8. Bootstrap confidence intervals

Question 1

How do we define “standard confidence interval”

Answer 1

The standard intervals are symmetric around theta, and not very accurate.

Question 2

Why is it wrong to say the true interval is with a probability of 95%?

Answer 2

We say confident because: in the example of height, we are working with real numbers, not random variables, were we could say probability.

Question 3

What is the Neymans construction and what do we use it for?

Answer 3

In the lecture we saw an example of calculating the confidence interval for a correlation between two variables.

Question 4

WHat is transformation invariance?

Answer 4

It is when even though we scale our variables, we want the confidence interval to stay the same.

Question 5

Explain the procentile method

Answer 5

You can use it any statistics, in our example we would use it for for example the studentized mean.
1. Generate bootstrap samples of a selected statistics
2 ….

Question 6

What propoerties does the procentile method hold?

Answer 6

Does not require knowing the transformation to normality φˆ = m(
ˆθ), it only assumes its existence.
* If the transformation exists, correct intervals in the Fisherian sense
* Standard intervals work fine once in the right frame—but they are not
transformation invariant.
* Bootstrap sample sizes B ≈ 2000 required
* There are even better methods than percentile methods: BC and BCa

Question 7

Explain the BC and BCa methods

Answer 7

• Ideally, the percentile method is applied for the normal bootstrapped statistic which is an unbiassed estimator with constant variance generated by the
  transformation φˆ = m(
  ˆθ).
• • If we do not have a direct access to unbiassed, constantly varying
    bootstrapped statistics, we may however remove the bias and take into
    account the changing variance.