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Stats Online Refresher Tutorial

Discussion in 'FAC & Stats Pack' started by Sarahlouise_23, Jan 1, 2014.

  1. Sarahlouise_23

    Sarahlouise_23 Member

    I'm going through the Stats Online Refresher tutorial at the moment and I've come to the estimation bit. Example question 13 says use the estimate for lambda calculated using the method of moments and the Cramer-Rao asymptotic result to obtain a 95% one sided confidence interval which provides an upper limit for lambda.

    The formula given is method of moments estimate + 1.6449 * square root of Cramer Rao estimate. What I don't understand is why the square root of the Cramer Rao estimate is not divided by square root of n. Can anyone explain why this isn't in the formula?

    Thanks.
     
    Sarahlouise_23, Jan 1, 2014
    #1
  2. John Lee

    John Lee ActEd Tutor Staff Member

    I think you're confusing this with a typical confidence interval based on the sample mean.

    The asymptotic distribution (on page 23 of the Tables) is:

    \(\hat{\theta} \backsim N(\theta,CRLB)\)

    Standardising this gives:

    \(\frac{\hat{\theta}-\theta}{\sqrt{CRLB}} \backsim N(0,1)\)

    Hence a 2-sided 95% CI would be:

    \(\theta = \hat{\theta} ± 1.96\sqrt{CRLB}\)


    For a normal distribution, we have:

    \(\hat{\mu} = \bar{x}\)
    \(CRLB = \frac{\sigma^2}{n}\)

    Hence, we would get:

    \(\mu = \hat{\mu} ± 1.96\sqrt{CRLB}\)


    \(\mu = \bar{x} ± 1.96 \frac{\sigma}{\sqrt{n}}\)
     
    Last edited: Jan 2, 2014
    John Lee, Jan 2, 2014
    #2
  3. Sarahlouise_23

    Sarahlouise_23 Member

    That's exactly what I was doing. It's been a while since I've looked at confidence intervals. Thanks John.
     
    Sarahlouise_23, Jan 2, 2014
    #3

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