We have seen a number of theorems concerning sufficiency and related concepts for exponential families. Theorem 5.2.11 gave the distribution of a statistic whose sufficiency is characterized in Theorem 6.2.10 and completeness in Theorem 6.2.25. But if the family is curved, the open set condition of Theorem 6.2.25 is not satisfied. In such cases, is the sufficient statistic of Theorem 6.2.10 also minimal? By applying Theorem 6.2.13 to T(x) of Theorem 6.2.10, establish the following:
a. The statistic (∑Xi, ∑Xi2) is sufficient, but not minimal sufficient, in the n(μ, μ) family.
b. The statistic ∑Xi2? is minimal sufficient in the n(μ, μ) family.
c. The statistic (∑Xi, ∑Xi2) is minimal sufficient in the n(μ, μ2) family.
d. The statistic (∑Xi, ∑Xi2) is minimal sufficient in the n(μ, σ2) family.