Efficiency and Cramér-Rao type lnequalities for convex loss functions

Andrzej Kozek

doi:10.1016/0047-259X(77)90034-3

Efficiency and Cramér-Rao type lnequalities for convex loss functions

Andrzej Kozek^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts ("Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,") and of Blyth (Ann. Statist.2, 464-473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.

Original language	English
Pages (from-to)	89-106
Number of pages	18
Journal	Journal of Multivariate Analysis
Volume	7
Issue number	1
DOIs	https://doi.org/10.1016/0047-259X(77)90034-3
Publication status	Published - 1977
Externally published	Yes

Keywords

Conjugate convex function
convex loss function
Cramér-Rao type inequality
efficiency
lower bound for risk
Orlicz spaces
subdifferential
unbiased estimates

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10.1016/0047-259X(77)90034-3

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title = "Efficiency and Cram{\'e}r-Rao type lnequalities for convex loss functions",

abstract = "A general method for obtaining inequalities of Cram{\'e}r-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cram{\'e}r-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts ({"}Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,{"}) and of Blyth (Ann. Statist.2, 464-473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.",

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author = "Andrzej Kozek",

year = "1977",

doi = "10.1016/0047-259X(77)90034-3",

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T1 - Efficiency and Cramér-Rao type lnequalities for convex loss functions

AU - Kozek, Andrzej

PY - 1977

Y1 - 1977

N2 - A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts ("Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,") and of Blyth (Ann. Statist.2, 464-473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.

AB - A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts ("Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,") and of Blyth (Ann. Statist.2, 464-473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.

KW - Conjugate convex function

KW - convex loss function

KW - Cramér-Rao type inequality

KW - efficiency

KW - lower bound for risk

KW - Orlicz spaces

KW - subdifferential

KW - unbiased estimates

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U2 - 10.1016/0047-259X(77)90034-3

DO - 10.1016/0047-259X(77)90034-3

M3 - Article

AN - SCOPUS:49349135497

SN - 0047-259X

VL - 7

SP - 89

EP - 106

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

IS - 1

ER -

Efficiency and Cramér-Rao type lnequalities for convex loss functions

Abstract

Keywords

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