1. Confidence Intervals
Chapter 11

2. What’s a 95% confidence interval?
Roughly, it is an interval that we are 95% confident contains the parameter of interest.
More carefully, it is the outcome of a random interval, where the random contains the parameter of interest with probability 95%.
Note that it is the random interval the contains the parameter of interest with probability 95%. The outcome of the random interval either does or does not contain it.
If we were to repeat the experiment many times independently, creating a 95% confidence interval each time, about 95% of these confidence intervals would contain the true value of the parameter.

3. Outline The basic steps to create any CI One sided CIs
Equal tailed CIs
Pivotal quantity
Location and scale parameters
CI for a function of a parameter
Approximate CIs

4. The basic steps to create any CI
Step 1: Write down a probability statement involving the parameter of interest.
Step 2: Isolate the parameter in the probability statement.
Step 3: Write down the random interval.
This is an interval estimator.
Step 4: Write down the outcome of the random interval.
This is an interval estimate, but we’ll just call in a confidence interval like everyone else.

5. Example 11. 1. 1 Data collected from EXP(μ) population
Example Data collected from EXP(μ) population. Outcome of sample mean is 37. Find a 90% equal tailed CI for μ.
Step 1: Write down a probability statement involving the parameter of interest.
Step 2: Isolate the parameter in the probability statement.
Step 3: Write down the random interval.
This is an interval estimator.
Step 4: Write down the outcome of the random interval.
This is an interval estimate, but we’ll just call in a confidence interval like everyone else.

6. A few variations: Repeat to find a 90% upper bound on μ.
Why might a person want this?
Repeat again to find a 90% lower bound on μ.
What about a funky, non-equal tail, two-sided CI?
Is there a ‘best’ confidence interval? What would that mean?

7. Pivotal quantity
A pivotal quantity is function of the random sample and the parameter of interest whose distribution does not depend on unknown stuff, e.g. the parameter of interest.
The first step in deriving a CI is to write down a probability statement. It is super convenient if this involves a pivotal quantity.
Example: Pivotal quantity based on a random sample from a normal population.

8. Finding pivotal quantities
A location parameter is one that shifts the ‘location’ of the distribution, i.e. f(x, θ) = f0(x-θ).
Example
Consider the density function for EXP(1, η): f(x, η) = e-(x-η) 1{x-η>0}.
Theorem If
Xi ~ i.i.d. f(x, θ)
θ is a location parameter
MLE is a maximum likelihood estimator for θ
Then MLE – θ is a pivotal quantity for θ.

9. Examples of finding pivotal quantities
Example 1: Find a CI for μ based on a random sample from N(μ, 1).
Example 2: Find a CI for η based on a random sample from EXP(1, η).
Example 3: Find a CI for σ2 based on a random sample from N(0, σ2).
Can it be done?

10. Finding pivotal quantities
A scale parameter is one that satisfies, i.e. f(x, θ) = (1/θ)f0(x/θ).
Theorem If
Xi ~ i.i.d. f(x, θ)
θ is a scale parameter
MLE is a maximum likelihood estimator for θ
Then MLE/θ is a pivotal quantity for θ.

11. Examples of finding pivotal quantities
Example 3: Find a CI for σ2 based on a random sample from N(0, σ2).
Example 4: Find a CI for μ based on a random sample from N(μ, σ2).
Can it be done?

12. Finding pivotal quantities
Theorem If
Xi ~ i.i.d. f(x, θ1, θ2)
θ1 is a location parameter
θ2 is a scale parameter
MLE1 and MLE2 are maximum likelihood estimators for θ1 and θ2, respectively.
Then
(MLE1 - θ1)/MLE2 is a pivotal quantity for θ1; AND
(MLE2)/θ2 is a pivotal quantity for θ2.
Why is (MLE1 - θ1)/ θ2 not pivotal for θ1?

13. Examples of finding pivotal quantities
Example 4: Find a CI for μ based on a random sample from N(μ, σ2).
Example 5: Find a CI for σ2 based on a random sample from N(μ, σ2).

14. Confidence interval for functions of parameters
(3, 4) is a 95% CI for λ
Can I find a 95% CI for eλ without doing hard work?
What about e-λ?
(-5, 10) is a 99% CI for μ
Can I find a CI for μ2 without doing hard work?

15. Approximate confidence intervals
Example
Xi ~ i.i.d. BER(p)
Sample size = 50
Find an equal tailed 90% CI for p
Is there a pivotal quantity?
What about an asymptotically pivotal quantity?

16. Approximate confidence intervals
Example
Xi ~ i.i.d. POI(λ)
Sample size = 50
Find an equal tailed 90% CI for λ
Use an asymptotically pivotal quantity.

17. IQ tested on 30 subjects on NZT and 40 different subjects on placebo
Example 1:
Xi ~ i.i.d. N(μ1, 1); N = 30
Yk ~ i.i.d. N(μ2, 2); N = 40
Xi is independent of Yk for all i, k.
Find a pivotal quantity for μ1-μ2.
Example 2:
Xi ~ i.i.d. N(μ1, σ12); N = 30
Yk ~ i.i.d. N(μ2, σ22); N = 40
Find a pivotal quantity for σ12/σ22.

18. IQ tested on 20 subjects on NZT and the same 20 subjects on placebo
Example 3:
Xi ~ i.i.d. N(μ1, 1); N = 20
Yi ~ i.i.d. N(μ2, 1); N = 20
Cov(Xi, Yi) = 1/2
Xi and Yi are measures of IQ before and after NZT for subject i.
Find a pivotal quantity for μ1-μ2.
Example 4:
Xi ~ i.i.d. N(μ, σ2); N = 20
Find a pivotal quantity for μ.

19. Hard example (11.4.1) – no pivotal quantity
Xi ~ i.i.d. f(x; θ) = (1/θ2)e-(x-θ)/θ21{x>θ}
N = 100
In the absence of a pivotal quantity, consider a sufficient statistic.
The sample mean and first order statistic are jointly sufficient.
Let’s try using X1:100.
Find a 90% CI for θ.
See lecture notes for details.

20. Interpretation
This comes from the main BATE paper: