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Improved Cramér–Rao Type Integral Inequalities or Bayesian Cramér–Rao Bounds

  • B. L. S. Prakasa Rao
Research Article

Abstract

New lower bounds on the mean square error for estimators of random parameter are obtained as applications of improved Cauchy–Schwarz inequality due to Walker (Stat Probab Lett 122:86–90, 2017).

Keywords

Bayesian Cramér–Rao bound Cauchy-Schwarz inequality Cramér–Rao type integral inequality Walker’s inequality 

References

  1. Babrovsky BZ, Mayer-Wolf E, Zakai M (1987) Some classes of global Cramér–Rao bounds. Ann Stat 15:1421–1438CrossRefzbMATHGoogle Scholar
  2. Borovkov AA, Sakhanenko AI (1980) On estimates for the average quadratic risk. Probab Math Stat 1:185–195 (In Russian)zbMATHGoogle Scholar
  3. Brown LD, Gajek L (1990) Information inequalities for the Bayes risk. Ann Stat 18:1578–1594MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brown LD, Liu RC (1993) Bounds on the Bayes and minimax risk for signal parameter estimation. IEEE Trans Inf Theory 39:1386–1394CrossRefzbMATHGoogle Scholar
  5. Chazan D, Ziv J, Zakai M (1975) Improved lower bounds on signal parameter estimation. IEEE Trans Inf Theory 21:90–93MathSciNetCrossRefzbMATHGoogle Scholar
  6. Gart John J (1959) An extension of Cramér–Rao inequality. Ann Math Stat 30:367–380CrossRefzbMATHGoogle Scholar
  7. Ghosh M (1993) Cramér-Rao bounds for posterior variances. Stat Probab Lett 17:173–178CrossRefzbMATHGoogle Scholar
  8. Gill RD, Levit Borris Y (1995) Application of the van Trees inequality: a Batesian Cramér–Rao bound. Bernoulli 1:59–79MathSciNetCrossRefGoogle Scholar
  9. Miller R, Chang C (1978) A modified Cramér–Rao bound and its applications. IEEE Trans Inf Theory 24:398–400CrossRefzbMATHGoogle Scholar
  10. Prakasa Rao BLS (1991) On Cramér–Rao type integral inequalities. Calcutta Stat Assoc Bull 40:183–205. Reprinted In: van Trees H, Bell KL (eds) Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IEEE Press, Wiley, New York. pp 900–922Google Scholar
  11. Prakasa Rao BLS (1992) Cramér–Rao type integral inequalities for functions of multidimensional parameter. Sankhya Ser A 54:53–73MathSciNetzbMATHGoogle Scholar
  12. Prakasa Rao BLS (1996) Remarks on Cramér–Rao type integral inequalities for randomly censored data. In: Koul HL, Deshpande JV (ed) Analysis of censored data. IMS Lecture Notes No. 27. Institute of Mathematical Statistics, pp 160–176Google Scholar
  13. Prakasa Rao BLS (2000) Cramér–Rao type integral inequalities in Banach spaces. In: Basu AK, Ghosh JK, Sen PK, Sinha BK (eds) Perspectives in statistical sciences. Oxford University Press, New Delhi, pp 245–260Google Scholar
  14. Prakasa Rao BLS (2001) Cramér–Rao type integral inequalities for general loss functions. TEST 10:105–120MathSciNetCrossRefzbMATHGoogle Scholar
  15. Schutzenberger MP (1957) A generalization of the Frechet–Cramér inequality to the case of Bayes estimation. Bull Am Math Soc 63:142Google Scholar
  16. Shemyakin ML (1987) Rao–Cramér type integral inequalities for estimates of a vector parameter. Theory Probab Appl 32:426–434MathSciNetCrossRefzbMATHGoogle Scholar
  17. Sudheesh K, Dewan I (2016) On generalized moment identity and its application: a unified approach. Statistics 50:1149–1160MathSciNetCrossRefzbMATHGoogle Scholar
  18. Targhetta M (1984) On Bayesian analogues to Bhattacharya’s lower bounds. Arab Gulf J Sci Res 2:583–590MathSciNetzbMATHGoogle Scholar
  19. Targhetta M (1988) On the attainment of a lower bound for the Bayes risk in estimating a parametric function. Statistics 19:233–239MathSciNetCrossRefzbMATHGoogle Scholar
  20. Targhetta M (1990) A note on the mixing problem and the Schutzenberger inequality. Metrika 37:155–161MathSciNetCrossRefzbMATHGoogle Scholar
  21. van Trees Harry L (1968) Detection, estimation and modulation theory part 1. Wiley, New YorkzbMATHGoogle Scholar
  22. van Trees Harry L, Bell Kristine L (2007) Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IEEE Press, Wiley, New YorkCrossRefzbMATHGoogle Scholar
  23. Walker SG (2017) A self-improvement to the Cauchy–Schwarz inequality. Stat Probab Lett 122:86–90MathSciNetCrossRefzbMATHGoogle Scholar
  24. Weinstein E, Weiss A (1985) Lower bounds on the mean square estimation error. Proc IEEE 73:1433–1434CrossRefGoogle Scholar
  25. Weiss A, Weinstein E (1985) A lower bound on the mean square error in random parameter estimation. IEEE Trans Inform Theory 31:680–682MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ziv J, Zakai M (1969) Some lower bounds on signal parameter estimation. IEEE Trans Inform Theory 15:386–391MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Indian Society for Probability and Statistics (ISPS) 2017

Authors and Affiliations

  1. 1.CR Rao Advanced Institute of MathematicsStatistics and Computer ScienceHyderabadIndia

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