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Suppose that $T_1$ is $100\gamma$ percent lower confidence limit for $\tau(\theta)$ and $T_2$ is $100\gamma$ percent uper confidence limit for $\tau(\theta)$. Further assume that $P_\theta(T_1<T_2)=1$. Find a $100(2\gamma-1)$ percent confidence interval for $\tau(\theta)$.(Assume $\gamma>\frac{1}{2})$

Using the same I idead from here Confidence Interval, uniform distribution. $$P(T_1\leq\tau(\theta)\leq T_2)=P(T_2\geq\tau(\theta))-P(T_1\geq\tau(\theta))$$ $$=P(T_2\geq\tau(\theta))-[1-P(T_1\leq\tau(\theta))]=\gamma-1+\gamma=2\gamma-1$$

Because by definition from Mood, in one sided interval $P(T_2\geq\tau(\theta))=\gamma$ and $P(T_1\leq\tau(\theta))=\gamma$.

But $P(T_1\leq\tau(\theta)\leq T_2)$ don't need to be $P(T_1\leq\tau(\theta)\leq T_2)=\gamma$?

And as I build the interval with such information?

Maybe I'm wrong understanding, but $(T_1,T_2)$ it would not be a confidence interval with confidence level $2\gamma-1$, assume that $\gamma>\frac{1}{2}$? Or I need to find $T_1,T_2$?

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    Also, do not model your reasoning after what is said in that link. They consider order statistics (max and min of the sample), which is completely different than what you have here. – StijnDeVuyst May 17 '15 at 21:55
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    If $[T_1,\infty]$ and $[-\infty,T_2]$ are both $100\gamma$%-CI for $\tau(\theta)$ *obtained from the same sample*, then it is indeed the case that $P(\tau(\theta) \in [T_1,T_2]) = 2\gamma-1$. If $T_1$ and $T_2$ are obtained from different samples, then the question does not provide sufficient information to calculate the probability $P(\tau(\theta) \in [T_1,T_2])$. – StijnDeVuyst May 18 '15 at 22:53

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