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

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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).

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References

  • Babrovsky BZ, Mayer-Wolf E, Zakai M (1987) Some classes of global Cramér–Rao bounds. Ann Stat 15:1421–1438

    Article  MATH  Google Scholar 

  • Borovkov AA, Sakhanenko AI (1980) On estimates for the average quadratic risk. Probab Math Stat 1:185–195 (In Russian)

    MATH  Google Scholar 

  • Brown LD, Gajek L (1990) Information inequalities for the Bayes risk. Ann Stat 18:1578–1594

    Article  MathSciNet  MATH  Google Scholar 

  • Brown LD, Liu RC (1993) Bounds on the Bayes and minimax risk for signal parameter estimation. IEEE Trans Inf Theory 39:1386–1394

    Article  MATH  Google Scholar 

  • Chazan D, Ziv J, Zakai M (1975) Improved lower bounds on signal parameter estimation. IEEE Trans Inf Theory 21:90–93

    Article  MathSciNet  MATH  Google Scholar 

  • Gart John J (1959) An extension of Cramér–Rao inequality. Ann Math Stat 30:367–380

    Article  MATH  Google Scholar 

  • Ghosh M (1993) Cramér-Rao bounds for posterior variances. Stat Probab Lett 17:173–178

    Article  MATH  Google Scholar 

  • Gill RD, Levit Borris Y (1995) Application of the van Trees inequality: a Batesian Cramér–Rao bound. Bernoulli 1:59–79

    Article  MathSciNet  Google Scholar 

  • Miller R, Chang C (1978) A modified Cramér–Rao bound and its applications. IEEE Trans Inf Theory 24:398–400

    Article  MATH  Google Scholar 

  • 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 900–922

  • Prakasa Rao BLS (1992) Cramér–Rao type integral inequalities for functions of multidimensional parameter. Sankhya Ser A 54:53–73

    MathSciNet  MATH  Google Scholar 

  • 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 160–176

  • 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 245–260

    Google Scholar 

  • Prakasa Rao BLS (2001) Cramér–Rao type integral inequalities for general loss functions. TEST 10:105–120

    Article  MathSciNet  MATH  Google Scholar 

  • Schutzenberger MP (1957) A generalization of the Frechet–Cramér inequality to the case of Bayes estimation. Bull Am Math Soc 63:142

    Google Scholar 

  • Shemyakin ML (1987) Rao–Cramér type integral inequalities for estimates of a vector parameter. Theory Probab Appl 32:426–434

    Article  MathSciNet  MATH  Google Scholar 

  • Sudheesh K, Dewan I (2016) On generalized moment identity and its application: a unified approach. Statistics 50:1149–1160

    Article  MathSciNet  MATH  Google Scholar 

  • Targhetta M (1984) On Bayesian analogues to Bhattacharya’s lower bounds. Arab Gulf J Sci Res 2:583–590

    MathSciNet  MATH  Google Scholar 

  • Targhetta M (1988) On the attainment of a lower bound for the Bayes risk in estimating a parametric function. Statistics 19:233–239

    Article  MathSciNet  MATH  Google Scholar 

  • Targhetta M (1990) A note on the mixing problem and the Schutzenberger inequality. Metrika 37:155–161

    Article  MathSciNet  MATH  Google Scholar 

  • van Trees Harry L (1968) Detection, estimation and modulation theory part 1. Wiley, New York

    MATH  Google Scholar 

  • van Trees Harry L, Bell Kristine L (2007) Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IEEE Press, Wiley, New York

    Book  MATH  Google Scholar 

  • Walker SG (2017) A self-improvement to the Cauchy–Schwarz inequality. Stat Probab Lett 122:86–90

    Article  MathSciNet  MATH  Google Scholar 

  • Weinstein E, Weiss A (1985) Lower bounds on the mean square estimation error. Proc IEEE 73:1433–1434

    Article  Google Scholar 

  • Weiss A, Weinstein E (1985) A lower bound on the mean square error in random parameter estimation. IEEE Trans Inform Theory 31:680–682

    Article  MathSciNet  MATH  Google Scholar 

  • Ziv J, Zakai M (1969) Some lower bounds on signal parameter estimation. IEEE Trans Inform Theory 15:386–391

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to B. L. S. Prakasa Rao.

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Prakasa Rao, B.L.S. Improved Cramér–Rao Type Integral Inequalities or Bayesian Cramér–Rao Bounds. J Indian Soc Probab Stat 19, 1–7 (2018). https://doi.org/10.1007/s41096-017-0030-z

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  • DOI: https://doi.org/10.1007/s41096-017-0030-z

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