Creating a Confidence Interval for θ

[twoflower]

Let's say we know this:

<br /> \sqrt{n}\left(\widehat{\theta} - \theta\right) \sim \mathcal{N}\left(0, \frac{1}{F(\theta)}\right)<br />

How do we get from this information to this expression of confidence interval for \theta?

<br /> \left( \widehat{\theta} \pm u_{1-\frac{\alpha}{2}}\frac{1}{\sqrt{nF\left(\widehat{\theta}\right)}}\right)<br />

Where u_{1-\frac{\alpha}{2}} is appropriate quantil of standard normal distribution.

Thank you.

 

[statdad]

If a is the value from Z (standard normal) with area {\alpha}/2 to its right, you know the value of

<br /> \Pr\left(-u &lt; \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) &lt; u) <br />

because of your stated approximate normality result. That means the event

<br /> -u &lt; \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) &lt; u<br />

has a known probability of occurring. What can you do with this inequality? (Try some work and include it with your next question if you are unsure.)