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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, ) family. (b) The statistic EX is minimal sufficient in the n(u, µ) family. (c) The statistic (EXi,X) is minimal sufficient in the n(u, µ²) family. (d) The statistic (Xi, EX?) is minimal sufficient in the n(u, o?) family.

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, ) family. (b) The statistic EX is minimal sufficient in the n(u, µ) family. (c) The statistic (EXi,X) is minimal sufficient in the n(u, µ²) family. (d) The statistic (Xi, EX?) is minimal sufficient in the n(u, o?) family.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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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 Theoren 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 (EXi, X?) is sufficient, but not minimal sufficient, in the n(u, H) family. (b) The statistic X is minimal sufficient in the n(u, µ) family. (c) The statistic (EXi, X) is minimal sufficient in the n(u, u²) family. (d) The statistic (Xi,X?) is minimal sufficient in the n(u, o?) family.
Transcribed Image Text: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 Theoren 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 (EXi, X?) is sufficient, but not minimal sufficient, in the n(u, H) family. (b) The statistic X is minimal sufficient in the n(u, µ) family. (c) The statistic (EXi, X) is minimal sufficient in the n(u, u²) family. (d) The statistic (Xi,X?) is minimal sufficient in the n(u, o?) family.
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