Section 6.5 EXERCISE 6.25 We have seen a number of theorems concer exponential families. Theorem 5.2.11 gave t ciency is characterized in Theorem 6.2.10 ar the family is curved, the open set condition cases, is the sufficient statistic of Theorem 6 6.2.13 to T(x) of Theorem 6.2.10, establish (a) The statistic (Xi, EX}) is sufficient family. (b) The statistic X? is minimal sufficien (c) The statistic (Xi, X) is minimal (d) The statistic (Xi, EX?) is minimal

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Section 6.5
EXERCISES
303
6.25 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 suffi-
ciency 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 Theorenm 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, EX?) is sufficient, but not minimal sufficient, in the n(u, µ)
family.
(b) The statistic X? is minimal sufficient in the n(u, µ) family.
(c) The statistic (Xi, X?) is minimal sufficient in the n(u, u²) family.
(d) The statistic (Xi, X?) is minimal sufficient in the n(u, o²) family.
Transcribed Image Text:Section 6.5 EXERCISES 303 6.25 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 suffi- ciency 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 Theorenm 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, EX?) is sufficient, but not minimal sufficient, in the n(u, µ) family. (b) The statistic X? is minimal sufficient in the n(u, µ) family. (c) The statistic (Xi, X?) is minimal sufficient in the n(u, u²) family. (d) The statistic (Xi, X?) is minimal sufficient in the n(u, o²) family.
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