Confidence interval

  1. Let's say we know this:

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

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

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

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

    Thank you.
     
  2. jcsd
  3. statdad

    statdad 1,455
    Homework Helper

    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

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

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

    [tex]
    -u < \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) < u
    [/tex]

    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.)
     
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