Let X1,..., Xn be a random sample from a n(μ, Ï2) population, where both μ and Ï2 are unknown. Each of the following methods of

Let X1,..., Xn be a random sample from a n(μ, σ2) population, where both μ and σ2 are unknown. Each of the following methods of finding confidence intervals for σ2 results in intervals of the form

but in each case a and b will satisfy different constraints. The intervals given in this exercise are derived by Tate and Klett (1959), who also tabulate some cutoff points.
Define fp(t) to be the pdf of a X2p random variable with p degrees of freedom. In order to have a 1 - α confidence interval, a and b must satisfy

but additional constraints are required to define a and b uniquely. Verify that each of the following constraints can be derived as stated.
a. The likelihood ratio interval. The 1 - α confidence interval obtained by inverting the LRT of H0: σ = σ0 versus H1: σ ‰  σ0 is of the above form, where a and b also satisfy fn+2(a) = fn+2(b).
b. The minimum length interval: For intervals of the above form, the 1 - α confidence interval obtained by minimizing the interval length constrains a and b to satisfy fn+3(a) = fn+3(b).
c. The shortest unbiased interval: For intervals of the above form, the 1 - α confidence interval obtained by minimizing the probability of false coverage among all unbiased intervals constrains a and b to satisfy fn+1(a) = fn+1(b). This interval can also be derived by minimizing the ratio of the endpoints.
d. The equal-tail interval: For intervals of the above form, the 1 - α confidence interval obtained by requiring that the probability above and below the interval be equal constrains a and b to satisfy

(This interval, although very common, is clearly non-optimal no matter what length criterion is used.)

2. ( 1 n-1(t) dt1-a, Jo fn-1(t) dt =- In-1 (t) dt =- 2 0