In Example 5.5.1, consider a confidence interval for 2 of the form I = [d1 n S2

In Example 5.5.1, consider a confidence interval for σ2 of the form I = [d−1 n S2 n, c−1 n S2 n], where S2 n = 

i(Xi − X¯)

2 and cn < dn are constants.

Subject to the level constraint, choose cn and dn to minimize the length of I.

Argue that the solution has shorter length that the uniformly most accurate one; however, it is biased and so does not uniformly improve the probability of covering false values. [The solution, given in Tate and Klett (1959), satisfies

χ2 n+3(cn) = χ2 n+3(dn) and dn cn χ2 n−1(y)dy = 1 − α, where χ2 n(y) denotes the Chisquared density with n degrees of freedom. Improvements of this interval which incorporate X¯ into their construction are discussed in Cohen (1972) and Shorrock

(1990); also see Goutis and Casella (1991).]

Section 5.6