Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 22 Issue 4

R eferences R éférences R eferencias © 2022 Global Journals 1 Year 2022 40 Global Journal of Science Frontier Research Volume XXII Issue ersion I V IV ( F ) On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial Distribution g ( θ ) is a monotonically increasing function of θ over the set S = (0 , 1) − { 0 . 5 } .Hence, γ 1 B ( X ) as above,presents a valid ’frequentist report’ as mentioned by Berger(1985). The results are summerized in the following: THEOREM.Let ( δ 1 B , γ 1 B ) be Bayes estimators of the unknown parameter θ of the binomial distribution under the loss function L ( θ, δ, γ ) = 1 { θ (1 − θ ) } ( θ − δ ) 2 γ − 1 2 + γ 1 2 and beta prior density with known parameters α and β .Then,the frequentist risk E θ L ( θ, δ 1 B , γ 1 B ) is finite for all values of α and β provided 0 < θ < 1.For α = β = 0, γ 1 B ( X ) is not conservatively biased. The estimator γ 1 B ( X ) is conservatively biased for α = β = 1 and for α = β > 1 satisfying α ≤ 1 + 2 θ (1 − θ ) (2 θ − 1) 2 , θ 6 = 0 . 5.However, if α = β > 1 , θ = 0 . 5, γ 1 B ( X ) is also conservatively biased. When, α = β > 1 , θ 6 = 0 . 5, E θ { γ 1 B ( X ) } ≥ R ( θ, δ 1 B ) (2.21) which holds if α ≤ 1 + g ( θ ) (2.22) .Where, g ( θ ) = 2 θ (1 − θ ) (2 θ − 1) 2 (2.23) When, α = β > 1 , θ = 0 . 5, E θ { γ 1 B ( X ) } ≥ R ( θ, δ 1 B ) (2.20) N otes 1. BERGER, J.(1985). The frequentist viewpoint and conditioning. In Proceedings of the Berkley Conference in Honor of Jerry Neyman and Jack Keifer, Ed. L. Lecam and R.Olshen, pp.15-44.Belmont, Cailf,: Wadsworth. 2. GUOBING FAN, (2016). Estimation of the Loss and Risk Functions of parameter of Maxwell's distribution. Science Journal of Applied Mathematics and Statistics. Vol.4, No.4, 2016, pp.129-133. doi: 10.11648/j.sjams.20160404.12 3. KEIFER, J.(1977). Conditional confidence statements and con fidence estimators. J. Am. Statist. Assoc. 72,789-827. 4. RANDHIR SINGH(2021). On Bayesian Estimation of Loss and Risk Functions. Science Journal of Applied Mathematics and Statistics Vol.9, No.3, 2021, pp.73- 77. doi: 10.11648/j.sjams.20210903.11 5. RUKHIN, A.L.(1988).Estimating the loss of estimators of a binomial parameter. Biometrika,75,1,pp.153-5.