical tests. The background will not be easily explained and I thought it is not really relevant. I just want to see whether it is feasible to find a function $G_0$ with given conditions. Thank you very much. $\endgroup$– user13154Sep 8, 2012 at 3:32
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