# Hypothesis testing for a Uniform distribution with both parameters unknown

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 $$RR_{\alpha}=\{x_{1},...x_{n}:X_{(n)}-X_{(1)} for a $$0?

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

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.

• 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. Oct 26, 2020 at 23:01