Transcript Chapter 7 Sufficient Statistics

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