Addison Fuller

2022-03-22

Let $X}_{1},\cdots ,{X}_{n$ random sample of $U[\theta -\frac{1}{2};\theta +\frac{1}{2}]$ .Consider $[{X}_{\left(1\right)};\text{}{X}_{\left(n\right)}]$ a confidence interval for $\theta$ . Find their confidence level and show that result is valid for any distribution symmetric around $\theta$

smekkleg5hhp

Beginner2022-03-23Added 8 answers

Step 1

If X's are i.i.d. and their density is symmetric around$\theta$ , then

$\mathsf{P}({X}_{\left(1\right)}\ge \theta )=\mathsf{P}({X}_{\left(n\right)}\le \theta )={\left(\mathsf{P}({X}_{1}\le \theta )\right)}^{n}={\left(\frac{1}{2}\right)}^{n}.$

Thus,

$1-\gamma =\mathsf{P}({X}_{\left(1\right)}\ge \theta )+\mathsf{P}({X}_{\left(n\right)}\le \theta )=2\times {\left(\frac{1}{2}\right)}^{n}.$

If X's are i.i.d. and their density is symmetric around

Thus,

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