Question

G2. This is a continuation of Exercise 8.46. (d) Use the pivotal quantity 2ye to derive a 90% upper confidence bound for θ (e) Use the pivotal quantity Y/θ as in Example 8.4 to derive a 90% upper confidence bound for θ (f) Which confidence interval is better (or are they equivalent)? The answer will imply in this case which pivotal quantity is more suitable (or equally suitable).

Reference:

Answer 1

Solution to example 8.46

Let U = 2Y/θ and let m_Y(t) denote the mgf for the exponential distributio with mean θ.

A) m_U(t) = E(e^tU) = E(e^t2Y/θ) = m_Y(2t/θ) = (1- 2t)^-1

This is the mgf for the chi-square distribution with one degree of freedom.

U has this distribution since mgfs are unique and since the distribution does not depend on θ, U is a piotal quantity.

B) The chi-square table with 2 degrees of freedom tells us that P(.102587 < 2Y/θ < 5.99147) = .90

SO (2Y/5.99147, 2Y/.102587) respresents a 90% confidence interval for θ.

Complimentary answer:

C) My guess is that they are the same but not sure.

Please give thumbs up to my answer...