Attainments of the Bayesian information bounds

Abstract

Necessary and sufficient conditions for the attainments of the Borovkov–Sakhanenko and the van Trees information bounds are given when the underlying distribution is an exponential family and the conjugate prior, and the estimand is a function of the parameter. Moreover, the attainments under the Jeffreys prior are also considered. Furthermore, some examples are given.

Keywords:

• Bayes risk
• Borovkov–Sakhanenko bound
• van Trees bound
• conjugate prior
• Jeffreys prior

AMS 2010 SUBJECT CLASSIFICATION:

• Primary 62F15
• Secondary 62F10

Acknowledgments

The author thanks the referee for the helpful comments and suggestions.

Notes

1 If $\varphi (\theta )$ and $\beta (\theta )$ are linearly dependent, then the conjugate prior of (6) cannot be expressed uniquely.

2 If a random variable p is distributed according to beta distribution $\text{Be}(c_1,c_2-c_1),$ the logit-transformed random variable $\theta =\hspace{0.17em}\text{log}\hspace{0.17em}\{p/(1-p)\}$ is distributed according to (34), and this distribution is a version of beta prime distribution (Johnson, Kotz, and Balakrishnan 1995).