Physics Forums

Physics Forums (
-   Calculus & Beyond Homework (
-   -   Confidence interval (

twoflower Sep11-08 03:02 PM

Confidence interval
Let's say we know this:

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

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

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

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

Thank you.

statdad Sep11-08 06:02 PM

Re: Confidence interval
If [tex] a [/tex] is the value from [tex] Z [/tex] (standard normal) with area [tex] {\alpha}/2[/tex] to its right, you know the value of

\Pr\left(-u < \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) < u)

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

-u < \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) < u

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.)

All times are GMT -5. The time now is 12:20 PM.

Powered by vBulletin Copyright ©2000 - 2014, Jelsoft Enterprises Ltd.
© 2014 Physics Forums