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I need some help, I need to make a hypothesis testing for a Uniform distribution with both parameters $(a,b)$ unknown for $\begin{align} H_{0}:b-a>2 \end{align}$

So. I know that I need a minimal sufficient statistic which is going to be a vector $(X_{(1)},X_{(n)})$, where $X_{(1)}=\min\{X1,...,X_{n} \}$ and $X_{(n)}=\max\{X1,...,X_{n} \}$,

Is my reject region going to be $RR_{\alpha}=\{x_{1},...x_{n}:X_{(n)}-X_{(1)}<C_{\alpha} \}$ for a $0<C_{\alpha}<1$?

How can I make the hypothesis testing on which the probability error type 1 and 2 don´t exceed $\alpha$ and $\beta$, where $\mathbb{P}(E.T.1)=\mathbb{P}$(Reject $H_{0}$|$H_{0}$ is true)=$\mathbb{P}(X_{(n)}-X_{(1)}\leq C_{\alpha}|b-a>2)$? Thanks

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Use your intuition. What matters for the hypothesis $$H_0 : b - a > 2 \quad \text{vs.} \quad H_1 : b - a \le 2$$ is the range of the sample; if it is too large, you obviously cannot reject $H_0$. The smaller the range of observed values, the more likely the alternative hypothesis. For instance, if the sample were $(2, 1, 5, 3, 2, 4)$, you would immediately fail to reject $H_0$ since $5 - 1 = 4 > 2$. But if the sample were $(2.1, 2.2, 2.3, 2.7, 1.9, 2.5)$, then the range is only $2.7 - 1.9 = 0.8$. Is that small enough to provide enough evidence that $b - a \le 2$? That's where you need to derive a rejection region that depends on $\alpha$.

So, your thought about using the first and last order statistics is on the right track but in fact, we only care about the statistic $Y = X_{(n)} - X_{(1)}$, not the specific values of these. And your rejection region clearly should be in the direction of the event $Y \le C_\alpha$, per our discussion above. So what is the sampling distribution of $Y$ under the (modified) null hypothesis $H_0 : b - a = 2$, and how do we find $C_\alpha$ such that $\Pr[Y \le C_\alpha \mid H_0] = \alpha$? Note we use a modified $H_0$ because under all other cases where $b - a > 2$, the Type I error is strictly smaller than $\alpha$ for the same $C_\alpha$ criterion.

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  • $\begingroup$ You're totally right, if my sample is too small I would reject $H_{0}$ if my samples are too small, I found in a book that I could use sample range (I'm not sure about this name, I found it in Spanish as "Rango Muestral"). Also I think that I have a problem finding the CDF of the random variable $Y$ in this case, but your thinking is really useful to me, thanks. $\endgroup$ Oct 26 '20 at 23:01

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