Chapter 7 Sufficient Statistics. 7.1 Measures of Quality of Estimators.

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Presentation transcript:

Chapter 7 Sufficient Statistics

7.1 Measures of Quality of Estimators

7.2 A Sufficient Statistic for a Parameter

7.3 Properties of a Sufficient Statistic

7.4 Completeness and Uniqueness

7.5 The Exponential Class of Probability Density Functions

7.6 Functions of a Parameter

Remark. We should like to draw the attention of the reader to a rather important fact. This has to do with the adoption of a principle, such as the principle of unbiasedness and minimum variance. A principle is not a theorem; and seldom does a principle yield satisfactory results.

7.7 The Case of Several Parameters

the extension of the notion of joint sufficient statistics for more than two parameters. the concept of a complete family of probability density functions. the exponential class of probability density functions of the continuous type. the exponential class of probability density functions of the discrete type.

7.8 Minimal Sufficient and Ancillary Statistics

minimal sufficient statistics are those that are sufficient for the parameters and are functions of every other set of sufficient statistics for those same parameters. Often, if there are k parameters, we can find k joint sufficient statistics that are minimal.

ancillary statistics : have distributions free of the parameters and seemly contain no information about those parameters. illustration: location-invariant statistic scale-invariant statistic location-and-scale-invariant statistic

7.9 Sufficiency, Completeness, and Independence