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\title{On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial Distribution By Randhir Singh}
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\begin{document}

             \author[1]{Randhir  Singh}

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\date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970}

\maketitle


\begin{abstract}
        


This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the binomial distribution under the loss function which is different from that given by Rukhin(1988). The estimation involves beta distribution, a natural conjugate prior density function for the unknown parameter. Estimators obtained are conservatively biased and have finite frequentist risk.

\end{abstract}


\keywords{Bayes Estimator, Loss Function, Risk Function, Binomial Distribution.}

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\section[{Issue ersion I}]{Issue ersion I}\par
V IV ( F ) \hyperref[b4]{Rukhin(1988)} introduced a loss function given by, L(?, ?, ?) = w(?, ?)? ? 1 2 + ? 1 2\par
(1.1)\par
Where,? is an estimator of the loss function w(?, ?) ,which is non-negative. Guobing(2016) used this loss function and derived estimates of the loss and risk function of the parameter of Maxwell's distribution. \hyperref[b3]{Singh (2021)} took various forms of w(?, ?) and derived estimates of the loss and risk function of the parameter of a continuous distribution which gives Half-normal distribution,Rayleigh distribution and Maxwell's distribution as particular cases. \hyperref[b4]{Rukhin(1988)} considered the Bayesian estimation of the unknown parameter ? of the binomial distribution by takingw(?, ?) = (? ? ?) 2 (1.2)\par
In this paper,Bayes estimate of the unknown parameter ? of the binomial distribution has been obtained by replacing w(?, ?) by w 1 (?, ?) given byw 1 (?, ?) = h(?)(? ? ?) 2 (1.3)\par
Where,h(?) = 1 \{?(1 ? ?)\} (1.4) 
\section[{Notes}]{Notes}\par
Summary-This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the binomial distribution under the loss function which is different from that given by \hyperref[b4]{Rukhin(1988)}. The estimation involves beta distribution, a natural conjugate prior density function for the unknown parameter. Estimators obtained are conservatively biased and have finite frequentist risk. Let the random variable X follows binomial distribution with parameters n and ?.Where ? is unknown satisfying 0 ? ? ? 1.The prior p.d.f of ?,denoted by ? 1 (?) is as follows: 
\section[{Global}]{Global}? 1 (?) = ? ??1 (1??) ??1 B(?,?) if ? ? 0,? ? 0,0 < ? < 1 0 Otherwise (2.1)\par
Under the assumption of prior probability density function (p.d.f.) for ? as above,Bayes estimates of ? derived by \hyperref[b4]{Rukhin (1988)} were as follows:For ? ? 0, ? ? 0 ? B (X) = (X + ?) (n + ? + ?) (2.2) ? B (X) = (X + ?)(n + ? ? X) (n + ? + ?) 2 (n + ? + ? + 1) (2.3)\par
and for ? = 0, ? = 0? 0 (X) = X n (2.4) ? 0 (X) = X(n ? X) n 2 (n + 1) (2.5)\par
It was shown thatE ? L(?, ? 0 , ? 0 ) = ? (2.6)\par
Under,w 1 (?, ?) as above, the corresponding Bayes estimate is given by, For ? ? 0, ? ? 0? 1B (X) = E\{?h(?)/X\} E\{h(?)/X\} (2.7) Or, ? 1B (X) = (X + ? ? 1) A ? 2 (2.8)\par
On simplification,provided,A = n + ? + ? > 2 and,? 1B (X) = E\{?h(?)/X\} ? \{? 1B (X)\} 2 E\{h(?)/X\} (2.9) 
\section[{Notes}]{Notes}\par
Estimation of Loss and Risk of the Parameter of Binomial Distribution II. 
\section[{Issue ersion I V IV ( F )}]{Issue ersion I V IV ( F )}\par
On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial DistributionE ? L(?, ? 1B , ? 1B ) = E ? [h(?)(? ? (X + ? ? 1)(A ? 2) ?1 ) 2 ](A ? 2) 1/2 + (A ? 2) ?1/2 (2.11)\par
Or,E ? L(?, ? 1B , ? 1B ) = [n+h(?)(1 ? ? + ?(? + ? ? 2)) 2 ](A?2) ?3/2 +(A?2) ?1/2 < ? (2.12) In this case, R(?, ? 1B ) = E ? \{h(?)(? ? ? 1B )\} 2 (2.13) Or, R(?, ? 1B ) = [n + h(?)\{1 ? ? + ?(? + ? ? 2)\} 2 ](A ? 2) ?2 (2.14)\par
As mentioned by \hyperref[b2]{Keifer (1977)},an estimator ?(X)is said to be conservatively biased if,E ? \{?(X)\} ? R(?, ?) = E ? \{w(?, ?)\} (2.15)\par
In the light of this condition,? 0 (X) as given by \hyperref[b4]{Rukhin (1988)} is not conservatively biased. In this case,E ? \{? 1B (X)\} = 1 A ? 2 (2.16)\par
Let ? 0B (X) and ? 0B (X)be values of ? 1B (X) and ? 1B (X) ,respectively when,? = ? = 0.If possible let ,E ? \{? 0B (X)\} ? R(?, ? 0B (2.17) which holds if, ?2? 2 + 2? ? 1 ? 0 (2.18)\par
which is a contradiction,since 0 < ? < 1 and maximum value of ?2? 2 + 2? ? 1 is? 1 2 which corresponds to ? = 1 2 .Moreover,?2? 2 + 2? ? 1 = ?1 for ? = 1 and ? = 0 Thus,? 0B (X) is not conservatively biased.\par
When ? = ? = 1,we have,E ? \{? 1B (X)\} = R(?, ? 1B ) = 1 n (2.19) ) Or, ? 1B (X) = 1 A ? 2 (2.10) on simplification,provided,A = n + ? + ? > 2.\par
We,see that, in this case ? 1B (X) does not depend upon X and is function of n,? and ? 
\section[{Notes}]{Notes}Issue ersion I V IV ( F )\par
On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial Distribution g(?) is a monotonically increasing function of ? over the set S = (0, 1) ? \{0.5\}.Hence, ? 1B (X) as above,presents a valid 'frequentist report' as mentioned by \hyperref[b0]{Berger(1985)}.\par
The results are summerized in the following: THEOREM.Let (? 1B , ? 1B ) be Bayes estimators of the unknown parameter ? of the binomial distribution under the loss function L(?, ?, ?) =1 \{?(1??)\} (? ??) 2 ? ? 1 2 +? 1 2\par
and beta prior density with known parameters ? and ?.Then,the frequentist risk E ? L(?, ? 1B , ? 1B ) is finite for all values of ? and ? provided 0 < ? < 1.For ? = ? = 0, ? 1B (X) is not conservatively biased. The estimator ? 1B (X) is conservatively biased for? = ? = 1 and for ? = ? > 1 satisfying ? ? 1 + 2?(1??) (2??1) 2 ,? = 0.5.However, if ? = ? > 1, ? = 0.5, ? 1B (X) is also conservatively biased.\par
When,? = ? > 1, ? = 0. \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.17149122807017542\textwidth}P{0.3578947368421052\textwidth}P{0.14912280701754385\textwidth}P{0.13421052631578947\textwidth}P{0.03728070175438596\textwidth}}
5,\tabcellsep \tabcellsep \tabcellsep \\
which holds if\tabcellsep \multicolumn{2}{l}{E ? \{? 1B (X)\} ? R(?, ? 1B )}\tabcellsep (2.21)\tabcellsep Notes\\
\tabcellsep \multicolumn{2}{l}{? ? 1 + g(?)}\tabcellsep (2.22)\\
.Where,\tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep g(?) =\tabcellsep 2?(1 ? ?) (2? ? 1) 2\tabcellsep (2.23)\end{longtable} \par
 
\caption{\label{tab_0}}\end{figure}
 			\footnote{© 2022 Global Journals} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Keifer ()]{b2}\label{b2} 	 		‘Conditional confidence statements and con fidence estimators’.  		 			J Keifer 		.  	 	 		\textit{J. Am. Statist. Assoc}  		1977. 72 p. .  	 
\bibitem[Rukhin ()]{b4}\label{b4} 	 		‘Estimating the loss of estimators of a binomial parameter’.  		 			A L Rukhin 		.  	 	 		\textit{Biometrika}  		1988. 75 p. .  	 
\bibitem[Fan ()]{b1}\label{b1} 	 		‘Estimation of the Loss and Risk Functions of parameter of Maxwell's distribution’.  		 			Guobing Fan 		.  		 \xref{http://dx.doi.org/10.11648/j.sjams.20160404.12}{10.11648/j.sjams.20160404.12}.  	 	 		\textit{Science Journal of Applied Mathematics and Statistics}  		2016. 2016. 4  (4)  p. .  	 
\bibitem[Singh ()]{b3}\label{b3} 	 		‘On Bayesian Estimation of Loss and Risk Functions’.  		 			Randhir Singh 		.  		 \xref{http://dx.doi.org/10.11648/j.sjams.20210903.11}{10.11648/j.sjams.20210903.11}.  	 	 		\textit{Science Journal of Applied Mathematics and Statistics}  		2021. 2021. 9  (3)  p. .  	 
\bibitem[Berger ()]{b0}\label{b0} 	 		‘The frequentist viewpoint and conditioning’.  		 			J Berger 		.  	 	 		\textit{Proceedings of the Berkley Conference in Honor of Jerry Neyman and Jack Keifer},  				 			L Lecam,  			R Olshen 		 (ed.)  		 (the Berkley Conference in Honor of Jerry Neyman and Jack KeiferBelmont, Cailf)  		1985. Wadsworth. p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
