Theory of Statistics - George Mason University

Risk
0.005 0.010 0.015 0.020 0.025
3.3 The Decision Theory Approach to Statistical Inference 273
randomized; α = 1 (n + 1)
randomized;
α = 0.05
MLE
0.0 0.2 0.4 0.6 0.8 1.0
Figure 3.1. Risk Functions for Squared-Error Loss in Binomial with n = 10 (Example
3.21).
inference, and the criteria by which the statistical methods to achieve those
objectives are to be evaluated.
A second objective in mathematical statistics is to develop techniques for
finding optimal methods in a particular setting. We have considered some of
these procedures above, and they will be major recurring topics throughout
the rest of this book.
Nonexistence of Optimal Methods
There are many criteria by which to evaluate a statistical method. In a given
setting there may not be a statistical procedure that is optimal with
respect to a given criterion.
The criteria by which to evaluate a statistical method include basic things
about the nature of the statistics used in the method, such as sufficiency, minimality,
and completeness. These properties are independent of the objectives
of the procedure and of the particular statistical method used. They depend
on the assumed distribution family and the nature of the available sample.
• sufficiency. There is always a sufficient statistic.
• minimal sufficiency. There is always a minimal sufficient statistic.
• completeness. There may not be a complete statistic. This depends on
the assumed family of distributions. (See Example 3.4.)
Theory of Statistics c○2000–2013 James E. Gentle
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