Confidence intervals

Question 1

What is a confidence interval?

Answer 1

A (point-)estimator θ̂ gives us an estimate of an unknown parameter value, but does not provide us with any direct information about the uncertainty in the estimate.

A confidence interval is an interval [ Θ̂𝐿 ,Θ̂𝑈 ] which with high probability will come to contain the value of the unknown parameter θ.

• Remember that e.g. a 95% conf. int. [ Θ̂𝐿 ,Θ̂𝑈 ] is made such that 𝑃( Θ̂𝐿 < θ < Θ̂𝑈)=0.95. This means that if we make many such intervals, 95% of them will contain the true parameter value θ.
• If we for instance get the observed interval [3.2, 5.6] we do not know whether θ is contained in the interval or not, but based on the way we have made the interval we have high confidence (NB! NOT 95% probability!) that the parameter is contained in the interval.

Question 2

How de we construct a confidence interval?

Answer 2

Question 3

The distribution of some functions of estimators
The estimator 𝝁̂=𝑿

Answer 3

Question 5

The distribution of some functions of estimators
The estimator 𝝈̂=𝑺𝟐

Answer 5

Question 6

The distribution of some functions of estimators
The estimator 𝜷̂ in the exponential distribution

Answer 6

Question 7

Wald interval / confidence interval based on MLE

Answer 7